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High Resolution Spectroscopy with the Neutron Resonance Spin Echo Method vorgelegt von Diplom-Physiker Felix Groitl aus Erlangen von der Fakultät II - Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. M. Kneissl Gutachter: Prof. Dr. D. A. Tennant Gutachter: Prof. Dr. P. Böni Gutachter: Dr. K. Habicht Tag der wissenschaftlichen Aussprache: 18.12.2012 Berlin 2013 D 83
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Page 1: High Resolution Spectroscopy with the Neutron Resonance Spin … · 2017. 10. 26. · trometers. Neutron spin echo (NSE) introduced by Mezei [1, 2] is one of the outstanding methods

High Resolution Spectroscopy with theNeutron Resonance Spin Echo Method

vorgelegt von

Diplom-Physiker

Felix Groitl

aus Erlangen

von der Fakultät II - Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

Dr. rer. nat.

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. M. Kneissl

Gutachter: Prof. Dr. D. A. Tennant

Gutachter: Prof. Dr. P. Böni

Gutachter: Dr. K. Habicht

Tag der wissenschaftlichen Aussprache: 18.12.2012

Berlin 2013

D 83

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Abstract

The first part of this thesis is dedicated to explore new territory for high resolution Neu-

tron Resonance Spin Echo (NRSE) spectroscopy beyond measuring lifetimes of elementary

excitations. The data analysis of such experiments requires a detailed model for the echo

amplitude as a function of correlation time. The model also offers guidance for planning

NRSE experiments in terms of a sensible choice of parameters and allows predicting quan-

titatively the information content of NRSE spectroscopy for line shape analysis or energy

level separation. Major generalizations of the existing formalism, developed in this thesis,

allow for violated spin echo conditions, arbitrary local gradient components of the dispersion

surface and detuned parameters of the background triple axis spectrometer (TAS) giving

rise to important additional depolarizing effects, which have been neglected before. Fur-

thermore, the formalism can now be applied to any crystal symmetry class. The model was

successfully tested by experiments on phonons in a high quality single crystal of Pb and the

results demonstrate the stringent necessity to consider second order depolarization effects.

The formalism was subsequently extended to analyze mode doublets. As a major step for-

ward, detuning effects for both modes are taken into account here. The model was verified

by NRSE measurements on a unique tunable double dispersion setup. The results prove the

potential of NRSE spectroscopy to resolve mode doublets with an energy separation smaller

than the typical energy resolution of a standard TAS.

The second class of NRSE experiments was dedicated to line shape analysis of temperature

dependent asymmetric line broadening. Inelastic NRSE spectroscopy was performed on two

systems, Cu(NO3)2·2.5D2O and Sr3Cr2O8. For this purpose high quality single crystals

of Cu(NO3)2·2.5D2O were grown in the course of this thesis. As a proof of principle the

results clearly show that the NRSE method can be used to detect temperature dependent

asymmetric line broadening. For the first time this effect was measured with NRSE.

The second major part of this thesis was the upgrade of the NRSE option of FLEXX at the

BER II neutron source at HZB, Berlin. Redesigned NRSE bootstrap coils allow for a more

efficient exploitation of the larger beam cross section, given due to the overall upgrade of

FLEXX. Higher accessible coil tilt angles enable measurements on steeper dispersions. The

newly designed spectrometer arms result in a more compact instrument, enabling direct

beam calibration measurements for the entire accessible wavevector range. In combination

with higher coil tilt angles the accessible Q-range in Larmor diffraction geometry is en-

hanced. Extensive calibration measurements were performed and the results clearly demon-

strate the reliable performance of the new NRSE option, now available for the broader user

community at FLEXX.

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Zusammenfassung

Der erste Teil dieser Arbeit erkundet Neuland für die hochauflösende Neutronen Resonanz

Spin-Echo (NRSE) Spektroskopie über die Messung von Lebensdauern elementarer Anre-

gungen hinaus. Die Datenanalyse solcher Experimente erfordert ein detailliertes Modell der

Echoamplitude als Funktion der Korrelationszeit. Das Modell bietet zudem eine Hilfestel-

lung für die Experimentplanung in Bezug auf die Wahl der Parameter. Des Weiteren erlaubt

es eine quantitative Vorhersage des Informationsgehaltes von NRSE Messungen, z.B. im Be-

reich der Linienformanalyse oder der Aufspaltung von Anregungsenergien. Wichtige, in die-

ser Arbeit entwickelte Verallgemeinerungen des existierenden Formalismus berücksichtigen

Depolarisationseffekte durch Spin-Echo-Bedingungen, die nicht exakt erfüllt sind. Lokale

Gradienten der Dispersion mit einer Orientierung, die nicht parallel zum Wellenvektor q

sein muss, und geringfügige Abweichungen der Parameter des Dreiachsen-Spektrometers

(DAS), welche zu zusätzlichen, zuvor vernachlässigten Depolarisationseffekten führen, wer-

den jetzt berücksichtigt. Ferner kann der Formalismus nun auf beliebige Symmetrieklassen

angewendet werden. Das Modell wurde erfolgreich mit Experimenten an Phononen in einem

Pb-Einkristall mit exzellenter Mosaizität überprüft. Die Ergebnisse demonstrieren die Not-

wendigkeit, Depolarisationseffekte zweiter Ordnung zu berücksichtigen.

Der Formalismus wurde dahingehend erweitert, die Analyse von Anregungsdubletts zu er-

möglichen. Dadurch werden nun Dejustageeffekte für beide Anregungen berücksichtigt. Das

Modell wurde durch elastische und inelastische NRSE Messungen an einem eigens dafür

entwickelten Aufbau, welcher künstlich aufgespaltene Moden realisiert, überprüft. Die Er-

gebnisse zeigen das Potenzial der NRSE Spektroskopie, Anregungsdubletts aufzulösen, deren

Energieaufspaltung unter der Energieauflösung eines Standard-DAS liegt.

Weitere hier durchgeführte NRSE Experimente widmeten sich der Linienformanalyse tem-

peraturabhängiger asymmetrischer Linienverbreiterungen. Dafür wurden inelastische NRSE

Messungen an Cu(NO3)2·2.5D2O sowie an Sr3Cr2O8 durchgeführt. Hierfür wurden eigens

hochwertige Cu(NO3)2·2.5D2O-Einkristalle gezüchtet. Die Ergebnisse zeigen deutlich, dass

die NRSE Methode in der Lage ist, eine temperaturabhängige asymmetrische Linienverbrei-

terung zu bestimmen. Erstmalig wurde dieser Effekt mit NRSE gemessen.

Im Zuge dieser Arbeit wurde außerdem die NRSE-Option des kalten Dreiachsen-Spektrome-

ters FLEXX an der Neutronenquelle BER II am HZB, Berlin, aufgerüstet. Die dafür neu ge-

fertigten NRSE Bootstrap-Spulen erlauben eine effektivere Ausnutzung des größeren Strahl-

querschnitts, der durch das FLEXX Upgrade zur Verfügung steht. Höhere erreichbare Spu-

lenkippwinkel bieten zusätzlich Zugang zu steileren Dispersionen. Das durch die neu ent-

wickelten Spektrometerarme kompaktere Instrument ermöglicht Kalibrationsmessungen im

direkten Strahl für den gesamten zugänglichen Wellenvektor-Bereich. In Kombination mit

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höheren Spulenkippwinkeln wird der zugängliche Q-Bereich in der Larmor Diffraktionsgeo-

metrie vergrößert. Umfangreiche Kalibrationsmessungen zeigen deutlich die Zuverlässigkeit

und Leistungsfähigkeit der neuen NRSE-Option, die nun einer breiten Nutzerschaft zu Ver-

fügung steht.

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Table of contents

1 Introduction 1

2 NRSE resolution theory 5

2.1 Principle of inelastic neutron spin echo spectroscopy . . . . . . . . . . . . . 5

2.2 Principle of neutron resonance spin echo . . . . . . . . . . . . . . . . . . . . 8

2.2.1 The π-coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 NRSE instrument with 4 π-coils . . . . . . . . . . . . . . . . . . . . 10

2.2.3 Spin echo phonon focusing . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Extended NRSE resolution function . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Generalized spin echo phase - violated spin echo conditions . . . . . 17

2.3.2 The τ dependence of the polarization . . . . . . . . . . . . . . . . . . 24

2.3.3 Quantitative description of depolarization due to sample imperfections 28

2.3.4 Quantitative description of depolarization due to curvature of the dis-

persion surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.5 Quantitative description of depolarization due to sample imperfections

and curvature of the dispersion surface . . . . . . . . . . . . . . . . . 35

2.3.6 Dispersion surface not coinciding with the center of the TAS resolution

ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.3.7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4 Experimental test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 NRSE investigations on split modes 47

3.1 Two modes within the TAS resolution ellipsoid - Simplified NRSE model . . 47

3.2 Second dispersion surface within the TAS resolution ellipsoid - General model 51

3.3 Experimental verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3.2 Niobium dispersion models . . . . . . . . . . . . . . . . . . . . . . . 55

i

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ii Table of contents

3.3.3 Elastic measurements on split modes . . . . . . . . . . . . . . . . . . 56

3.3.4 Inelastic measurements on split modes . . . . . . . . . . . . . . . . . 61

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 NRSE line shape analysis 69

4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O . . . . . . . . . . 70

4.1.1 Properties of Cu(NO3)2·2.5D2O . . . . . . . . . . . . . . . . . . . . . 70

4.1.2 Sample deuteration and growth of single crystals . . . . . . . . . . . 71

4.1.3 Inelastic NRSE measurements . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Asymmetric line shape of excitations in Sr3Cr2O8 . . . . . . . . . . . . . . . 82

4.2.1 Properties of Sr3Cr2O8 . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.2 Inelastic NRSE measurements . . . . . . . . . . . . . . . . . . . . . . 84

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 Upgrade of the NRSE option at FLEXX 89

5.1 Bootstrap coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.1.1 B0 coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.1.2 Cooling circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.1.3 RF coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 Spin echo instrument arms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2.1 Magnetic shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2.2 Coupling coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2.3 Motors and encoders . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.3 Calibration of the new NRSE option at FLEXX . . . . . . . . . . . . . . . . 102

5.3.1 Calibration of currents and HF voltage . . . . . . . . . . . . . . . . . 103

5.3.2 Spin echo curve and echo point . . . . . . . . . . . . . . . . . . . . . 106

5.3.3 Phase stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.3.4 Calibration of coil tilt angles . . . . . . . . . . . . . . . . . . . . . . 112

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6 Conclusion and perspectives 119

Appendix 123

References 139

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Chapter 1

Introduction

To investigate the structure and dynamics of matter on the atomic scale, neutron physics

and the field of neutron scattering have produced many distinguished methods and instru-

ments. The unique properties of thermal neutrons - the electrical charge neutrality, the

magnetic moment, the wavelength being of the order of inter-atomic distances, a high pene-

tration power for many materials and an energy of the order of excitations in condensed

matter - emphasizes it as an excellent probe for structure and dynamics in matter.

Energy resolving neutron scattering techniques have made essential contributions to the un-

derstanding of matter, e.g. the development of the triple axis spectrometer (TAS) enabled

measuring dispersive excitations. Allowing a reasonable intensity for inelastic experiments,

the energy resolution of a conventional TAS is limited to 0.1meV. However, the investigations

of linewidths of dispersive excitations and slow dynamics require much higher resolution in

the range of µeV. This is only accessible using specially designed, highest resolution spec-

trometers. Neutron spin echo (NSE) introduced by Mezei [1, 2] is one of the outstanding

methods providing the necessary energy resolution in the µeV-range. The method of neutron

resonance spin echo (NRSE), developed by Gähler and Golub [3], in combination with TAS,

as suggested by Mezei [5], allows to measure linewidths of dispersive excitations with highest

resolution. This is still a young technique getting more and more established. Furthermore,

Rekveldt [44] pointed out that the technique can be used to perform high precision mea-

surements in Larmor diffraction geometry.

A major part of this thesis focuses on methodological development and experiments, explor-

ing new territory of NRSE and pushing the method beyond “standard” linewidth measure-

ments. One class of experiments, which demands highest resolution, are those seeking to

resolve mode doublets with an energy separation smaller than the typical energy resolution

of standard TAS. The data analysis of such experiments requires a model for the echo ampli-

1

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2 Introduction

tude as a function of the correlation time. A model first developed in [6, 7] expands the spin

echo phase to second order and includes second order effects arising from the TAS resolution

function, sample imperfections and the curvature of the dispersion surface. However, the

model is restricted to cubic systems. In this thesis a more generalized formalism is devel-

oped allowing for all crystal symmetry classes. Furthermore, violated spin-echo conditions,

arbitrary oriented local gradients of the dispersion surface and a detuning of the instrument

parameters are included in the generalized description. The derivation of the model, the

resolution function and experimental tests on phonons in Pb are discussed in chapter 2.

Subsequently the model is extended to include the interesting case of mode doublets. Exper-

imental tests were performed on Nb as a well understood model system. A unique tunable

double dispersion setup, allowing to generate artificially split modes, was realized. The

model and the experimental verifications are presented in chapter 3.

Another class of new NRSE-TAS experiments investigates temperature dependent asymme-

tric line broadening. This effect has recently been observed in Cu(NO3)2·2.5D2O, a model

system for a 1-D bond alternating Heisenberg chain [53], and Sr3Cr2O8, a 3-D gapped quan-

tum spin dimer [65], using time-of-flight (ToF) and triple axis spectrometer (TAS) neutron

scattering instruments. In the course of this thesis inelastic NRSE spectroscopy was per-

formed on both material systems. For the first time the effect of temperature dependent

asymmetric line broadening was measured with NRSE. The particular advantage of the

NRSE method is the direct access to the line shape, since there is no convolution of the sig-

nal with the resolution function of the spectrometer. The results are discussed in chapter 4.

The instrumental part of this thesis was the upgrade of the NRSE option available at the

cold TAS FLEXX at BER II at HZB, Berlin. The upgrade of the NRSE option benefits from

the new features available at FLEXX after its major upgrade. In the course of this thesis

the NRSE spectrometer arms were redesigned to enable a more compact construction of the

overall NRSE option. This gives access to higher scattering angles, allowing a larger Q-range

for Larmor diffraction experiments. In combination with the increased instrument area, the

direct beam geometry is now accessible for the entire wavelength-range of the NRSE-TAS

instrument. Furthermore, new NRSE bootstrap coils were manufactured in collaboration

with the Max Planck Institute For Solid State Research, Stuttgart. The new design of the

NRSE coils allows for larger beam cross sections exploiting the increased polarized neutron

flux. In addition, larger accessible coil tilt angles enable measurements of steeper dispersions.

All necessary calibration measurements for the NRSE option were performed with the new

instrument components. The individual experimental tests demonstrate the high quality,

the high manufacturing accuracy and the excellent performance of the new coils. Calibration

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Introduction 3

tables for currents and impedance matching, essential for a successful user operation, were

measured and implemented successfully. Finally, spin echo measurements were performed,

demonstrating high polarization for a wide range of spin echo times virtually independent

of coil tilt angle and thus excellent performance of the NRSE option. All measurements are

presented in chapter 5.

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4 Introduction

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Chapter 2

NRSE resolution theory

In this chapter the principles of classical neutron spin echo [5], the neutron resonance spin

echo technique [3, 4] and the so called “phonon focusing” method [5], used to investigate

dispersive excitations, will be introduced. The theory of inelastic neutron spin echo spec-

troscopy will be extended to second order including depolarizing effects [6]. In order to

develop a more general resolution function, the existent formalism will be extended here

by dropping limiting restrictions. The formalism will be subsequently extended to include

a detuning of the background TAS spectrometer. At the end of this chapter experimental

tests of the model performed at the cold triple axis spectrometer V2/FLEX at BER II at

HZB, Berlin, will be discussed.

A classical point of view is sufficient to develop the resolution function for inelastic neutron

spin echo. Therefore, all principles introduced in this chapter are discussed in a classical

frame.

2.1 Principle of inelastic neutron spin echo spectroscopy

In neutron spin echo (NSE), first introduced by Mezei [5], the velocity change of neutrons

due to scattering is measured by comparing their Larmor precession angle in magnetic fields

before and after the sample. The experimental setup for a classical NSE spectrometer is

shown in Fig. 2.1.

A polarizer P and an analyzer A are placed before and after the spectrometer, respectively.

Propagating in the y-direction the monochromatic neutron beam (∆λ/λ = 10% − 20%) is

polarized in the x-direction. Neutrons enter the first precession field region of length L1,

with a well defined homogenous magnetic field B1 oriented in z-direction perpendicular to

the polarization vector of the beam. In the magnetic field the magnetic moments of the

neutrons start to precess with the Larmor frequency ω1 = γB1, where γ = 2π · 2.913 · 103HzG

5

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6 2 NRSE resolution theory

L1 L2

Ppolarized

beam

B1

A

Det

ecto

r

y

z

x

0

-1

1

Pola

riza

tion

B2Sample

Fig. 2.1: Top: Schematic NSE setup. The beam is polarized by a polarizerP. The average polarization decays in the first spectrometer arm. For elasticscattering and equal field integrals the polarization is restored in a spin echoafter the second spectrometer arm (black) and analyzed by an analyzer A.For the inelastic case (red) the polarization at the echo point decreases asa function of the spin echo time τ and the phase is shifted. Bottom: Theaverage beam polarization as a function of the instrument length for theelastic case (black) and the inelastic case (red).

is the gyromagnetic ratio of the neutron. A neutron with a velocity v1 needs the time

t1 = L1/v1 to pass the first region, where its magnetic moment Larmor precesses by the

angle Φ1 = ω1t1. Due to the velocity spread of the incoming beam the distribution of the

Larmor precession angles becomes more and more spread out. For a typical configuration of

B = 300G, L = 1m, λ = 6Å and a wavelength distribution of ∆λλ = 10% the spread dΦ is of

the order of 102 · 2π. Since the polarization of the beam is the average of the analyzed spin

component and given by P = 〈cosΦ〉, it is zero after the first field region. By scattering at

the sample the neutron velocity changes to v2. The orientation of the second magnetic field

B2 is opposite to B1. Here B2 and B1 denote the magnitudes of the magnetic fields. By

passing through the second field region, the magnetic moment of the neutron precesses by

an angle of φ2 = −ω2t1, resulting in a total angle of

ΦNSE =γB1L1

v1− γB2L2

v2(2.1)

after the second spectrometer arm.

For elastic scattering (v1 = v2), a symmetric instrument configuration (|B1L1| = |B2L2|)and ~B1 = − ~B2, the phase φNSE will be zero for all velocities after the second field region.

Since all magnetic moments are then oriented in the original direction, this effect is called

“spin echo”. As pointed out by Mezei [1, 2] for inelastic scattering processes, the idea of

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2.1 Principle of inelastic neutron spin echo spectroscopy 7

neutron spin echo is the use of ΦNSE to determine the energy transfer

~ω =m

2

(v21 − v22

). (2.2)

To get a quantitative relation between ΦNSE and ω, the scattering function S (ω) is consid-

ered to be independent of the momentum transfer Q and distributed around a mean energy

transfer of ~ω0. The energy and the velocity of the incoming neutron beam then read:

ω = ω0 + dω and v1 = v1 + dv1 (2.3)

Using equation (2.2) v2 can be replaced in equation (2.1):

ΦNSE =γB1L1

v1 + dv1− γB2L2√

(v1 + dv1)2 − 2~

m (ω0 + dω)(2.4)

Expanding equation (2.4) to first order in dv1 and dω yields:

ΦNSE = ΦNSE,0 −(γB1L1

(v1)2 − γB2L2v1

(v2)3

)dv1 −

~

m

γB2L2

(v2)2 dω (2.5)

where ΦNSE,0 corresponds to the mean velocities v1 and v2

ΦNSE,0 =γB1L1

v1− γB2L2

v2(2.6)

and v2 is the mean neutron velocity after the sample

v2 =

√(v1)

2 − ~

2mω0. (2.7)

If the so called spin echo condition

B1L1

B2L2=

(v1)3

(v2)3 (2.8)

holds, the phase ΦNSE becomes independent of the velocity spread dv1 and dΦNSE is

proportional to dω:

dΦNSE = ΦNSE − ΦNSE,0 = −τNSEdω (2.9)

The instrumental constant

τNSE =~γB1L1

m (v1)3 =

~γB2L2

m (v2)3 (2.10)

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8 2 NRSE resolution theory

is called “spin echo time”.

After the second field region the neutron beam passes an analyzer transmitting neutrons

polarized in the x-direction. The intensity at the detector is given by:

I =I02(1 + P ) (2.11)

where I0 is the intensity of the incident neutron beam. The mean beam polarization P at

the analyzer, i.e. the echo amplitude AE , is the cosine Fourier transform of the scattering

function S (ω)

AE = 〈cosΦNSE〉 =∫

S (dω) cos (ΦNSE,0 − dω · τNSE) d (dω) . (2.12)

As a classical example from quasi elastic scattering let the scattering function S (ω) be a

Lorentzian line with a linewidth of Γ:

S (ω) ∝ Γ

Γ2 + ~2ω2(2.13)

Equation (2.12) then yields:

AE (τNSE) =

∫S (ω) cos (ωτNSE) dω∫

S (ω) dω= e−

ΓτNSE~ (2.14)

2.2 Principle of neutron resonance spin echo

The method of neutron resonance spin echo (NRSE) was suggested by Gähler and Golub

[3, 4] in 1987. NRSE replaces each of the long static fields of the NSE method by two

resonance spin flippers with sharp field boundaries placed in a zero field region.

2.2.1 The π-coil

The basic component of NRSE is a resonance flipper called π-coil. Within a resonance spin

flipper two magnetic fields are superimposed: a static field ~B0 pointing in the z-direction

and a rotating field ~BRF in the xy-plane with the frequency ωRF . The superposed magnetic

field is described by the vector

~B1 (t) = (BRF cos (ωRF t) , BRF sin (ωRF t) , B0) . (2.15)

To understand the movement of the magnetic moment of a neutron entering the coil with an

initial orientation of the magnetic moment in the xy-plane, it is useful to change the reference

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2.2 Principle of neutron resonance spin echo 9

B1

x

z

y

Fig. 2.2: Principle of the π-coil. The incoming neutron polarization vector(green) precesses around the effective magnetic field B1 and leaves the coilwith a new orientation (blue). Setting B1 = πv0

γdthe polarization vector

performs a π-flip and stays in the scattering plane

frame to a system rotating with the frequency ωRF around the z-axis. The effective field

will be [10]:

~B1 =

(BRF , 0, B0 −

ωRF

γ

). (2.16)

If the frequency of the rotating field is tuned to the Larmor frequency of the static field

(ωRF = γB0), the z-component vanishes and the static field BRF remains. Thus, the

magnetic moment of the neutron will precess around BRF with the frequency ωRF = γBRF .

The magnitude of BRF is chosen such that the magnetic moment precesses by an angle π

while the neutron passes the coil:

BRF =πv0γd

(2.17)

where v0 is the velocity of the neutron and d is the thickness of the coil. Therefore, after

leaving the coil the magnetic moment is again in the xy-plane (see Fig 2.2).

In the lab-system the neutron enters the coil at a time t1. At this time the rotating field

has a phase of ΦRF = ωRF t1 and while passing the coil, the magnetic moment precesses an

angle π around ~B1. Simultaneously ~B1 precesses around the z-axis by an angle of ωRFdv0

.

Thus, after leaving the coil the phase of the magnetic moment is:

Φ1 = 2ωRF t1 + ωRFd

v0− Φ0 (2.18)

where Φ0 is the angle between the magnetic moment and the x-axis.

Since the realization of a rotating field setup is challenging, it is favorable to use a linearly

oscillating field instead. The oscillating field B1 can be divided into two counter rotating

parts B+1 and B−

1 with an amplitude of B12 . Only the part rotating in the same direction as

the magnetic moment of the neutron contributes to the π-flip. The counter rotating part

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10 2 NRSE resolution theory

can be neglected for B0 ≫ B1. This causes a small shift of the resonance frequency, the so

called “Bloch-Siegert-Shift” [11]∆ω

ω=

B2RF

4B20

, (2.19)

which can be neglected for frequencies higher than 50kHz.

The resonance condition given by equation (2.17) is only valid for one v0. The magnetic

moments of faster (slower) neutrons will precess less (more) than an angle of π. Therefore,

the magnetic moment is oriented slightly out of the xy-plane after the coil and the average

polarization decreases. The estimates in [12] show that for small velocity spreads the echo

amplitude is:

AE = 1−Mπ2

4

(∆v

v0

)2

(2.20)

where M is the number of π-flippers used. For a velocity spread of 1.5%, which is typical

for a TAS using a crystal monochromator, and 8 flippers, the loss of average polarization

will be < 1%.

2.2.2 NRSE instrument with 4 π-coils

Using a NRSE instrument with a setup of 4 π-coils (see Fig. 2.3) provides the same function

as a conventional NSE spectrometer with the same length and the static magnetic fields twice

as strong.

The two coils before the sample region (A and B) have a static field ~B1 and are driven with

a frequency of ω1 = γB1 while the coils C and D after the sample region have the static

field ~B2 with a frequency of ω2 = γB2. The two fields ~B1 and ~B2 are oriented in opposite

directions. Therefore, the precession direction of the magnetic moment of the neutron and

the rotating fields is opposite to each other. The regions between the coils are zero field

regions and the stray fields of the coils have to be minimized.

A neutron polarized in x-direction (Φ0 = 0 in equation (2.18)) with a velocity v1 enters the

first coil at the time tA. After passing the coil, the phase angle of the magnetic moment is

ΦA = 2ω1tA + ω1d

v1. (2.21)

In the zero field region, between the coils A and B, the magnetic moment of the neutron does

not change its phase angle and enters the second coil at the time tB = tA + L1v1

. Therefore,

the phase angle of the magnetic moment after the second coil is:

ΦB = 2ω1tB + ω1d

v1− ΦA = 2ω1

L1

v1. (2.22)

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2.2 Principle of neutron resonance spin echo 11

Spin

x

y

z

L1 L2

A B C D

zero-field regions

sample regiond

Fig. 2.3: Schematic drawing of a NRSE setup with 4 π-coils: The orienta-tion of the magnetic moment of the neutron entering each coil is illustratedin green. Within the coil the magnetic moment precesses around the ef-fective magnetic field Beff (red) and has a new orientation (blue) leavingeach coil. The flight path between the coils are regions of zero magneticfield.

ΦB becomes independent from the time tA and only depends on the time of flight L1v1

between

the two coils. Compared to a conventional NSE instrument with the same magnitude of the

magnetic field and the same length, one obtains twice the phase angle. In other words, the

two π-coils give the same Larmor precession angle as a NSE spectrometer arm with length

L1 and magnetic field 2B1.

During the scattering process the neutron velocity changes to v2. Since the sample region

is a field free region, the magnetic moment enters coil C with the phase ΦB at time tC and

leaves coil C with an angle of

ΦC = −2ω2tC − ω2d

v2− ΦB. (2.23)

Hence, after the last coil D the phase is

ΦD = −2ω2tD − ω2d

v2− ΦC

= 2ω1L1

v1− 2ω2

L2

v2

= ΦNRSE = 2ΦNSE .

(2.24)

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12 2 NRSE resolution theory

Therefore, the previous discussion on spin echo applies directly to NRSE if B1 and B2

are substituted by 2ω1γ and 2ω2

γ , respectively. However, in a typical NRSE experiment the

minimum accessible spin echo time is determined by a frequency of 50 kHz in the π-coils,

since for lower frequencies the effect of the above mentioned Bloch-Siegert-Shift (see equation

(2.19)) increases leading to a strong depolarization.

Enhancement of NRSE by the bootstrap technique

The so called bootstrap technique was first introduced in [4]. By replacing each π-coil by a

pair of resonance coils with the static fields of each pair oriented in opposite directions, the

effective precession angle and hence the resolution is doubled. Within this configuration the

first coil within a bootstrap coil provides a return path for the magnetic flux of the second

coil and vice versa. This reduces the stray field outside the coils significantly. The phase

angle after four bootstrap coils, each containing a series of N resonance coils, is

ΦN = 2N

(ω1

L1

v1− ω2

L2

v2

)= NΦNRSE (2.25)

All existing NRSE spectrometers operate with double bootstrap coils (N = 2). A discussion

of the limits of the bootstrap technique can be found in [4].

2.2.3 Spin echo phonon focusing

A triple axis spectrometer (TAS) is a classic instrument to determine dispersion curves

in the entire Brillouin zone. However, the instrumental resolution of a TAS is in rare

cases sufficient to measure the natural linewidth of excitations and in addition needs to

be known very accurately at the inelastic signal. NSE and NRSE spectrometers provide

the required high resolution in the µeV range. Mezei [5] suggested to combine TAS and

spin echo spectrometers and to use the tilted field technique, which is necessary to measure

linewidths of dispersive excitations.

For this technique to work, the field boundaries need to be perpendicular to the flight path of

the neutrons. Then the phase Φ (ω) is proportional to the energy transfer ω and independent

of the momentum transfer q. In the quasielastic case with a mean excitation energy of zero

and a broad wavelength distribution ∆λλ = 10%, a relaxed q resolution does not affect the

energy resolution. However, for dispersive excitations where the excitation energy ω is a

function of q, ω = ω (q), a finite momentum resolution results in a spread in ω and hence

in Φ (ω) even for sharp excitations with zero linewidth. To allow measurements of the

intrinsic linewidth of excitations, the spin echo phase needs to be tuned to the slope ~∇qω

of the dispersion (see Fig. 2.4). Hence, all scattering events lying within the resolution

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2.2 Principle of neutron resonance spin echo 13

ω

q

ω0

q0

Fig. 2.4: The resolution function of the TAS selects a certain region inthe (q, ω)-space. The lines of constant Larmor phase (Φ = const, paleblue) have to be parallel to the dispersion curve. This can be reached byinclining the precession field boundaries.

ellipsoid of the background TAS and on the dispersion surface have the same spin echo

phase Φ (ω,q). As pointed out by Mezei [5] the spin echo phase can be tuned to the slope

of the dispersion surface by tilting the boundaries of the precession fields relative to the

neutron flight path. In such a configuration, for a linear dispersion with zero linewidth, all

scattered neutrons have the same total Larmor precession phase. A finite linewidth would

cause a spread in Φ. If the lineshape F (ω) within the resolution ellipsoid is independent

of q (S (q, ω) = S (q0)F (ω)) and significantly smaller than the width of the resolution

ellipsoid, Φ is proportional to the Fourier transform of the line shape as explained above.

Spin echo conditions

In this section the spin echo conditions for the phonon focusing technique are introduced

briefly in a classical picture. The discussion of an alternative quantum mechanical framework

where Φ is a phase shift between two spin states can be found in [8, 13]. The quantum

mechanical approach shows that the spin echo time τNSE is equal to the correlation time in

the time dependent van Hove density-density correlation function [9].

Assuming that the variation of the scattering law on the dispersion surface depends on

ω (q) only, S (q, ω) can be considered as a function of energy deviation from the dispersion

surface:

S (q, ω) = S (ω (q)− ω0 (q)) . (2.26)

In order to measure equation (2.26) the Larmor precession angle needs to be proportional

to (ω (q)− ω0 (q)):

Φ− Φ0 = −τ (ω (q)− ω0 (q)) = −τ∆ω. (2.27)

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14 2 NRSE resolution theory

kF

B2

B1

kI

nF

nI

i2

j2

l2

j1

i1

l1

L2

precession fieldregions

sample

L1

kI

ni

ti

Q

sf

si

kF

l0

j0

i0

nI

nF

nf

tf

Θ2

Θ1

Fig. 2.5: Schematic drawing of the tilted precession field regions in aninelastic NSE setup. The defined Cartesian coordinate systems are usedthroughout.

Assuming a planar dispersion and expanding the dispersion to first order in (q− q0) yields:

∆ω = ω (q)− ω0 (q) = ω (q)− [ω0 (q0) + (q− q0) · ∇qω (q0)] . (2.28)

Using momentum and energy conservation with mean wavevectors kI and kF before and

after the scattering:

ki = kI +∆ki and kf = kF +∆kf

ω(q) =~2

2m

(k2i − k2

f

)and ω0(q) =

~2

2m

(k2I − k2

F

),

(2.29)

∆ω is expressed as a function of the deviations from the mean wavevectors ∆ki and ∆kf :

∆ω = ω (q)− ω0 (q) =~

m[(kI ·∆ki)− (kF ·∆kf )] . (2.30)

Using the relation for the momentum in a perfect crystal

q = ki − kf −G0 and q0 = kI − kF −G0 (2.31)

where G0 is the reciprocal lattice vector and combining equations (2.28) and (2.30) yields:

∆ω = ∆ki

[~

mkI −∇qω0(q0)

]−∆kf

[~

mkF −∇qω0(q0)

](2.32)

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2.2 Principle of neutron resonance spin echo 15

The Larmor precession angle after two tilted field regions is

Φ = Φ0 +∆Φ =m

~

(ωL1L1 cosΘ1

ki cosΘ1− ωL2L2 cosΘ2

kf cosΘ2

)

=m

~

(ωL1L1 cosΘ1

ki · ni− ωL2L2 cosΘ2

kf · nf

)=

A1

ki · ni− A2

kf · nf

(2.33)

where ni,f are the unit vectors normal to the precession field boundaries before and after

the sample tilted by an angle Θ1,2 w.r.t. the neutron flight path (see Fig. 2.5). The Larmor

frequency is ωL1,2 = NγB1,2 with N = 1 for conventional NSE, N = 2 for NRSE with 4

flipper coils and N = 4 for NRSE in bootstrap mode (8 flipper coils). Using

Φ0 =A1

kI · ni− A2

kF · nf

(2.34)

and expanding the total Larmor precession angle to first order yields

Φ− Φ0 = − A1

(kI · ni)2 (∆ki · ni) +

A2

(kF · nf )2 (∆kf · nf ) . (2.35)

For equation (2.27) to hold, the coefficients in equations (2.32) and (2.35) have to be com-

pared. The result for the normal vectors of the precession field boundaries is:

ni,f =~

mkI,F −∇qω0(q0)∣∣ ~mkI,F −∇qω0(q0)

∣∣ =~

mkI,F −∇qω0(q0)

NI,F(2.36)

and for the spin echo time

τ =A1,2

(kI,F · ni,f )∣∣ ~mkI,F −∇qω0(q0)

∣∣ =A1,2

(kI,F · ni,f )NI,F. (2.37)

The condition for the adjustment of the field integrals is then:

ωL1L1

ωL2L2=

cosΘ2 (kI · ni)2∣∣ ~mkI −∇qω0(q0)

∣∣

cosΘ1 (kF · nf )2∣∣ ~mkF −∇qω0(q0)

∣∣ =cosΘ2 (kI · ni)

2NI

cosΘ1 (kF · nf )2NF

. (2.38)

If the spin echo conditions given by equations (2.36) and (2.38) are satisfied, τ is the same

for both spectrometer arms and the total Larmor phase is independent of the momentum

transfer in the presence of a finite dispersion. Note that this is only true for a first order

approximation of the total Larmor phase.

In the general case the echo amplitude, measured by a spin echo spectrometer combined

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16 2 NRSE resolution theory

with a background TAS, is given by [6]

AE =1

N

∫S (Q, ω)RTAS (ki,kf ) e

iΦ(ki,kf)d3kid3kf + c.c. (2.39)

where RTAS (ki,kf ) is the transmission function of the background TAS and Φ (ki,kf ) is the

sum of Larmor precession angles before and after the sample. Assuming that the linewidth

and S (Q) are independent of Q within the TAS resolution ellipsoid, the integral over the

momentum components yields

AE ∝∫

S (∆ω)RTAS (∆ω) eiΦd∆ω + c.c. (2.40)

with ∆ω given by equation (2.28). The application of spin echo spectroscopy is only rea-

sonable if the linewidth S (∆ω) of the excitation is much smaller than the energy resolution

T (∆ω) of the background TAS. Therefore, RTAS (∆ω) is assumed to be constant. Thus,

the result for the echo amplitude is again the cosine Fourier transform of the line shape:

AE ∝∫

S (∆ω) cos (τ∆ω) d∆ω (2.41)

2.3 Extended NRSE resolution function

In this section the resolution function for a NRSE spectrometer with a background TAS is

extended by expanding the spin echo phase to second order following the approach of [6]

and expressing the resolution function in a covariance matrix formalism. This will include

second order effects arising from the TAS resolution function, sample imperfections, the

curvature of the dispersion surface and a detuning of the instrument parameters. Major

differences between the treatment given in [6] and the treatment given in the present thesis

are the following generalizations [14, 15]:

1. Allow for violated spin echo conditions

2. The formalism is extended to treat systems with lower crystallographic symmetry than

cubic

3. The local gradient of the dispersion surface ∇ω(qc0) may have components out of the

scattering plane, i.e. the center of the TAS resolution ellipsoid must not necessarily be

located at a point with high crystallographic symmetry. Here qc0 is the momentum

transfer expressed in a Cartesian coordinate system attached to the reciprocal lattice

system. q0 is transformed into qc0 by the so called B-matrix [16, 17]. The UB matrix

formalism is explained in Appendix A.

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2.3 Extended NRSE resolution function 17

One advantage of the generalizations made here is the applicability of the formalism to a

broader range of systems. The allowance of violated spin echo conditions enables the treat-

ment of cases where a second excitation is present within the TAS resolution ellipsoid. In

such cases the spin echo conditions are at least violated for one excitation. Investigations on

split modes using NRSE are treated in chapter 3. Without these generalizations, depolar-

ization effects would be neglected. This would result in an overestimation of the linewidth Γ

of the excitation.

In addition minor errors in [6] in the part discussing lattice imperfections and the curvature

of the dispersion surface are corrected.

2.3.1 Generalized spin echo phase - violated spin echo conditions

A simplified situation is discussed by making the following assumptions:

1. A single dispersion surface is located within the TAS resolution ellipsoid.

2. The center of the TAS resolution ellipsoid coincides with the dispersion surface.

3. The precession field boundaries are exactly perpendicular to the scattering plane.

4. The precession field boundaries can only be tilted around axes that are perpendicular

to the scattering plane.

5. All instrument components are ideal, i.e. field boundaries are ideal planes, no stray

fields are present and RF flippers provide exact π-flips.

Following [6] and expanding equation (2.35) to second order yields:

Φ (ki,kf )− Φ0 =− A1

(kI · ni)2 (∆ki · ni) +

A2

(kF · nf )2 (∆kf · nf )

+A1

(kI · ni)3 (∆ki · ni)

2 − A2

(kF · nf )3 (∆kf · nf )

2 .

(2.42)

The aim is to express the total Larmor precession angle in terms of the variable vector

J = (∆ω,∆kin, y1, y2, z1, z2) with

∆ki = x1i1 + y1j1 + z1l1 (2.43)

∆kf = x2i2 + y2j2 + z2l2 (2.44)

∆kin = ∆ki · ni (2.45)

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18 2 NRSE resolution theory

where the variables of J are defined as in [6]. According to Fig. 2.5 i, j and l are the basic

vectors of a right handed coordinate system with i1 and i2 pointing along the direction of

the mean wavevectors kI and kF , respectively. First, the dispersion relation is expanded to

second order

ω (qc) = ∆ω + ω (qc0) + ∆qc0 · ∇ω (qc0) +1

2∆qT

c Hc|qc0∆qc (2.46)

assuming that the line broadening is the same for every qc within the TAS resolution

ellipsoid, i.e. ∆ω is independent from qc. Here Hc is the curvature matrix of the dispersion

surface at qc0, which is defined in section 2.3.4. In a real crystal lattice imperfections are

present and therefore variations of the lattice vector Gc need to be included. The following

relation

∆q′c = ∆qc −∆Gc = ∆ki −∆kf −∆Gc (2.47)

with the total wavevector transfer

Qc = Gc + qc (2.48)

and

Gc = Gc0 +∆Gc (2.49)

leads to

ω (qc) =∆ω + ω (qc0) + ∆ki · ∇ω (qc0)−∆kf · ∇ω (qc0)

−∆Gc · ∇ω (qc0) +1

2∆q′T

c Hc|qc0∆q′c

. (2.50)

Note again that all variables with the index c are expressed in a Cartesian coordinate system

attached to the reciprocal lattice. The relation between these quantities and their expression

in the basis of the reciprocal lattice are explained in Appendix A. For now the last two

terms will be left unchanged, since effects from sample imperfections and curvature of the

dispersion surface will be treated separately in sections 2.3.3 and 2.3.4, respectively.

The energy conservation in second order reads

ω (qc) =~

2m

(k2i − k2

f

)(2.51)

= ω (qc0) +~

mkI ·∆ki −

~

mkF ·∆kf

+~

2m∆k2

i −~

2m∆k2

f (2.52)

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2.3 Extended NRSE resolution function 19

using

qc0 = kI − kF (2.53)

and

ω (qc0) =~

2m

(k2I − k2

F

). (2.54)

Combining equations (2.50) and (2.51) yields

∆ω =

(~

mkI −∇ω (qc0)

)·∆ki −

(~

mkF −∇ω (qc0)

)·∆kf

+~

2m∆k2

i −~

2m∆k2

f +∆Gc · ∇ω (qc0)−1

2∆q′T

c Hc|qc0∆q′

c. (2.55)

Now unit vectors are defined as

ǫi =~

mkI −∇ω (qc0)∣∣ ~mkI −∇ω (qc0)

∣∣ =~

mkI −∇ω (qc0)

NI(2.56)

ǫf =~

mkF −∇ω (qc0)∣∣ ~mkF −∇ω (qc0)

∣∣ =~

mkF −∇ω (qc0)

NF. (2.57)

Since these unit vectors, defined by the crystal properties, in general may have components

out of the scattering plane, they cannot be assumed to be equal with the normal vectors

of the precession field boundaries ni,f , which are defined by the instrument settings. As

mentioned above in this subsection kI,F correspond to a point on the dispersion surface

coinciding with the center of the TAS resolution ellipsoid. This assumption will be dropped

later in section 2.3.6. In the next step the spin echo times in both spectrometer arms τ1,2

given by the instrument parameters are defined:

τ1,2 =A1,2

(kI,F ·ni,f )2∣∣ ~mkI,F −∇ω (qc0)

∣∣ =A1,2

(∆kI,F ·ni,f )2NI,F

. (2.58)

Here τ1,2 are not longer identical to the correlation time τ in the time-dependent van Hove

density-density correlation function. The inclusion of instrument alignment errors in the

tilt angle of the precession field boundaries and the ratio of the field integrals means that in

general τ1,2 are not equal in the two spectrometer arms. Multiplying equation (2.55) with

τ2 yields:

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20 2 NRSE resolution theory

τ2∆ω =A2

(kF ·nf )2NF

(~

mkI −∇ω (qc0)

)·∆ki

− A2

(kF ·nf )2

~

mkF −∇ω (qc0)

NF·∆kf

+~

2mτ2∆k2

i −~

2mτ2∆k2

f

+τ2∆Gc · ∇ω (qc0)−1

2τ2∆q′T

c Hc|qc0∆q′

c (2.59)

=A2

(kF ·nf )2

NI

NFǫi ·∆ki −

A2

(kF ·nf )2 ǫf ·∆kf

+~

2mτ2∆k2

i −~

2mτ2∆k2

f

+τ2∆Gc · ∇ω (qc0)−1

2τ2∆q′T

c Hc|qc0∆q′

c. (2.60)

In the general case the unit vectors ǫi,f are expressed in a right handed Cartesian coordinate

system with the basis vectors ni,f , ti,f and si,f according to Fig. 2.5. The unit vectors ni,f ,

ti,f and si,f are the same as u1,2, v1,2 and w1,2 in [6].

ǫi,f = ei,f1ni,f + ei,f2ti,f + ei,f3si,f (2.61)

Here ni,f and ti,f are the normal vector and the tangent of the precession field boundaries

lying in the scattering plane, respectively. si,f is the unit vector perpendicular to the

scattering plane defined by si,f = ni,f × ti,f . Note that the unit vectors si,f are identical

since they are both perpendicular to the scattering plane. The components ei,f1, ei,f2 and

ei,f3 are defined by

ei1 = ǫi · ni ei2 = ǫi · ti ei3 = ǫi · si (2.62)

ef1 = ǫf · nf ef2 = ǫf · tf ef3 = ǫf · sf . (2.63)

As a generalization compared to [6], the case where the gradient vector of the dispersion

∇ω (qc0) has a component out of the scattering plane is included. Inserting equation (2.61)

in equation (2.59) yields

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2.3 Extended NRSE resolution function 21

A2

(kF ·nf )2∆kf ·nf = −τ ′′2∆ω + τ ′′2NIei1∆kin

+τ ′′2NIei2

(−∆kin tan θ1 + y1

1

cos θ1

)

+τ ′′2NIei3z1 − τ ′′2NF ef3z2

−τ ′′2NF ef21

cos θ2y2

+~

2mτ ′′2∆k2

i −~

2mτ ′′2∆k2

f

+τ ′′2∆Gc · ∇ω (qc0)−1

2τ ′′2∆q′T

c Hc|qc0∆q′

c (2.64)

with

τ ′′2 =τ2

(ef1 − ef2 tan θ2). (2.65)

For the calculation of equation (2.64) the following relations for ∆ki,f expressed in the basis

of ni,f , ti,f and si,f were used:

si ·∆ki = z1 (2.66)

sf ·∆kf = z2 (2.67)

ni ·∆ki = ∆kin. (2.68)

∆ki = (x1 cos θ1 + y1 sin θ1)ni + (−x1 sin θ1 + y1 cos θ1) ti + z1si (2.69)

= ∆kinni +

(−∆kin tan θ1 + y1

1

cos θ1

)ti + z1si (2.70)

∆kf = (∆kf ·nf )nf +

(− (∆kf ·nf ) tan θ2 + y2

1

cos θ2

)tf + z2sf . (2.71)

ti ·∆ki = −∆kin tan θ1 + y11

cos θ1(2.72)

and

tf ·∆kf = − (∆kf ·nf ) tan θ2 + y21

cos θ2. (2.73)

Substituting equation (2.64) into equation (2.42) introduces the energy deviation from the

dispersion surface due to the linewidth broadening ∆ω into the Larmor phase. This is re-

quired to provide the Fourier transform of the scattering function S (Q, ω):

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22 2 NRSE resolution theory

φ(ki,kf )− φ0 = −τ1NI∆kin − τ ′′2∆ω + τ ′′2NIei1∆kin

+τ ′′2NIei2

(−∆kin tan θ1 + y1

1

cos θ1

)+ τ ′′2NIei3z1

−τ ′′2NF ef21

cos θ2y2 − τ ′′2NF ef3z2

+τ ′′2∆Gc · ∇ω (qc0)−1

2τ ′′2∆q′T

c Hc|qc0∆q′

c

+~

2mτ ′′2∆k2

i −~

2mτ ′′2∆k2

f

+τ1NI

kI · ni∆k2in − τ2

NF

kF ·nf(∆kf ·nf )

2 . (2.74)

The Larmor phase now takes into account second order effects and the above mentioned

generalizations. Contributions from components of the gradient vector of the dispersion

surface, which are out of the scattering plane are considered by the terms proportional to

ei2,3 and ef2,3. If the gradient vector of the dispersion surface lies within the scattering

plane, these terms will vanish since then ei2,3 = 0 and ef2,3 = 0. The terms proportional to

∆Gc and Hc account for contributions arising from sample imperfections and the curvature

of the dispersion surface, respectively.

The term (∆kf · nf )2 is now substituted in equation (2.74) by using equation (2.64). Since

only second order effects are considered and higher order terms are neglected, it is sufficient

to consider equation (2.64) to first order only:

∆kf ·nf = − 1

CfNF∆ω +

NICi

NFCf∆kin

+NI

CfNFei2

1

cos θ1y1 −

1

Cfef2

1

cos θ2y2

+NI

CfNFei3z1 −

1

Cfef3z2 +

1

CfNF∆G′

c · ∇ω (qc0) (2.75)

using

τ2 =A2

(kf · nf )2NF

(2.76)

Ci = ei1 − ei2 tan θ1 (2.77)

Cf = ef1 − ef2 tan θ2. (2.78)

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2.3 Extended NRSE resolution function 23

The term + 1CfNF

∆G′c ·∇ω (qc0) considers only the first order terms arising from the lattice

imperfections. Since lattice imperfections have been introduced from the beginning, it is

considered here. This term introduces cross terms between the lattice imperfection variables

∆η, ∆ν and ∆Gc and the variables of the 6 component vector J = (∆ω,∆kin, y1, y2, z1, z2).

∆G′c is defined in equation (2.99). ∆k2

i and ∆k2f are substituted in equation (2.74) using

equations (2.69) and (2.71):

∆k2i = ∆k2in +

(−∆kin tan θ1 + y1

1

cos θ1

)2

+ z21 (2.79)

∆k2f = (∆kf ·nf )

2 +

(− (∆kf ·nf ) tan θ2 + y2

1

cos θ2

)2

+ z22 . (2.80)

Inserting equations (2.79) and (2.80) into equation (2.74) and using again equation (2.75)

to substitute all ∆kf · nf terms allows to express the total Larmor precession angle as a

function of squared and cross terms of the six variables (∆ω,∆kin, y1, y2, z1, z2). The total

Larmor phase can conveniently be expressed in matrix notation [13]:

φ(ki,kf )− φ0 = τ ′′2TTJ− 1

2τ ′′2 J

TΨJ

+τ ′′2∆Gc · ∇ω (qc0)

−1

2τ ′′2∆q′T

c Hc|qc0∆q′

c +X(∆Gc) (2.81)

Here X(∆Gc) denotes all cross terms introduced by ∆η, ∆ν and ∆Gc. Effects from sample

imperfections are treated in section 2.3.3. The components of the 6-dimensional column

vector T are:

T1 = −1 T2 = NI

(Ci −

τ1τ2Cf

)T3 =

NIei2cos θ1

(2.82)

T4 = −NF ef2cos θ2

T5 = NIei3 T6 = −NF ef3. (2.83)

The elements of the symmetric (6× 6) matrix Ψ are given in Appendix B.

For the special case, where the spin echo conditions are satisfied, and the gradient of the

dispersion surface lies in the scattering plane:

ei1 = 1 ei2 = ei3 = 0 (2.84)

ef1 = 1 ef2 = ef3 = 0 (2.85)

Ci = Cf = 1 (2.86)

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24 2 NRSE resolution theory

and thus

τ ′′2 = τ2 (2.87)

τ1 = τ2. (2.88)

The matrix Ψ then reduces to the matrix given in [6]. Since only T1 = −1 remains non-zero,

the terms, which are linear in J, vanish, leaving only the desired term −τ∆ω in the spin

echo phase.

2.3.2 The τ dependence of the polarization

In this section the τ dependence of the polarization for violated spin echo conditions for a

sample without lattice imperfections (∆Gc = 0) and a planar dispersion (Hc|qc0 = 0) is

derived. With these assumptions equation (2.81) reduces to

φ(ki,kf )− φ0 = τ ′′2TTJ− 1

2τ ′′2 J

TΨJ. (2.89)

The TAS transmission function is derived, following the approach of Popovici [51]. In order

to have a consistent nomenclature, this approach is summarized in Appendix C. It reads:

RTAS (ki,kf ) = exp

(−1

2JTLTASJ

). (2.90)

Substituting the equations (2.89) and (2.90) into the fundamental equation (2.39) yields:

AE =1

N

∫S (Q, ω) exp

(iτ ′′2T

TJ)exp

(−1

2JTLIJ

)d6Jn + c.c. (2.91)

with

LI = LTAS + iτ ′′2Ψ. (2.92)

Terms in ∆ω higher than linear can be neglected, since the integral over the energy coordi-

nate will be dominated by the term exp (−iτ∆ω). The only reasonable application of spin

echo is the situation where ∆ω is very small and hence S (∆ω) is very narrow compared to

the TAS energy resolution. Therefore, rewriting equation (2.91) yields:

AE =1

N

∫S (Q, ω) exp

(−iτ ′′2∆ω

)d∆ω

×∫

exp(iτ ′′2 T

T J)exp

(−1

2JT LI J

)d5Jn (2.93)

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2.3 Extended NRSE resolution function 25

where J = (∆kin, y1,y2, z1, z2) and T are the 5-D sub vectors of J and T without the energy

variable ∆ω, respectively, and LI is the corresponding symmetric (5× 5) sub matrix of LI .

Following [6] and using the general matrix theorem [18]

∞∫

−∞

exp(KTJ

)exp

(−1

2JTMJ

)dnJn

=(2π)n/2

(detM)1/2exp

(−1

2KTM−1K

)(2.94)

the resolution function is obtained

FI

(τ ′′2)=

∣∣∣∣∣

√det LI (τ ′′2 = 0)

det LI (τ ′′2 )exp

(−1

2τ ′′22 TT L−1

I

(τ ′′2)T

)∣∣∣∣∣ . (2.95)

For strongly violated spin echo conditions the exponential term will dominate the decay of

the polarization. For perfectly satisfied spin echo conditions this term will be unity since T

becomes zero.

Example 1: Numerical calculations for RbMnF3

Fig. 2.6 and 2.7 show numerical examples for depolarization effects arising from a detuning

of the tilt angles Θ1,2 and a detuning of the frequency f1,2. Here the echo amplitude is

normalized to 1 for the optimum value for each parameter. The numerical calculations were

done for RbMnF3 at the zone boundary excitation Q = [0.5 0.5 −1], E = 8.287meV, for

two different spin echo times τ = 12.4ps (Fig. 2.6) and τ = 80ps (Fig. 2.7). Details about

the dispersion relation properties of RbMnF3 are listed in Appendix D. At the maximum of

the dispersion the slope of the dispersion is equal to zero and hence Θ1,2 = 0. Due to the

symmetry of the dispersion at the maximum and the first order dependence on the cosine

of the coil tilt angles, the depolarizing effects arising from a detuning of Θ1,2 are symmetric

with respect to the optimum value. Note that this is different for other values of Q, where

the slope of the dispersion is different from zero. The depolarizing effect is slightly stronger

for a detuning of Θ1. For the case of Θ1,2 = 0 any detuning of the coil tilt angles increases

the path length of the neutron within the π-coil. Therefore, a higher Larmor phase is ob-

tained and the depolarizing effect due to detuned coil tilt angles increases with increasing

frequencies. Since f1 > f2 for the inelastic case, the depolarizing effect is stronger for the

first spectrometer arm. In the case of the minimum experimentally accessible τ = 12.4ps a

detuning of dΘ1,2 = ±9 would decrease the echo amplitude by about 3%.

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26 2 NRSE resolution theory

−40 −20 0 20 400

0.2

0.4

0.6

0.8

1

dΘ1 [deg]

echo

am

plitu

de

a−40 −20 0 20 400

0.2

0.4

0.6

0.8

1

dΘ2 [deg]

echo

am

plitu

de

b

−40 −20 0 20 400

0.2

0.4

0.6

0.8

1

df1 [kHz]

echo

am

plitu

de

c−40 −20 0 20 400

0.2

0.4

0.6

0.8

1

df2 [kHz]

echo

am

plitu

de

d

Fig. 2.6: Depolarization effects due to a detuning of spin echo parametersat τ = 12.4ps corresponding to f2 = 50kHz. The detuned Parameters are:Θ1 (a), Θ2 (b), f1 (c) and f2 (d).

In contrast, the depolarizing effect arising from a frequency detuning is more sensitive to a

detuning of the second spectrometer arm. Since f1 > f2 the relative detuning is smaller for

higher frequencies and therefore the depolarizing effect decreases with increasing frequency.

The calculated frequency ratio of this numerical example is f1f2

≈ 3. For τ = 12.4ps a de-

tuning of df1 = ±25kHz and df2 = ±8kHz would decrease the echo amplitude by about 3%.

The detuning of f1 needs to be a factor of 3 higher compared to a detuning of f2 in order to

obtain the same effect. This is in very good agreement with the calculated frequency ratio.

The depolarizing effect arising from detuned coil tilt angles increases with increasing fre-

quency. Hence, the echo amplitude is more sensitive to a detuning of Θ1,2 for higher spin

echo times (see Fig. 2.7 a and b). For τ = 80ps a detuning of dΘ1,2 = ±1.5 would de-

crease the echo amplitude by about 3%. In contrast the echo amplitude gets less sensitive

for a detuning of the frequencies (see Fig. 2.7 c and d), since the frequency increases with

increasing spin echo time τ . For τ = 80ps a detuning of ±30kHz for f1 and ±10kHz for f2

would decrease the echo amplitude by about 3%.

A special case of detuning occurs if no spin-flip scattering occurs in the sample while the

second precession field region is tuned to spin-flip-scattering. This case is easily treated if

the second frequency f2 is assumed to be detuned to −f2. A numerical example for the zone

boundary excitation Q = [0.5 0.5 −1], E = 8.287meV, in RbMnF3 is shown in Fig. 2.8.

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2.3 Extended NRSE resolution function 27

−40 −20 0 20 400

0.2

0.4

0.6

0.8

1

dΘ1 [deg]

echo

am

plitu

de

a−40 −20 0 20 400

0.2

0.4

0.6

0.8

1

dΘ2 [deg]

echo

am

plitu

de

b

−40 −20 0 20 400

0.2

0.4

0.6

0.8

1

df1 [kHz]

echo

am

plitu

de

c−40 −20 0 20 400

0.2

0.4

0.6

0.8

1

df2 [kHz]

echo

am

plitu

de

d

Fig. 2.7: Depolarization effects due to a detuning of spin echo parametersat τ = 80ps. The detuned Parameters are: Θ1 (a), Θ2 (b), f1 (c) and f2(d).

It is obvious that the non-spin-flip signal is already completely depolarized at the lowest

accessible spin echo time τ = 12.4ps. Note that this is not the general case and the signal

arising from the occurrence of non-spin-flip scattering might be not completely depolarized

at the lowest accessible spin echo time for other materials.

0 5 10 15 200

0.2

0.4

0.6

0.8

1

τ [ps]

echo

am

plitu

de

Fig. 2.8: Depolarization effects for spin-flip scattering without reversingthe second field region. The example shown applies to the zone boundaryexcitation Q = [0.5 0.5 −1],E = 8.287meV, in RbMnF3.

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28 2 NRSE resolution theory

Example 2: Transverse acoustic phonon in Pb

Using equation (2.95) and assuming an excitation with zero linewidth and perfectly satisfied

spin echo conditions the instrumental resolution function for the transverse acoustic phonon

[2 ±0.1 0] in Pb can be calculated numerically (see Fig. 2.9). For the calculations an energy

of ~ω0 = 0.88meV, a slope of the dispersion of |~∇qω0 (q0)| = 6.9meVÅ at T = 290K and

an incident wavevector of ki = 1.7Å−1 were assumed. The resolution function changes only

slightly with different TAS configurations. This is due to the small dependence of the beam

divergence on the scattering senses. For this specific phonon the spin echo time for the

NRSE option of the cold TAS V2/FLEX at BER II is limited to τ = 230ps. Therefore, the

depolarizing effects arising from the scattering senses of the background TAS do not play a

role in the accessible τ -range of the NRSE spectrometer.

0 1000 2000 3000 40000

0.2

0.4

0.6

0.8

1

τ [ps]

echo

am

plitu

de

SM=−1 SS=−1 SA=−1SM=−1 SS=−1 SA=+1SM=−1 SS=+1 SA=−1SM=−1 SS=+1 SA=+1

Fig. 2.9: Calculated instrumental resolution for the [2 ±0.1 0] TA phononin Pb assuming zero linewidth and perfectly satisfied spin echo conditions.The depolarization is calculated for different scattering senses. The smalldifference for different instrument settings arise from small changes of thebeam divergence due to the scattering senses. Note that for this specificphonon the spin echo time for the NRSE option of the TAS V2/FLEX atBER II is limited to τ = 230ps.

2.3.3 Quantitative description of depolarization due to sample imperfec-

tions

In this section the additional term +τ ′′2∆Gc · ∇ω (qc0) in the total Larmor phase given by

equation (2.81) arising from sample imperfections is considered in more detail. As pointed

out by Pynn [19] the mosaicity of the sample introduces a further limit to the resolution.

Fig. 2.10 shows a schematic drawing of this effect for a transverse phonon. For each lattice

vector G within the mosaic spread the dispersion will have a different orientation in q-space.

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2.3 Extended NRSE resolution function 29

This smearing of the dispersion leads to a broadening of the linewidth, which is different

from the intrinsic linewidth of the excitation.

ω

q

Δω

Δq

lattice vector G

Fig. 2.10: Effect of the mosaic spread of the sample on the width of thedispersion. An angular variation in the lattice vector G (mosaic) leadsto a variation of the linewidth of the dispersion and thus to an artificialbroadening of the linewidth.

Correction for sign errors and using the correct second order term rather than that stated

in [6] yield:

∆Gc = Gc −Gc0 (2.96)

=

(Gc0 +∆Gc) cos∆ν cos∆η

(Gc0 +∆Gc) cos∆ν sin∆η

(Gc0 +∆Gc) sin∆ν

. (2.97)

Expanding small variations in the lattice vector Gc and neglecting terms higher than second

order leads to

∆Gc =

∆Gc − 12Gc0

(∆ν2 +∆η2

)

Gc0∆η +∆Gc∆η

Gc0∆ν +∆Gc∆ν

. (2.98)

Since the scattering function S (Q, ω) is proportional to (Q · ξ)2 where ξ is the phonon

polarization vector, the optimum choice for q0 for transverse phonons is an orientation

perpendicular to G0, whereas for a longitudinal phonon the optimum choice for q0 is parallel

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30 2 NRSE resolution theory

to G0. Equation (2.98) shows that for longitudinal phonons the mosaic spread contributes

only in second order. Therefore, the sample can have a rather large mosaic spread before

these effects limit the resolution. For a more detailed discussion see [6]. Since this vector

has linear and quadratic terms in the variables ∆η, ∆ν and ∆Gc the linear and quadratic

parts are defined separately:

∆G′c =

∆Gc

G0c∆η

G0c∆ν

(2.99)

∆G′′c =

−12G0c

(∆ν2 +∆η2

)

∆Gc∆η

∆Gc∆ν

. (2.100)

The complex resolution matrix can now be written as

LM = I−1LII+N+W, (2.101)

with the non-zero elements of the (6× 9) matrix I:

I11 = I22 = I33 = I44 = I55 = I66 = 1, (2.102)

I17 = Cx, I18 = CyG0c, I19 = CzG0c (2.103)

with the definition C = ∇qω (qc0). The non-zero elements of the (9× 9) matrix N are:

N77 =1

Υ2S

, N88 =1

η2S, N99 =

1

ν2S. (2.104)

Here ΥS is the 1σ standard deviation for the Gaussian distribution of lattice vectors. The

quadratic terms in ∆η, ∆ν and ∆Gc are taken into account by the symmetric (9× 9) matrix

W. Its non-zero elements are:

W78 = W87 = −iτ ′′2Cy, W79 = W97 = −iτ ′′2Cz (2.105)

W88 = +iτ ′′2CxG0c, W99 = +iτ ′′2CxG0c. (2.106)

Note that equation (2.106) is the corrected version of equation (83) in [6]. The linear terms

in ∆η, ∆ν and ∆Gc arising from ∆G′c can be written as +iτ ′′2T

Tg JM , with the column

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2.3 Extended NRSE resolution function 31

vectors

Tg = (0, 0, 0, 0, 0, 0, Cx, CyG0c, CzG0c) (2.107)

JM = (∆ω,∆kin, y1, y2, z1, z2,∆Gc,∆η,∆ν) . (2.108)

All linear terms can be taken into account by introducing

TM =

−1, NI

(Ci − τ1

τ2Cf

), NIei2cos θ1

,−NF ef2cos θ2

, NIei3,−NF ef3,

Cx,−CyG0c,−CzG0c

. (2.109)

Analog to the treatment in the previous subsection, terms in ∆ω higher than linear are

neglected, leading to

AE =1

N

∫S (Q, ω) exp

(−iτ ′′2∆ω

)d∆ω (2.110)

×∫

exp(iτ ′′2 T

TM JM

)exp

(−1

2JTM LM JM

)d8Jn, (2.111)

where JM = (∆kin, y1,y2, z1, z2,∆Gc,∆η,∆ν) and TM are the 8-dimensional sub vectors of

JM and TM , respectively. LM is the corresponding symmetric (8× 8) sub matrix of LM .

Applying the general matrix theorem (2.94), leads to the resolution function in the form

FM

(τ ′′2)=

∣∣∣∣∣

√det LM (τ ′′2 = 0)

det LM (τ ′′2 )exp

(−1

2τ ′′22 TT

M L−1M

(τ ′′2)TM

)∣∣∣∣∣ , (2.112)

which includes general lattice imperfections.

Figure 2.11 shows numerical calculations for the depolarizing effects of different mosaic

spreads using (2.112). The calculations were done for the [2 0.1 0] TA phonon in Pb at

the energy E = 0.879meV, assuming a zero linewidth. Note that the mosaic spread ηs is

the FWHM. As discussed above, the higher the mosaic spread of the sample, the faster the

polarization decreases.

2.3.4 Quantitative description of depolarization due to curvature of the

dispersion surface

In this subsection the additional term −12τ

′′2∆q′T

c Hc|qc0∆q′

c in the total Larmor phase given

by equation (2.81), arising from the curvature matrix of the dispersion surface, is discussed.

A curved dispersion surface within the TAS resolution ellipsoid leads to additional depo-

larization effects, since neutrons corresponding to different points on the dispersion surface

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32 2 NRSE resolution theory

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ [ps]

echo

am

plitu

de

η

S=1’

ηS=2.5’

ηS=5’

Fig. 2.11: Depolarization effects due to mosaic spread calculated for the[2 0.1 0] TA phonon in Pb. The mosaic spread ηs is the FWHM.

will have a different spin echo phase Φ. The formalism discussed here is also applicable for

crystallographic systems with a symmetry lower than cubic. The Hessian matrix Hc is ex-

pressed in the Cartesian coordinate system related to the reciprocal lattice by the B-matrix

[16]. The elements of the Hessian H are defined by

Hijc =∂2

∂qic∂qjcω (qc) i, j = 1, 2, 3. (2.113)

If the dispersion ω (q) is differentiated in the frame of the reciprocal lattice with basis vectors

b1, b2 and b3, i.e. the elements of H are defined as

Hij =∂2

∂qi∂qjω (q) , (2.114)

and the Hessian H needs to be transformed into Cartesian coordinates using the B matrix:

Hc = BHB−1. (2.115)

Here only second order curvature terms and no sample imperfections are considered. Thus

∆Gc = 0, ∆q′c reduces to ∆qc and LI and J are used. Terms arising from sample imper-

fections will be included in the next subsection. Following [6], the modified matrix reads:

LC = LI + iτ ′′2 I−1C Θ−1

C Hc|qc0ΘCIC (2.116)

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2.3 Extended NRSE resolution function 33

where the matrix ΘCIC transforms the Hessian Hc into the coordinate space of LI . The

matrix ΘC describes the transformation

∆qθ = ΘCY (2.117)

with the column vector Y = (x1, y1, z1, x2, y2, z2). The variation of the wavevector ∆qθ =

∆ki−∆kf is expressed in the basis of the Cartesian system (i0, j0 and l0). In this coordinate

system the total wavevector transfer Qθ is parallel to i0 (see Fig. 2.5):

Qθ =

QM

0

0

. (2.118)

Note that (i0, j0 and l0) are identical to the coordinate system as defined as θ-coordinate

system in [17]. Here the same notation is adopted. In order to express the wavevector

variation ∆qθ in the θ-coordinate system, ∆ki and ∆kf have to be rotated into the Qθ-

frame. Using equations (2.43), (2.44) and the definitions made in Fig. 2.5 yields

∆ki = (x1 cosφ− y1 sinφ) i0 + (x1 sinφ+ y1 cosφ) j0 + z1l0 (2.119)

∆kf = (x2 cos Ξ− y2 sin Ξ) i0 + (x2 sin Ξ + y1 cos Ξ) j0 + z2l0 (2.120)

where φ is defined as the angle between Qθ and ki and Ξ is defined as the angle between

Qθ and kf . For equation (2.117) to hold

ΘC =

cosφ − sinφ 0 − cos Ξ sinΞ 0

sinφ cosφ 0 − sin Ξ − cos Ξ 0

0 0 1 0 0 −1

. (2.121)

Note that here the definitions of φ and Ξ are different to [6] and the sign errors in the

definition of ΘC are corrected. In order to evaluate the expression ∆qTc Hc|qc0

∆qc, the

Hessian Hc needs to be rotated into Hθ. According to [17] the transform of the vector Q is

defined as

Qθ = ΩMNUBQ (2.122)

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34 2 NRSE resolution theory

and that of the matrix H is

Hθ = ΩMNUBHB−1U−1N−1M−1Ω−1, (2.123)

since QTθ HθQθ = QTHQ must be invariant. Equation (2.123) is the generalization of the

matrix transform given by equation (91) in [6].

In order to evaluate the resolution matrix, the Hessian needs a further transformation

to the variable space of the six dimensional vector J. The matrix IC relates Y and

J = (∆ω,∆kin, y1, y2, z1, z2) in the linear transformation [6]

Y = ICJ. (2.124)

Since the aim is an expression for the linear relation between the variable vectors J and

Y, the linear term in the expansion of the dispersion relation in combination with energy

conservation relation (see equation 2.55) is used:

∆ω =

(~

mkI −∇ω (qc0)

)·∆ki −

(~

mkF −∇ω (qc0)

)·∆kf (2.125)

= NIǫi ·∆ki −NF ǫf ·∆kf . (2.126)

The substitution of ki,f and ǫi,f by using equations (2.69), (2.71) and (2.61) yields

∆ω = x1 (NIei1 cos θ1 −NIei2 sin θ1)

+y1 (NIei1 sin θ1 +NIei2 cos θ1) + ei3NIz1

−x2 (NF ef1 cos θ2 −NF ef2 sin θ2)

−y2 (NF ef1 sin θ2 −NF ef2 cos θ2)− ef3NF z2 (2.127)

and

∆kin = x1 cos θ1 + y1 sin θ1. (2.128)

According to equation (2.124) the transformation reads

J = I−1c Y. (2.129)

The elements of the matrices I−1C and IC are defined in Appendix B. For perfectly satisfied

spin echo conditions equations (2.84) and (2.85) hold and IC reduces to the matrix given

in [6].

With the equations (2.124) and (2.117) the additional term in the Larmor phase due to a

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2.3 Extended NRSE resolution function 35

curved dispersion surface is given by

∆qTc Hc|qc0

∆qc = YTΘ−1C Hc|qc0

ΘCY = JT I−1C Θ−1

C Hc|qc0ΘCICJ. (2.130)

The resolution matrix as given in equation (2.116) can now be evaluated. Following the

procedure described in section 2.3.2 the polarization can be written as

AE =1

N

∫S (Q, ω) exp

(−iτ ′′2∆ω

)d∆ω

×∫

exp(iτ ′′2 T

T J)exp

(−1

2JT LC J

)d5Jn. (2.131)

Using equation (2.94) the resolution function taking into account curvature effects and

neglecting contributions from sample imperfections is then given by

FC

(τ ′′2)=

∣∣∣∣∣

√det LC (τ ′′2 = 0)

det LC (τ ′′2 )exp

(−1

2τ ′′22 TT L−1

C

(τ ′′2)T

)∣∣∣∣∣ . (2.132)

2.3.5 Quantitative description of depolarization due to sample imperfec-

tions and curvature of the dispersion surface

The results of the previous two subsections can now be combined to give the resolution

matrix

LMC = ITLCI+N+W (2.133)

The resolution function can be expressed as

FMC

(τ ′′2)=

∣∣∣∣∣

√det LMC (τ ′′2 = 0)

det LMC (τ ′′2 )exp

(−1

2τ ′′22 TT

M L−1MC

(τ ′′2)TM

)∣∣∣∣∣ (2.134)

This resolution function FMC includes general lattice imperfections and the curvature of

the dispersion surface. Here, the advantages of the matrix formalism are obvious, since

cross terms arising from sample imperfections and the curvature are automatically taken

into account by using (2.133)

∆q′Tc Hc|qc0

∆q′c =

(∆qc −∆G′

c

)THc|qc0

(∆qc −∆G′

c

)(2.135)

= ∆qTc Hc|qc0

∆qc −∆G′Tc Hc|qc0

∆G′c

−2∆qTc Hc|qc0

∆G′c (2.136)

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36 2 NRSE resolution theory

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ [ps]

echo

am

plitu

de

only instrinstr + sampleinstr + curvinstr + sample + curv

Fig. 2.12: Numerical results for different depolarizing effects for the [20.1 0] TA phonon in Pb. Considering only instrumental resolution (black),instrumental resolution and sample imperfections (red), instrumental res-olution and curvature of the dispersion surface (blue). The depolarizationdue to all effects is shown by the green curve.

A comparison of the depolarization arising from the different effects discussed in the previous

subsections is shown in Fig. 2.12. Since experimental tests on the numerical model discussed

here were performed with Pb (see section 2.4), the numerical calculations were done for the

[2 0.1 0] TA phonon in Pb, making the same assumptions as in section 2.3.2. Within the

considered τ -range the depolarizing effects arising from the instrumental configuration of

the background TAS (black) are negligible. The depolarizing effects due to sample mosaicity

(red) are stronger compared to effects arising from the curvature of the dispersion surface

(blue). However, the green curve shows, that it is important to consider all effects.

2.3.6 Dispersion surface not coinciding with the center of the TAS reso-

lution ellipsoid

In this subsection the simplifying assumption that the center of the TAS resolution ellipsoid

coincides with the dispersion surface of the excitation is dropped. The TAS is set such that

the wavevector Q0 satisfies the scattering condition for a given set of wavevectors kI0 and

kF0 and an energy of ~ω0TAS . The wavevector transfer is

Q0 = kI0 − kF0 (2.137)

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2.3 Extended NRSE resolution function 37

and the TAS resolution ellipsoid is centered at the energy of (Q0, ω0TAS), where

ω0TAS (kI0,kF0) =~

2m

(k2I0 − k2

F0

). (2.138)

This center does not need necessarily to coincide with the energy of the excitation ~ω0S (Q0),

which is given by the dispersion relation

ω0S (Q0) = ω0 (qc0) . (2.139)

As a consequence all “mean” quantities, used in the derivation of the spin echo phase so far,

have to be taken with respect to the sample dispersion. Since the TAS resolution ellipsoid is

offset, the TAS resolution function needs to take into account the finite energy shift ∆ΩTAS :

∆ω → ∆ω −∆ΩTAS (2.140)

where

∆ΩTAS = ω0TAS − ω0S (Q0) . (2.141)

The resolution matrix reads

LMC = ITLCI+N+W (2.142)

= IT(LTAS + iτ ′′2Ψ+ iτ ′′2 I

−1C Θ−1

C Hc|qc0ΘCIC

)I+N+W. (2.143)

Since only the TAS transmission function changes, the following substitution has to be made

JT ITLTASIJ → JTTASI

TLTASIJTAS (2.144)

with the substitution

JTAS = J− J′=(∆ω −∆ΩTAS ,∆kin, y1, y2, z1, z2), (2.145)

where J′ = (∆Ω, 0, 0, 0, 0, 0). Therefore, the new TAS resolution matrix is defined as

LS = ITLTASI. (2.146)

The Matrix LS is symmetric and therefore J′TLSJ = JTLSJ′. Using this symmetry yields

JTTASLSJTAS = JTLSJ− 2JTLSJ

′ + J′TLSJ′ (2.147)

= JTLSJ− 2∆ΩTAS (LS)1n J+ (LS)11∆Ω2TAS . (2.148)

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38 2 NRSE resolution theory

where (LS)1n defines the 6 dimensional row vector of the matrix LS and (LS)11 is the

(1,1) element of the matrix LS . The term proportional to (LS)11∆Ω2TAS only produces a

constant, which can be absorbed in the normalization factor N. With

−1

2JTTASLMC,TASJTAS = −1

2JTLMCJ+∆ΩTAS (LS)1n J

−1

2(LS)11∆Ω2

TAS (2.149)

the modified expression for the polarization reads

AE =1

N

∫S (Q, ω) exp

(−iτ ′′2∆ω +∆ΩTAS (LS)11∆ω

)d∆ω

×∫

exp(TT

TASJ)exp

(−1

2JT LMC J

)d5Jn. (2.150)

Here the definition

TTAS = iτ ′′2 TM +∆ΩTAS

(LS

)

1n. (2.151)

was used. Therefore, the final result for the resolution function, including its normalization

to 1 at τ = 0, is

FMC,TAS

(τ ′′2)

=

∣∣∣∣∣∣∣∣∣∣∣

√det LMC(τ ′′2 =0)det LMC(τ ′′2 )

× exp(12T

TTASL

−1MC (τ ′′2 ) TTAS

)

× exp(−1

2TTTAS (0) L−1

MC (τ ′′2 = 0) TTAS (0))

∣∣∣∣∣∣∣∣∣∣∣

. (2.152)

Since

exp(TT

TASL−1MCTTAS

)= exp

(−τ ′′22 TT

M L−1MCTM

)

× exp

(+∆Ω2

TAS

(LS

)T1n

L−1MC

(LS

)

1n

)

× exp(+i2τ ′′2∆ΩTASTM L−1

MC

(LS

)

1n

), (2.153)

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2.3 Extended NRSE resolution function 39

equation (2.152) yields

FMC,TAS

(τ ′′2)

=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

√det LMC(τ ′′2 =0)det LMC(τ ′′2 )

× exp

(12∆Ω2

TAS

(LS

)T1n

L−1MC (τ ′′2 )

(LS

)

1n

)

× exp

(−1

2∆Ω2TAS

(LS

)T1n

L−1MC (τ ′′2 = 0)

(LS

)

1n

)

× exp(−1

2τ′′22 TT

M L−1MC (τ ′′2 ) TM

)

× exp(iτ ′′22 ∆ΩTAST

TM L−1

MC (τ ′′2 )(LS

)

1n

)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

. (2.154)

Note that the phase term arising from

exp

(−1

2TT

TAS (0) L−1MC

(τ ′′2 = 0

)TTAS (0)

)(2.155)

= exp

(−1

2∆Ω2

TAS

(LS

)T1n

L−1MC

(τ ′′2 = 0

) (LS

)

1n

)(2.156)

is of no interest in this context, since the term is independent from τ ′′2 and will be absorbed

in the normalization factor.

The Fourier transform is no longer as simple as in the previous subsections, since there is

an additional term linear in ∆ω

∫exp (∆ΩTAS (LS)11∆ω)S (∆ω) exp

(−iτ ′′2∆ω

)d∆ω. (2.157)

This term

exp (∆ΩTAS (LS)11∆ω) (2.158)

multiplied with the scattering function introduces a small asymmetry. However, in practical

cases

∆ΩTAS (LS)11∆ω ≪ 1 (2.159)

Thus, the exponential

exp (∆ΩTAS (LS)11∆ω) (2.160)

is only slowly varying with ∆ω and close to 1 over the ∆ω-range of the scattering function

S (∆ω). Therefore, this factor can be neglected.

Note that it is very important to determine the correct ∆ΩTAS with respect to the exci-

tation energy. Since any detuning of the background triple axis spectrometer changes the

effective Q0 vector for the sample, all the excitation parameters might change, which leads

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40 2 NRSE resolution theory

automatically to a detuning of the spin echo parameters. A detuning of the TAS effectively

corresponds to a new Q′0, which can be derived using the UB matrix formalism described

in Appendix A. Q′0 then gives rise to a new excitation energy, dispersion gradient and

curvature matrix. Considering the change in the excitation energy one can easily derive the

correct ∆ΩTAS . Numerical examples for detuning effects arising from detuned instrument

parameters are discussed in the next section.

2.3.7 Numerical examples

The numerical examples presented here show the depolarizing effects arising from a detuning

of the background TAS. Using the results of the previous sections, all numerical calcula-

tions were performed for the zone boundary excitation Q = [0.5 0.5 −1], E = 8.287meV,

in RbMnF3. This excitation has been measured in previous NRSE experiments (see for

example [23]). The investigated TAS parameters were:

• A3: A detuning in A3 corresponds to a rotation of the sample around the z-axis.

• A4: The scattering angle A4 is the angle between the wavevectors kI and kF .

• ν and µ: These parameters are the angular variables of the sample goniometers.

• ∆ΩTAS : A detuning in ∆ΩTAS corresponds to a shift of the center of the TAS reso-

lution ellipsoid in energy w.r.t. the nominal energy of the excitation, while the total

wavevector transfer Q is kept fixed.

The zone boundary excitation corresponds to the maximum of the dispersion. Here the

slope of the dispersion is zero and thus, no depolarizing effects due to sample mosaicity

are expected. However, by considering second order effects, the curvature of the dispersion

surface leads to a depolarization even for the case of a tuned instrument. If the instrument

is detuned, the total wavevector transfer Q changes in general. Thus, the local dispersion

parameters, i.e. energy, slope and curvature matrix, need to be recalculated. As a result

the slope of the dispersion becomes non-zero. Hence, depolarizing effects due to sample

mosaicity contribute to the total effect. Since in the case of RbMnF3 the dispersion around

the maximum is flat, small changes in Q lead only to small changes in the excitation energy

and the slope of the dispersion. Hence, depolarization effects arising from a detuning of

the spin echo parameters are negligible (see section 2.3.2). Note that a detuning at steeper

parts of the dispersion will increase the depolarization effects. A decrease of the intensity

at the detector due to a detuning of the TAS parameters is not taken into account here.

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2.3 Extended NRSE resolution function 41

Detuning the sample angle A3 and the scattering angle A4

0 20 40 60 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ [ps]

echo

am

plitu

de

dA3 = 0°dA3 = 0.1°dA3 = 0.5°dA3 = 1°

0 20 40 60 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ [ps]

echo

am

plitu

de

dA4 = 0°dA4 = 0.1°dA4 = 0.5°dA4 = 1°

Fig. 2.13: Calculated depolarization effects for RbMnF3 at the zoneboundary excitation Q = [0.5 0.5 −1], E = 8.457meV, arising from adetuned A3 angle (left) and a detuned scattering angle A4 (right).

Fig. 2.13 (left) shows the depolarizing effects arising from a detuning of A3. The depolariz-

ing effects arising from a detuning of the scattering angle A4 are shown in Fig. 2.13 (right).

The depolarizing effect increases with an increasing detuning of the angles. A comparison

between the two plots shows that a detuning of A3 and A4 give almost the same depolar-

izing effects (see Fig. 2.16). Due to the small changes in Q the depolarizing effects due to

a detuning of the spin echo parameters are negligible.

Detuning the sample goniometers

0 20 40 60 800

0.2

0.4

0.6

0.8

1

τ [ps]

echo

am

plitu

de

dν=0°dν=0.5°dν=1°dν=2°

0 20 40 60 800

0.2

0.4

0.6

0.8

1

τ [ps]

echo

am

plitu

de

dµ=0°dµ=0.5°dµ=1°dµ=2°

Fig. 2.14: Calculated depolarization effects for RbMnF3 at the zoneboundary excitation Q = [0.5 0.5 −1], E = 8.457meV, arising from adetuning of the sample goniometers.

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42 2 NRSE resolution theory

Fig. 2.14 shows the depolarizing effects arising from a detuning of the sample goniometers.

Here the rotation axis µ was perpendicular to the [1 1 0] direction (left) while the rotation

axis of ν was perpendicular to the [0 0 1] direction (right). The depolarizing effects due

to a misalignment in µ (ν) increase with increasing detuning. However, the depolarizing

effects are small. Again the changes in Q are small, allowing to neglect effects arising from

detuned spin echo parameters.

Detuning the center of the TAS resolution ellipsoid in energy

0 20 40 60 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ [ps]

echo

am

plitu

de

∆ΩTAS

= 0 meV

∆ΩTAS

= 0.1 meV

∆ΩTAS

= 0.5 meV

∆ΩTAS

= 1 meV

Fig. 2.15: Calculated depolarization effects for RbMnF3 at the zoneboundary excitation Q = [0.5 0.5 −1], E = 8.287meV, arising from a shiftof the center of the TAS resolution ellipsoid in energy.

Fig. 2.15 shows the depolarizing effects arising from a detuning of the TAS resolution

ellipsoid in energy. The depolarizing effects increase with increasing detuning. However,

a detuning of ∆ΩTAS = 0.1meV has almost no effect on the echo amplitude. A resulting

systematic error arising from a slight detuning of the TAS resolution ellipsoid can therefore

be neglected. This is simultaneously an indicator for the required accuracy of the dispersion

data. The required accuracy of the center of the TAS resolution ellipsoid in energy can easily

be achieved for a triple axis spectrometer. By increasing the detuning to ∆ΩTAS = 0.5meV

the depolarizing effects become significant. For ∆ΩTAS = 1meV the echo amplitude drops

drastically. However, the signal is not completely depolarized for accessible spin echo times

τ , which are above τ ≈ 12ps for fRF = 50kHz frequency of the RF coils. If the instrument

is tuned to a certain mode, a strong second mode can still contribute to the echo amplitude

for small spin echo times τ , even if the instrument is strongly detuned for the second mode

(see chapter 3 and section 4.1.3 where it is exactly this effect, which plays a dominant role

in phase sensitive NRSE measurements).

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2.4 Experimental test 43

Comparison of different detuned TAS parameters

0 20 40 60 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ [ps]

echo

am

plitu

de

no detuningdΩ=0.5meVdA3=0.5°dA4=0.5°

Fig. 2.16: Depolarization effects for RbMnF3 at the zone boundary exci-tation Q = [0.5 0.5 −1], E = 8.287meV, for different detuned TAS param-eters.

Fig. 2.16 shows the numerically calculated depolarization effects arising from a detuning

of different TAS parameters. It is obvious that a shift of the TAS resolution ellipsoid in

energy (∆ΩTAS) has the largest effect. A rotation of the sample around the z-axis (A3) and

a detuning of the scattering angle A4 have smaller but still significant effects. A detuning

of the center of the TAS resolution ellipsoid in energy causes a decay of the echo amplitude

different from the expected behavior as seen for a detuning of A3 and A4. The deviations

of the echo amplitude are similar to the spin echo signal corresponding to an asymmetric

line shape of the scattering function (see section 4.1.3). Since the energy resolution of the

background TAS is multiplied with the signal a detuning of the TAS resolution ellipsoid

introduces an artificial asymmetry and thus modifies the decay of the echo amplitude.

2.4 Experimental test

In this section the results of experimental tests [20] performed on the extended resolution

model described in the previous section are discussed. The results presented here have been

published in [21]. If the spin echo parameters of an inelastic NRSE experiment are kept fix

while the sample is rotated, the instrumental settings are effectively detuned, which can be

described by the treatment given in the previous subsection. By rotating the sample, the

dispersion surface is moved through the resolution ellipsoid of the background TAS giving

rise to an energy offset ∆ΩTAS . Therefore, the instrument probes different portions of the

dispersion, leading to a decay of the polarization according to the resolution function given

by equation (2.152) and a phase shift due to additional second order terms in the spin echo

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44 2 NRSE resolution theory

phase. These additional terms predict a non-linear behavior of the phase, compared to a

linear relation in the first order expansion of the Larmor phase.

Fig. 2.17: The cold triple axis spectrometer V2/FLEX at the BER IIof HZB, Berlin, with its spin echo option mounted (before the upgrade in2010-2011).

An experiment was performed at a spin echo time τ = 20ps focusing on the phase shift arising

from the detuning of the apparatus. For the present purpose it was beneficial to perform an

experiment on a simple, well understood model system. Therefore, the [2 0.1 0] TA phonon

in a large single crystal of Pb was chosen. This phonon has an energy of E = 0.97meV, well

suited to a cold TAS. The measurements were performed with the NRSE option of the cold

neutron TAS V2/FLEX at the BER II reactor of HZB, Berlin [76] (see Fig. 2.17) at a fixed

incident wavevector kI = 1.7Å−1. The sample was kept at a temperature of T = 100K to

enhance the echo amplitude and the polarization, respectively. At T > 100K the lifetime is

reduced due to phonon-phonon interactions, while for T < 100K the intensity is reduced by

the Bose factor. Since the intensity changes with moving the dispersion surface through the

TAS resolution ellipsoid, the only way to experimentally access the phase, is to record a full

spin echo curve for each rotation angle of the sample. For a more accurate determination

of the phase angle about 2.5 periods of the spin echo signal were recorded for each rotation

angle. As an example, Fig. 2.18 shows the measured spin echo signal for τ = 20ps at

∆A3 = 0. Here ∆A3 is the deviation of the sample angle A3 (corresponding to a rotation

around the z-axis) from the tuned position of A3 corresponding to the excitation at [2 0.1 0].

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2.4 Experimental test 45

10 15 20 25100

120

140

160

180

200

220

240

dl2 [mm]

Cou

nts

per

mon

itor

Fig. 2.18: NRSE scan at τ = 20ps for ∆A3 = 0.

Fig. 2.19 shows the total accumulated Larmor phase as a function of the rocking angle ∆A3

using the extended resolution model. The experimental data is in full agreement with the

“parameter free” calculations from our model (black) and shows the non-linear behavior of

the Larmor phase as predicted.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2−600

−500

−400

−300

−200

−100

0

100

200

∆ A3 [°]

phas

e sh

ift [°

]

Fig. 2.19: Phase shift of the [2 0.1 0] TA phonon while detuning the setupwith rotating the sample around ∆A3. Each phase angle is obtained froma full NRSE scan at each rocking angle. The blue line shows a fit to alinear function. The black line is obtained from the extended model and isin full agreement with the experimental data.

Using a simplified approach of a first-order expansion of the Larmor phase, allows to extract

the slope of the dispersion dωdq from the linear coefficient of the phase dependence. A linear

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46 2 NRSE resolution theory

fit [6] gives dωdq = 6.42 (14)meVÅ. Though using a full force-constant parameterization of

the dispersion including three-body interactions [22] gives a significantly higher value ofdωdq = 7.43meVÅ. This result demonstrates that higher order terms in the total Larmor

phase are generally significant and important to consider.

2.5 Summary

The theory of inelastic neutron spin echo spectroscopy has been introduced and extended to

second order. As a result depolarizing effects arising from the TAS instrumental resolution,

sample imperfections and curvature of the dispersion surface are included. By allowing

for violated spin echo conditions and crystal symmetries lower than cubic, the existing

formalism has been generalized. A further important extension of the formalism includes

a detuning of the background TAS spectrometer. In order to test the extended resolution

theory, phase sensitive measurements were performed at the cold triple axis spectrometer

V2/FLEX at BER II at HZB, Berlin. By rotating the sample, the background TAS is

automatically detuned and the single dispersion surface is effectively moved through the

resolution ellipsoid of the background TAS. The results are in good agreement with the

extended resolution model and show that second order effects need to be considered. A

resolution model, which accounts for a violation of the spin echo conditions for inelastic

scattering, is appropriate for high resolution spin echo measurements on mode doublets.

Such experiments and the required theoretical framework will be discussed in the next

chapter.

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Chapter 3

NRSE investigations on split modes

The high resolution of NRSE spectroscopy potentially allows to resolve excitations separated

in energy, which are unresolved by standard neutron scattering techniques [21, 23]. Pos-

sible applications are for example splittings of magnon excitations as observed by Náfrádi

et al. [24], hybridized magnon-phonon modes, which are existent in multiferroics [25, 26], or

excitations with small energy separations, which are found in orbital Peierls systems [27].

However, prior to application of the method to complex systems with interesting physical

properties, a basic understanding of the potential of the method is required. Since it is not

possible for two excitations separated in energy to satisfy the spin echo conditions simulta-

neously, unavoidable depolarization effects are introduced.

First, a simplified model describing the signature of split modes, a modulation of the echo

amplitude, is introduced. In the next section the results obtained in the previous chapter

will be used to develop a numerical model for the most general case. To test this model,

experimental tests were performed with a so called tunable double dispersion setup. The

results of inelastic and elastic measurements are presented here and have been published

in [21].

3.1 Two modes within the TAS resolution ellipsoid - Simplified

NRSE model

A first approach considers a simplified NRSE model with two modes within the TAS reso-

lution ellipsoid [23]. In order to describe the echo amplitude as a function of the spin echo

time τ , the spin echo conditions are assumed to be perfectly satisfied for both excitations.

Although this is strictly not possible, this assumption can be useful in practical cases, where

the violation of the spin echo conditions is very small. In this minimal model the echo am-

47

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48 3 NRSE investigations on split modes

plitude AE is described by the Fourier transform of two Lorentzians. Including the relevant

energy offsets yields:

|AE | =∣∣∣∣1

N

∫S (∆ω − ω0S1)RTAS (∆ω − ω0TAS) exp (−i (∆ω − ω0NRSE) τ) d∆ω

∣∣∣∣ . (3.1)

Here S (∆ω − ω0S1) is the scattering function describing two Lorentzian profils, given by

equation (3.3). The NRSE instrument is tuned to ω0NRSE and N is a normalization factor

ensuring that AE = 1 for τ = 0. As in chapter 2.2 it is assumed that the scattering function

S (Q, ω) does not vary in Q within the TAS resolution ellipsoid. Therefore, the scattering

function can be factorized. S (Q) then only contributes to the absolute intensity and can

be neglected. As an additional weighting function the energy resolution of the background

TAS is

RTAS (∆ω − ω0TAS) = exp

(−4 ln 2

~2 (∆ω − ω0TAS)

2

E2TAS

), (3.2)

where the TAS resolution ellipsoid is centered at ω0TAS with a FWHM of ETAS . The

simplified model function considers no dispersion surface at all and effects arising from TAS

resolution in Q-space and NRSE resolution are neglected. Two excitations are allowed and

parameterized as

S (∆ω − ω0S1) = A1Γ1

Γ21 + (∆ω − ω0S1)

2 +A2Γ2

Γ22 + (∆ω − ω0S1 −∆Ω)2

, (3.3)

where A1,2 are amplitude factors representing the relative intensity weight of the Lorentzians.

The linewidths are given as HWHM Γ1,2. The two excitations are separated in energy

by ∆Ω. Note that the scattering function is centered at the first excitation with an energy

of ~ω0S1.

In an inelastic scattering process the total Larmor phase will have a constant term propor-

tional to ω0NRSE . This energy will be equal to the mean energy of the first excitation ω0S1,

if the apparatus is tuned to this mode. The Fourier transform of the echo amplitude then

reduces to

AE =1

N

∫S (∆ω)RTAS (∆ω − (ω0TAS − ω0S1)) exp (−i∆ωτ) d∆ω, (3.4)

representing the fact that the scattering function is centered at the first mode and the center

of the TAS resolution is at ω0TAS − ω0S1. The normalization factor is then given by

N =

∫S (∆ω)RTAS (∆ω − (ω0TAS − ω0S1)) d∆ω. (3.5)

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3.1 Two modes within the TAS resolution ellipsoid - Simplified NRSE model 49

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

2

4

6

8

10

∆ ω [meV]

S(∆

ω)

[a.u

.]

0 5 10 15 20 25 30

10−1

100

τ [ps]

echo

am

plitu

de

Fig. 3.1: Top: Scattering function S (∆ω) (blue solid), assuming Γ = Γ1 =Γ2 = 20µeV, A1=1, A2=2 and ∆Ω = 0.5meV, scattering function multi-plied with the TAS energy resolution function assuming ETAS = 1.25meV(red solid) and scattering function convoluted with the TAS resolution func-tion (grey dashed). Bottom: Single exponential decay corresponding to Γ(black solid), Fourier transform of S (∆ω) (blue solid), Fourier transformof R ·S (∆ω) (red) and the approximation of equation (3.6) (green dashed)

Fig. 3.1 (top) shows the scattering function described by equation (3.3), the scattering

function multiplied and convoluted with the resolution function of the background TAS (see

equation (3.2)) and the corresponding Fourier transform (bottom). Note that the maxima

of the modulation follow the single exponential decay. For the case of identical linewidths

the echo amplitude can be described by the analytical expression of an exponential decay

modulated by a cosine term [23]

AE =|AM |+ (1− |AM |)

∣∣∣cos(2π

τ

T

)∣∣∣exp

(−τΓ

~

), (3.6)

where the period of the modulation is inversely proportional to the separation in energy

T =4π~

∆Ω(3.7)

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50 3 NRSE investigations on split modes

and the amplitude of the modulation is determined by the intensity ratio of the modes A1A2

AM =A′

1 −A′2

A′1 +A′

2

(3.8)

with

A′1 = A1RTAS (−ω0TAS) (3.9)

A′2 = A2RTAS (∆Ω− ω0TAS) . (3.10)

Another approximation is

AE =|AM |+ (1− |AM |) cos2

(2π

τ

T

)exp

(−τΓ

~

), (3.11)

which describes the Fourier transform of the scattering function almost as good as equation

(3.6).

To demonstrate the effects of a change in the energy separation and the amplitude ratio

of the two modes on the echo amplitude, Fig. 3.2 shows numerical examples for different

scattering functions and the corresponding Fourier transforms. As seen from equation (3.7),

the period T of the modulation of the echo amplitude decreases with increasing energy sepa-

ration of the excitations (comparing top to middle in Fig. 3.2). Whereas the amplitude AM

of the modulation given by (3.8) reaches a minimum if the amplitude ratio of both modes

equals 1 (comparing middle to bottom in Fig. 3.2).

Beside the scattering probability in this simple model the amplitude parameters A1 and A2

should also contain the information about the fraction of the signal giving rise to a polarized

signal in the detector. As mentioned in [23] two limiting cases can be seen directly:

First, if the spin echo conditions are strongly violated for the second excitation, the depo-

larized background, independent from the spin echo time τ , will increase and A2 can be set

to zero. For the case that the signal from the second excitation is completely depolarized

for the smallest accessible spin echo time, the initial polarization for τ = 0, P0, will be less

than 1. In the second case the spin echo conditions are perfectly satisfied for both modes.

The amplitudes A1,2 are then determined by the structure factor. In this case the contrast of

the modulation is independent from the spin echo time τ . Note that in general the contrast

of the modulation is a function of the spin echo time τ . If the depolarizing effects arising

from violated spin echo conditions are different for both modes, the amplitude ratio A1A2

, i.e.

the contrast, becomes τ -dependent.

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3.2 Second dispersion surface within the TAS resolution ellipsoid - General model 51

Any intermediate case is beyond this model. A general model taking into account the decay-

ing contributions of the excitations arising from violated spin echo conditions is discussed

in the next subsection.

−0.2 −0.1 0 0.1 0.20

5

10

15

∆ ω [meV]

S(∆

ω)

[a.u

.]

0 20 40 60

10−1

100

τ [ps]

echo

am

plitu

de

−0.2 −0.1 0 0.1 0.20

5

10

15

∆ ω [meV]

S(∆

ω)

[a.u

.]

0 20 40 60

10−1

100

τ [ps]

echo

am

plitu

de

−0.2 −0.1 0 0.1 0.20

10

20

∆ ω [meV]

S(∆

ω)

[a.u

.]

0 20 40 60

10−1

100

τ [ps]

echo

am

plitu

de

Fig. 3.2: Calculated examples for different model scattering functionsS (∆ω) (left) and the corresponding Fourier transform (right), assumingΓ = Γ1 = Γ2 = 20µeV. Top: A1 = A2, ∆Ω = 0.1meV. Middle: A1 = A2,∆Ω = 0.2meV. Bottom: A1 = 2 ·A2, ∆Ω = 0.2meV.

3.2 Second dispersion surface within the TAS resolution ellip-

soid - General model

In this section the general case of a second dispersion surface within the TAS resolution

ellipsoid is discussed. The model introduced in [23] will be generalized such that the center

of the TAS resolution ellipsoid ωTAS is not required to coincide with neither the energy

of the first dispersion surface ω0S1 (qc0) nor with the second ω0S2 (qc0). In addition, the

spin echo conditions may be violated for both modes. The general expression for the echo

amplitude is given by

AE =1

N

∫S1 (Q, ω)R1(ki,kf )e

iφ1(ki,kf )d3kid3kf

+

∫S2 (Q, ω)R2(ki,kf )e

iφ2(ki,kf )d3kid3kf

. (3.12)

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52 3 NRSE investigations on split modes

The echo amplitude at the spin echo point will be given by |AE |. Using the results from

section 2.3.6 AE is

AE =1

N

∫S1 (∆ω)RTAS(∆ω −∆Ω1, Jn)e

iφ1(∆ω,Jn)d∆ωd5Jn

+

∫S2 (∆ω)RTAS(∆ω −∆Ω2, Jn)e

iφ2(∆ω,Jn)d∆ωd5Jn

, (3.13)

where

∆Ω1 = ω0TAS − ω0S1 (qc0) (3.14)

∆Ω2 = ω0TAS − ω0S2 (qc0) . (3.15)

The scattering function does not contain the energy offsets:

S1 (∆ω) = A1Γ1

Γ21 +∆ω2

(3.16)

S2 (∆ω) = A2Γ2

Γ22 +∆ω2

. (3.17)

Substituting the TAS resolution function RTAS in equation (3.13), the most general case

reads:

AE =1

N

∫S1 (∆ω) exp

(−iτ ′′2,1∆ω +∆Ω1 (LS)11∆ω

)d∆ω

×∫

exp(TT

TAS1J)exp

(−1

2JT LMC1J

)d5Jn

+

∫S2 (∆ω) exp

(−iτ ′′2,2∆ω +∆Ω2 (LS)11∆ω

)d∆ω

×∫

exp(TT

TAS2J)exp

(−1

2JT LMC2J

)d5Jn

(3.18)

where

TTAS1,2 = iτ ′′2,1,2TM1,2 +∆Ω1,2

(LS

)

1n. (3.19)

In principle this formalism can be extended to a multiple dispersion case, which can be

useful for mode multiplets with more than 2 excitations. However, this is beyond the scope

of this thesis.

Any mixture of spin-flip and non-spin-flip scattering is included in the general expression

given by equation (3.18). In that case there would be no energy offset for any of the two

excitations. The TAS resolution ellipsoid would be centered and the spin echo conditions

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3.2 Second dispersion surface within the TAS resolution ellipsoid - General model 53

for one of the two modes would be violated by a sign reversal in τ2, resulting in a rapid de-

polarization of the “wrong” spin state. An example for this was already given in section 2.3.2.

In order to test the developed general model, existing experimental data of RbMnF3 [23, 15]

were analyzed. The investigated crystal consisted of two grains of comparable size. The

relative orientation of the two crystallites has been obtained by standard procedures [15].

The measurement was performed at a fixed kf = 2.51Å−1 and a temperature T = 3.18K

for the zone boundary magnon at Q = [0.5 0.5 −1], E = 8.46meV. The data points and the

calculated time dependence of the echo amplitude given in [23] is shown in Fig. 3.3 (dashed

blue). Note that in [23] it has been assumed that one mode is perfectly tuned while the

instrument is detuned for the second mode. The black line displays the calculations for the

time dependence of the echo amplitude according to the general model developed in this

thesis. Here, depolarization effects for both excitations due to a detuning of the instrument

are considered and minor sign errors in [6] were corrected for. The calculations give a much

better description of the modulated signal compared to the results in [23]. This proves

the importance of taking depolarization effects for all excitations into account. The energy

separation between the two modes was calculated to be 0.386meV using the UB matrix

formalism. However, as stated in [23], for a fit of the energy split of the two modes more

data points are required.

0 5 10 15 20 25 3010

−1

100

τ [ps]

echo

am

plitu

de

Fig. 3.3: Experimental data measured on the zone boundary magnonQ = [0.5 0.5 −1], E = 8.46meV, T = 3.18K, in RbMnF3 at TRISP, FRM-II [23]. The black line shows the calculations according to the generalmodel developed in this thesis. Calculations assuming several simplifica-tions [23] are given by the dashed blue line. The general model gives abetter description of the data and demonstrates the importance of takingdepolarization effects for both excitations into account.

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54 3 NRSE investigations on split modes

3.3 Experimental verification

Before applying the general model for NRSE spectroscopy on multiple modes to complex

systems it is crucial to gain a basic understanding of the potential of the method on a well

understood model system. Therefore, phonon excitations in Nb were chosen as a simple

system to verify the numerical model described in the previous section experimentally.

3.3.1 Experimental setup

The experiments were performed using the NRSE option of the cold triple axis spectrometer

V2/FLEX at the BER II of HZB, Berlin [76] (see Fig. 2.17). In order to study a mode dou-

blet with NRSE, a unique tunable double dispersion setup was realized using 2 Nb crystals

(see Fig. 3.4). In this setup the lower crystal was rigidly fixed, while the upper crystal was

mounted on a stack of piezoelectric devices (purchased from Attocube Systems AG [28]),

consisting of two goniometers and a rotational stage. Thus, the setup allowed for rotating

the upper crystal with respect to the lower crystal and hence provided the opportunity to

generate artificially split modes with a tunable separation in energy. The whole setup was

mounted on a sample stick and both crystals were simultaneously illuminated by the neu-

tron beam. Using the two goniometers of the Attocube setup and the sample goniometers

of the background TAS, both crystals were oriented in the (h h l) scattering plane.

Rotation stage: ( A3)ΔΩ Δ

Goniometer 1: tiltμ

Goniometer 2: tiltν

Nb crystal 1: movablecylinder axis (100)

Nb crystal 2: fixedcylinder axis (110)

0 0 11 1 0

μ

ν

Ω

Fig. 3.4: Left: The double crystal setup mounted on a 3-stage Attocube[28] module (1 rotational stage and 2 goniometers) and attached to thesample stick. The Nb crystals are oriented in the (h h l) scattering plane.Right: Sketch of the sample rotations provided by the module.

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3.3 Experimental verification 55

3.3.2 Niobium dispersion models

For the extended resolution model the dispersion parameters enter into the calculation of

the depolarization in the case of detuned instrument parameters. Hence, a numerical model

for the dispersion of Nb is required. Several different models for the dispersion of transition

metals such as Nb are described in the literature (see [29, 30, 32, 33, 34, 35] and a brief

overview is given in [36]). The approaches differ in their ability to reproduce features of the

Nb dispersion such as Kohn anomalies [38]. However, for small values of q all models are

well suited to describe the dispersion relation. Since the inelastic experiments were carried

out for a small q (q = 0.05r.l.u.) the following models were implemented:

• CGW model: This model, proposed by Clark, Gazis and Wallis [39], is based on

angular forces. Modifications suggested by Behari et al. [41, 42] to consider electron-

ion interactions were included by Bose et al. [29].

• DAF model: This model, proposed by deLaunay [40], is based on angular forces.

Modifications suggested by Behari et al [41, 42] to include electron-ion interactions

are realized in [29]. It is shown that the modified CGW and the modified DAF model

give the same results for bcc metals such as Nb [29].

• Modified axially-symmetric model: This model discussed by Bajpai et al. [30]

considers ion-ion and electron-ion interactions. In order to correct smaller typos in

the formulas [37] was used.

0 0.05 0.1 0.15 0.20

1

2

3

4

5

6

q=[0 0 l] [r.l.u]

E [m

eV]

DAFCGWBaypai

Fig. 3.5: Comparison of the dispersion data obtained from the differentimplemented models for Nb. The wavevector of the excitation q is directedalong the [0 0 l] direction. For small values of q the models give almost thesame result.

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56 3 NRSE investigations on split modes

Fig. 3.5 shows the dispersion values obtained from the different models for the wavevector of

the excitation q directing along the [0 0 1] direction. For small values of q, here q = 0.05r.l.u.,

all implemented models give the same result within 0.3% except for the (3,3)-component of

the curvature matrix where the value given by the Bajpai is smaller (see Tab. 3.1). There-

fore, for the following analysis of the experimental results only one model, namely the DAF

model, was used.

Model E[meV]

Slope[meV·Å]

H(1,1)[meV·Å2]

H(2,2)[meV·Å2]

H(3,3)[meV·Å2]

CGW 1.1670 12.4389 541.91 541.97 5.61

DAF 1.1656 12.4244 542.60 542.60 5.61

Bajpai 1.1661 12.4017 542.09 542.09 4.74

Tab. 3.1: Dispersion parameters for Nb extracted from the different mod-els for q = 0.05r.l.u. . Elements of the curvature matrix H not listedare zero. The elements of H are expressed in the Cartesian system of thereciprocal lattice.

3.3.3 Elastic measurements on split modes

Since separated Bragg peaks also cause a modulation of the echo amplitude in the spin

echo length domain, accompanying elastic measurements with a mosaic-sensitive Larmor

diffraction setup [44] were performed. This allows for an independent accurate calibration

of the attocube parameters, which determine the dispersion splitting in the inelastic case.

Larmor diffraction

For completeness a short summary of the principles of Larmor diffraction is given here.

A more detailed discussion can be found in [8, 44, 45]. In contrast to the main idea of

neutron spin echo spectroscopy that the Larmor precessions can be used to cancel for each

velocity separately to preserve the polarization, Rekveldt [44, 45] pointed out that useful

measurements can be done by adding the Larmor precessions instead of reversing them.

The key point of this so called Larmor diffraction is that all neutrons fulfilling the Bragg

condition will have the same Larmor precession angle. If the precession field boundaries

are oriented parallel to the diffraction planes (see Fig. 3.6 left), every neutron satisfying

Bragg’s law will have the same perpendicular component ~k⊥ and therefore accumulate the

same Larmor precession angle, while passing through the fields before and after the sample.

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3.3 Experimental verification 57

kI

L

θ

kF

k

kI

θ

kF

Fig. 3.6: Left: Larmor diffraction setup. The precession field boundariesare parallel to the lattice planes and the magnetic fields are oriented inthe same direction. All neutrons fulfilling the Bragg condition have thesame k⊥ and therefore undergo the same Larmor precession. This setupis sensitive to a spread of the lattice spacing. Right: Mosaicity sensitiveLarmor diffraction geometry. The lattice planes of the crystal are slightlytilted with respect to each other. Hence, neutrons acquire a different totalLarmor precession angle, as their paths depend on which crystallite theyare scattering off. The magnetic fields of the precession regions are orientedantiparallel.

It can be shown that for parallel oriented magnetic fields the total Larmor phase reads:

Φtot =2mωLL

π~d (3.20)

where d is the lattice spacing of the crystal, ωL is the Larmor frequency and L is the distance

between the precession field boundaries (see Fig. 3.6 left). Therefore, a change in the lattice

spacing results in a phase shift

∆Φ = Φtot∆d

d. (3.21)

Since Φtot can be up to 105rad, a relative resolution of ∆dd

∼= 10−6 can be realized. Analog

to neutron spin echo spectroscopy in Larmor diffraction, the echo amplitude at the analyzer

is the cosine Fourier transform of the scattering function S(Q, ω) w.r.t. Q. This directly

gives the normalized distribution of the lattice spacing variations f (∆d). Note that the

total Larmor precession angle is to first order independent of a tilt of the diffraction planes

(due to mosaicity) with respect to the field boundaries.

If the magnetic fields are oriented in opposite directions as in conventional spin echo mea-

surements, the setup becomes sensitive to mosaic spread, rather than to the distribution of

lattice constants [45] (see Fig. 3.6 right).

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58 3 NRSE investigations on split modes

Experimental results

The elastic part of the measurements using the tunable double dispersion setup was per-

formed in a mosaicity sensitive Larmor diffraction setup, i.e. all precession field boundaries

were kept perpendicular to the reciprocal lattice vector QB and were oriented antiparallel.

The spin echo signal was measured using the [1 1 0] Bragg peaks of the two crystals with

an incident wavevector of kI = 1.9Å−1.

First, the [1 1 0] Bragg peak of each crystal was measured individually. This was realized

by rotating the other crystal by more than 4 in A3. In Fig. 3.7 the echo amplitude of

the individual Bragg peaks is shown as a function of the spin echo length δ [46], which is

identical to the van-Hove correlation length [9]. The data were corrected for instrumental

depolarization using direct beam calibration measurements. A Gaussian distribution was

fitted to each data set (see Fig. 3.7) resulting in a FWHM of 564Å (fixed crystal, black)

and 1026Å (movable crystal, blue). This corresponds to a mosaic spread of 5.12arcmin and

7.82arcmin, respectively. The results are in good agreement with previous gamma diffrac-

tion measurements performed at HZB, Berlin.

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

δ [Å]

echo

am

plitu

de

Fig. 3.7: Gaussian distribution fitted to the elastic NRSE data measuredat the [1 1 0] Bragg peak of the individual crystals. The black and blueline correspond to the fixed crystal and the movable crystal, respectively.

In order to investigate the spin echo signal of two Gaussian mosaic distributions the Bragg

peaks of the two crystals were then separated by a small angle ∆A3 in the scattering plane.

The echo amplitude of the exact Fourier transform of the two Bragg peaks can be approxi-

mated with a Gaussian, modulated by a cosine term [21]:

AE =

∣∣∣∣|AM |+ (1− |AM |) cos(2π

δ

∆s

)∣∣∣∣ exp(−4ln2

δ2

w2S

), (3.22)

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3.3 Experimental verification 59

0 200 400 600 8000

0.2

0.4

0.6

0.8

1

δ [Å]

echo

am

plitu

de

Fig. 3.8: Elastic NRSE data measured at the [1 1 0] Bragg peaks of two Nbcrystals. The red (green) data points were measured at a nominal relativeangle of 0.3 (0.39) between the [1 1 0] Bragg peaks. The model describedby equation (3.22) is fitted to the data.

where δ is the spin echo length. The modulation of the echo amplitude is inversely propor-

tional to the difference in sample rotation angle ∆A3 and the magnitude of the reciprocal

lattice vector QB

∆s =4π

QB∆A3. (3.23)

The FWHM wS of the distribution function in correlation length may be obtained from the

FWHM of the sample mosaicity ηS .

wS =2π

QBηS. (3.24)

In a mosaicity sensitive Larmor diffraction setup using NRSE, the correlation length δ reads:

δ = 4πm

~

LtanΘ

kIfeff

1

QB, (3.25)

where L is the distance between the NRSE coils operated with effective frequencies feff

and Θ is the tilt angle of the NRSE coils with respect to the incident wavevector with

magnitude kI .

The A3-positions of the Bragg peaks were determined by TAS rocking scans. In order to

obtain a good contrast of the modulation of the echo amplitude the measurements were

performed at a sample angle A3 close to the center between the two Bragg peaks.

Fig. 3.8 shows the results of Larmor diffraction measurements for nominal angular sepa-

rations of 0.30 (red) and 0.39 (green). As expected, the modulation period of the signal

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60 3 NRSE investigations on split modes

decreases with increasing angular separation. The approximation described by equation

(3.22) is fitted to the data. Modulation periods of ∆s = 611(1)Å (red) and ∆s = 486(1)Å

(green) are obtained from the fits. According to equation (3.23) a corresponding angular

peak separation of ∆ω = 0.357(1) (red) and ∆ω = 0.449(1) (green) can be extracted. This

is in agreement with an independent calibration of the angular encoders of the Attocube

device using standard TAS scans at larger peak separations. The results demonstrate that

the echo amplitude in the spin echo length space is in good agreement with the prediction.

The development of the contrast of the modulation as a function of the amplitude ratio

of both [1 1 0] Bragg peaks was investigated. Elastic measurements were performed for

different TAS sample angles A3 in Larmor diffraction geometry with an angular separation

between the two Bragg peaks of 0.39. By varying the TAS sample angle A3 the respective

contributions of the Bragg peaks, i.e. the amplitude ratio, changes. Hence, the contrast of

the modulation changes. With a decreasing amplitude ratio the contrast of the modulation

of the echo amplitude should also decrease. Fig. 3.9 shows the accompanying TAS rocking

scan for the experiment. Two Gaussian functions (dashed blue lines) are fitted to the data.

The sum of both functions is shown by the black curve. Larmor diffraction measurements

were performed for A3 values of 45.05, 45.10 and 45.15. Corresponding amplitude ratios

of A1A2 = 0.63, A1

A2 = 0.16 and A1A2 = 0.04 are obtained from the Gaussian fits. Thus, the

contrast of the modulation should decrease with increasing A3.

44 44.5 45 45.5 460

200

400

600

800

1000

1200

rocking angle A3 [°]

coun

ts /

1 s

Fig. 3.9: TAS rocking scan at an angular separation of 0.39 betweenthe two Bragg peaks. Two Gaussians (dashed blue lines) are fitted too thedata. The black line shows the superimposed fit.

Fig. 3.10 shows the result of the Larmor diffraction measurements for A3 values of 45.05

(green), 45.10 (red) and 45.15 (black). As expected, the modulation decreases for larger

values of A3. The approximation given by equation (3.22) is fitted to the data. Correspond-

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3.3 Experimental verification 61

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

δ [Å]

echo

am

plitu

de

Fig. 3.10: Elastic NRSE data measured at the [1 1 0] Bragg peaks of twoNb crystals with an angular separation of 0.39. The measurements wereperformed for A3 values of 45.05 (green), 45.10 (red) and 45.15 (black).The approximation (3.22) is fitted to the data. With increasing A3 theamplitude ratio, i.e. the contrast of the modulation, decreases.

ing amplitude ratios A1A2 of 0.553(3), 0.109(4) and 0.066(3) are obtained from the fits. The

results are in good agreement with the amplitude ratios extracted from the TAS rocking

scan. The first minimum of the modulation is reasonably well fitted by the simplified ap-

proximation. However, for lower amplitude ratios the approximation fails to describe the

rise of the echo amplitude beyond δ = 700Å. Thus, the approximation described by equa-

tion (3.22) is suitable to determine the splitting of the Bragg peaks only if A1A2 > 0.5. For

smaller amplitude ratios the approximation fails. Hence, a model including a more detailed

description of the echo amplitude for Larmor diffraction geometry would be needed. How-

ever, a discussion of such a model is beyond the scope of this thesis.

3.3.4 Inelastic measurements on split modes

In this subsection the results from the inelastic measurements performed on the tunable

double dispersion setup during two beamtimes on Nb single crystals are presented. The

simplified model as discussed in section 3.1 and the general model of section 3.2 have been

fitted to the data. A discussion of the results obtained includes a comparison between the

two approaches.

The cold triple axis spectrometer V2/FLEX was operated in a configuration with scat-

tering senses SM = −1 (monochromator), SS = −1 (sample) and SA = +1 (analyzer)

with an experimental transverse Q-resolution of about 0.006Å−1 FWHM at fixed incident

ki = 1.9Å−1 [43]. The two Nb crystals were aligned with [h h l] plane as the scattering

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62 3 NRSE investigations on split modes

plane. The energy of the investigated excitation [1 1 0.05] phonon was experimentally found

at E = 1.144(8)meV at the temperature T = 65K (see Fig. 3.11). This is in good agreement

with the excitation energy obtained from the DAF model. The energy scan was performed

at [1 1 -0.05] since this is the focused TAS configuration leading to higher intensities. Due to

resolution effects of the background TAS the signal due to incoherent scattering is not cen-

tered around zero. The required tilt angles of the precession field boundaries, Θ1 = −31.60

and Θ2 = +27.27, were calculated using SERESCAL [15].

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.20

200

400

600

800

1000

1200

1400

1600

E [meV]

Inte

nsity

[Cou

nts

/ 15m

in]

Fig. 3.11: TAS energy scan to determine the energy of the [1 1 -0.05]excitation for T = 68K. The scan was done for [1 1 -0.05] since this is thefocused TAS configuration leading to higher intensities. The energy of theexcitation is E = 1.144(8)meV. Due to resolution effects of the backgroundTAS the incoherent scattering signal is not centered around zero.

In order to calibrate the Attocube modules, several TAS energy scans for different angular

separations of the crystals were performed for each module. An examples for the separation

44 44.5 45 45.5 46 46.5 470

500

1000

1500

2000

A3 [°]

Inte

nsity

[Cou

nts

/ 1s]

−0.5 0 0.5 1 1.5 20

500

1000

1500

E [meV]

Inte

nsity

[Cou

nts

/ 12.

3min

]

Fig. 3.12: Left: A3 rocking scan giving a split in A3 of −0.84. The mov-able peak (A3 = 45.72) is defined as the [1 1 0] Bragg peak. Right: Thecorresponding TAS energy scan at [1 1 -0.05] gives an energy of 1.607meVfor the shifted excitation resulting in an energy split of 0.463meV.

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3.3 Experimental verification 63

of the two crystals in the rocking angle Ω (A3) is shown in Fig. 3.12. The movable peak

(A3 = 45.73) was defined as the [1 1 0] Bragg peak leading to a split of ∆A3 = −0.84

for the fixed peak. The left plot displays the A3 rocking scan to determine the separation

in A3, while the right plot shows the corresponding TAS energy scan. The energy of the

shifted excitation is fitted to 1.60(2)meV. Here a negative tilt in A3 results in a shift towards

higher energies. Note that the relation between the sign of the split in A3 and the sign of

the energy shift is reversed for the unfocused TAS configuration, which is used for the NRSE

scans at [1 1 0.05].

0 5 10 15 20 25 300.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ [ps]

echo

am

plitu

de

exponential decayextended model

Fig. 3.13: Inelastic NRSE data of the fixed peak at [1 1 0.05], E =1.144meV. A single exponential decay of the spin echo time τ (red dashed)and the extended resolution model (blue solid) are fitted to the data. Theextended resolution model gives a linewidth, which is zero within error. Incontrast, the linewidth obtained from the exponential decay would suggestan unphysical linewidth of Γ = 40(3)µeV.

In order to determine the linewidth of the excitation at T = 65K, inelastic NRSE measure-

ments were performed on the fixed crystal, while the movable crystal was rotated in A3 by

more than 3 ensuring that only the inelastic signal from the fixed crystal contributes to

the NRSE signal. The data were corrected using tilt angle calibration scans of the direct

beam (see section 5.3.4) allowing to fix P0 = P (τ = 0) = 1 in the fits. Since the disper-

sion at [1 1 0.05], E = 1.144meV, is very steep and close to the Bragg peak the curvature

parameters of the dispersion surface at this point in the (Q, ω)-space have large values.

Therefore, as it can be calculated by the formalism outlined in the previous chapter strong

depolarization effects arising from curvature effects are expected. In a first step to verify

these assumption, a simple exponential decay was fitted to the data resulting in a very broad

linewidth of Γ = 40(3)µeV. A second fit using the extended resolution model, i.e. equation

(2.134), and the DAF model, to calculate the dispersion parameters, gives Γext = 2(3)µeV.

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64 3 NRSE investigations on split modes

This is within error in agreement with a vanishing intrinsic linewidth or decay mechanism

(anharmonic interactions, i.e. phonon-defect interactions) giving rise to a linewidth of less

than 3µeV. Both fits are plotted in Fig. 3.13. This demonstrates again that higher order

terms in the Larmor phase are generally significant and important to consider. As the ex-

ample of Nb shows, the depolarization effects increase for steeper and more strongly curved

dispersion surfaces.

For the sake of completeness both models were fitted to the data, leaving P0 as an additional

free parameter. The exponential decay then gives a linewidth of Γ = (26.46± 6.67)µeV and

P0 = (0.71±0.11). Since the data is corrected using the calibration scans, a P0 significantly

below 1 does not make any physical sense. The fit of the general model however, gives a

linewidth of Γ = (4.24 · 10−7 ± 0.008)µeV and P0 = (0.94± 0.17). The results for Γ of both

fits using the general model agree within the error. Furthermore, leaving P0 as a free fit

parameter gives a result close to 1 confirming the validity of the model assumed.

44 44.5 45 45.5 46 46.50

100

200

300

400

500

600

700

800

A3 [°]

Inte

nsity

[Cou

nts

/ 1s]

Fig. 3.14: TAS rocking scan to determine the splitting in A3 of the twomodes.

For a second set of inelastic measurements, both crystals were separated in A3 providing

an artificially split dispersion. The period of the modulation of the echo amplitude should

decrease with increasing separation of the two modes (see equation (3.7)). Since the inelastic

signal decays very fast due to the depolarizing effects arising from the dispersion parame-

ters, a rather large separation in A3 was chosen to ensure that at least one minimum of the

modulated echo amplitude is within the accessible range of spin echo times τ . Therefore,

the crystals have been aligned to have a separation of ∆A3 = 0.5, as determined by a

TAS rocking scan (see Fig. 3.14). Since the contrast of the modulation should be largest

if both modes contribute with the same intensity, the NRSE measurements were done with

an nominal TAS energy of E = 1.28meV. This energy value was obtained as the center

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3.3 Experimental verification 65

between the two excitations using the chosen Attocube parameters and the independent

TAS calibrations scans of the Attocube modules.

Simplified model

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

τ [ps]

echo

am

plitu

de

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

τ [ps]

echo

am

plitu

de

Fig. 3.15: The inelastic data at [1 1 0.05], E = 1.28meV, at T = 68K showsa clear modulation of the echo amplitude. Left: Fit of the simple model(equation (3.6)). Right: Fit of the two other approximations (equation(3.11), red dotted, and equation (3.26), green).

The results of the inelastic measurements with the split dispersion are shown in Fig. 3.15.

The data were corrected using direct beam calibration measurements and show a clear

modulation of the echo amplitude. As a first step, the simple model discussed in section 3.1

(see equation (3.6)) was fitted to the data (left). In addition the second approximation (see

equation (3.11)) and the similar approximation

AE =∣∣∣|AM |+ (1− |AM |) cos

(2π

τ

T

)∣∣∣ exp(−τΓ

~

)(3.26)

were fitted to the data (right). The results of the different fits are listed in Tab. 3.2. All

simplified models give almost the same result for the energy separation ∆Ω of the two exci-

tation of about 260µeV, which is in good agreement with the value of 280µeV, as obtained

from ∆A3 and the calculated dispersion values. The results for the linewidth Γ are far too

high compared to the fit of of the extended model to the single dispersion data. This is

similar to the result from the exponential decay fitted to the data of the single excitation.

The first two simplified models in Tab. 3.2 give an amplitude ratio A1A2

below 0.5, which

is incompatible with the accompanying TAS scans giving an amplitude ratio close to 1.

Whereas the result of the last simplified model is at least closer to 1 compared to the other

models. However, due to the low count rate, the statistics of the data is low, which obscures

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66 3 NRSE investigations on split modes

the contrast of the modulation. Since the amplitude ratio is derived from the contrast, this

contributes to a large error in A1A2

.

Model (equation) Amplitude ratio A1A2 Γ [µeV] ∆Ω [meV] χ2

(3.6), blue 0.49± 0.09 47.0± 3.7 0.26± 0.01 0.67

(3.11), red dotted 0.41± 0.08 44.4± 4.0 0.26± 0.01 0.74

(3.26), green 0.74± 0.08 25.5± 8.2 0.27± 0.01 1.54

Tab. 3.2: Fit results of the different approximations of the simple modelcorresponding to Fig. 3.15.

In conclusion, the simple models can be used to determine the split in energy of the two

dispersions. However, the deduced amplitude ratio is very inaccurate. Since the simplified

models do not take into account any depolarizing effects arising from sample imperfections

and curvature of the dispersion surface, the obtained results for the linewidth Γ appear

unphysically large. Note that here it is assumed that the two excitations have the same

linewidth.

General model

In order to test the extended resolution model, the results from section 3.2 were fitted to

the data. Since there is only one maximum of the modulation of the echo amplitude within

the accessible τ -range, it would be very difficult to extract the linewidth Γ out of the data.

Therefore, Γ was fixed to zero, which is in good agreement with the result obtained from the

single dispersion fit with the extended resolution model. Note that it is again assumed that

both excitations have the same linewidth, since in order to extract linewidths out of inelastic

measurements on split modes, many more maxima of the echo amplitude would be needed

within the accessible τ -range to perform proper fits. From the TAS calibration scans using

the UB matrix formalism and the DAF model for the Nb dispersion parameters the energy

separation was calculated to ∆ΩUB = 0.294meV. The result of the fit of the extended model

is shown in Fig. 3.16 (left). The split in energy gives ∆Ω = 0.274± 0.026meV, which is in

very good agreement with the calculated value. The amplitude ratio gives A1A2

= 0.35±0.86.

The large error in the amplitude ratio is a consequence of the rather large error of the echo

amplitude in the minima. The amplitude ratio is extracted from the contrast of the mod-

ulation of the echo amplitude. However, the value obtained for the amplitude ratio is in

agreement with the TAS calibration scans within the error.

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3.4 Summary 67

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

τ [ps]

echo

am

plitu

de

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

τ [ps]ec

ho a

mpl

itude

single dispersiondouble dispersion

Fig. 3.16: Left: The general model discussed in section 3.2 fitted to theinelastic data. The fit is in very good agreement with the data and gives aseparation in energy of the modes of ∆Ω = 0.274 ± 0.026meV. Right: Fitresults of the single and the double dispersion using the extended model.The maxima of the modulation follow the decay of the single dispersionsignal.

For comparison the fit curves for the single dispersion and the split dispersion using the

extended resolution model are shown in Fig. 3.16 (right). The plot shows that the maxima

of the modulation of the echo amplitude follow the decay of the single mode as mentioned

in section 3.1.

The results show that the NRSE method can be used to resolve modes split in energy. How-

ever, the achievable resolution depends strongly on the dispersion of the excitation. In the

case of Nb and small q only larger splittings can be resolved, since the depolarization arising

from the dispersion parameters masks the larger modulation periods for smaller separations.

Both models presented here, the simplified and the general model, give comparable results

for the energy splitting of the two excitations. The general model considers depolarizing

resolution effects and describes the linewidth of the excitation properly in contrast to the

phenomenological function of the simplified model. However, since the computation time

for a fit of the general model is much longer, the simplified model is sufficient and preferable

for a fast estimation of the energy splitting. For a proper physical interpretation of the

splitting the generalized model should be chosen.

3.4 Summary

The resolution model discussed in chapter 2 accounts for a violation of the spin echo condi-

tions for inelastic scattering and is appropriate for high resolution spin echo measurements

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68 3 NRSE investigations on split modes

on mode doublets. A simplified model describing the signature of split modes, a modula-

tion of the echo amplitude, was discussed and the results obtained in the previous chapter

were used to develop a model for the most general case. In order to test the models a

unique setup was realized, allowing to tune a double crystal arrangement to any desired

splitting of elastic or inelastic TAS and NRSE signals. Echo amplitudes for this tunable

double dispersion setup were measured in Larmor diffraction geometry for two neighboring

Bragg peaks. The results demonstrate the echo modulation in spin echo length space to

be in good agreement with the predictions. Inelastic NRSE spectroscopy on an effectively

split dispersion clearly shows the modulation and agrees with the simplified and the general

model, indicating persistence of the modulation over the entire spin echo time range, probed

by the experiment.

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Chapter 4

NRSE line shape analysis

Another class of experiments, where the NRSE method with its high resolution opens up

entirely new perspectives, is dedicated to line shape analysis. In particular, this is relevant

for phenomena related to temperature dependent asymmetric line broadening, as has been

studied in this thesis. Here, the high resolution of NRSE allows to resolve deviations of the

scattering function from symmetric Lorentzian line shape by directly accessing correlations

in the time domain.

The phenomenological function [53]

S (ω) =1

π

1

1 +(ωΓ − α

(ωΓ

)2+ γ

(ωΓ

)3)2 (4.1)

takes the form of a modified Lorentzian and describes an asymmetric line shape. Here, the

usual argument is replaced by a polynomial, that includes an asymmetry term α(ωΓ

)2 and

a damping term γ(ωΓ

)3. A numerical example for the NRSE fingerprint of an asymmetric

line shape using equation (4.1) is shown in Fig. 4.1. The particular advantage of the NRSE

method is its direct access to the line shape in the time domain, since there is no convolu-

tion of the signal with the resolution function of the spectrometer. From the methodological

point of view the challenge of the experiments presented in this chapter is to explore new

territory for NRSE beyond standard linewidth measurements.

A quasi-particle, which is interacting with other thermally excited quasi-particles, will have

a limited lifetime. Conventional theory describes the loss of correlation in the time-domain

as an exponential decay, i.e. the scattering function is a Lorentzian in the energy domain.

Recent time-of-flight (ToF) experiments performed on OSIRIS, ISIS Facility, Rutherford

Appleton Laboratory, UK, on Cu(NO3)2·2.5D2O (copper nitrate), a model system for a 1-D

bond alternating Heisenberg chain, have established thermal development of an asymmetric

69

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70 4 NRSE line shape analysis

−0.2 −0.1 0 0.1 0.2 0.30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

∆ω [meV]

Sca

tterin

g fu

nctio

n

0 20 40 60 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ [ps]

echo

am

plitu

deFig. 4.1: Left: Model scattering function with an asymmetric line shape(black) using the phenomenological function (see equation (4.1)) and anormal Lorentzian scattering function (dashed blue). Right: Calculatedecho amplitude for the asymmetric line shape (black) and the Lorentzian(dashed blue).

continuum of scattering, which differs strongly from the proposed universality of Lorentzian-

type linewidths for one-dimensional quantum systems ([53] and references therein). Recent

theoretical work [54] relates this asymmetric line broadening to hard-core constraints and

quasi-particle interactions. This is proposed to apply to a broad range of quantum systems

[53].

Investigations on Sr3Cr2O8, a gapped 3-dimensional antiferromagnet, have shown that

magnons in this compound likewise develop a temperature dependent asymmetric line broad-

ening [65]. The results of [65] prove that the effect of asymmetric thermal line broadening is

not only confined to the special case of a highly gapped alternating chain like copper nitrate,

but is realized for the whole system of dimerized magnets, independent of dimensionality.

A major part of this thesis was to explore the potential of NRSE as a method to resolve such

effects. High resolution inelastic NRSE measurements were performed for both compounds,

Cu(NO3)2·2.5D2O and Sr3Cr2O8, at TRISP at the FRM-II, Garching. For the first time,

this effect was measured with NRSE.

4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O

4.1.1 Properties of Cu(NO3)2·2.5D2O

High field magnetization and inelastic neutron scattering measurements have been per-

formed in the past [56, 57, 58, 60, 55] and provide a consistent picture of Cu(NO3)2·2.5D2O

as a 1-D dimerized spin-12 antiferromagnet. The compound was shown to have a monoclinic

structure with space group I12/c1 [56, 57] and the low temperature (T = 3K) lattice pa-

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4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O 71

rameters a = 16.1Å, b = 4.9Å, c = 18.8Å and β = 92.9 [58]. Copper nitrate closely realizes

the alternating Heisenberg chain with the spin-1/2 moments of its Cu2+ ions (see Fig. 4.2).

Equivalent chains lie along the [12 ,12 ,

12 ] and [12 ,−1

2 ,12 ] directions and project onto the same

direction on the (h 0 l)-plane. The dimerization gives rise to a singlet ground state and the

elementary excitation is a triplet of spin-1 states. The low alternation parameter ensures an

energy separation between the spin-1 bound state and the 2-magnon continuum [59]. The

inter-dimer coupling allows the excitation to hop from site-to-site along the chain. For the

dominant exchange couplings (J = 0.443meV, J ′ = 0.101meV) the magnon bandwidth is

small compared to the gap and due to the smallness of the alternation ratio α = J ′

J ≈ 0.227,

there is a clear energy separation of about 0.5meV between single magnon excitations and

two magnon continuum even at higher temperatures [53].

chain direction

JJ´

d

ρ

Fig. 4.2: Alternating chain layout with the dimer coupling strength pro-vided by J and the intradimer coupling provided by J ′. ρ donates theseparation between dimer spins and the chain repeat vector is given by d

[62].

4.1.2 Sample deuteration and growth of single crystals

In order to increase the intensity of the scattered signal large single crystals are needed.

Since impurities can mask the effect of asymmetric line broadening in the spin echo signal,

the investigated single crystals must be of high purity. High quality single crystals of copper

nitrate were grown from powder samples of Cu(NO3)2·2.5H2O. In its original composition

Cu(NO3)2·2.5H2O copper nitrate produces a high incoherent cross section due to the large

amount of associated crystal water. To avoid a large incoherent contribution to the total

cross section hydrogen needs to be replaced with deuterium, as the incoherent cross section

of deuterium is comparably low: σincH = 80.26b and σinc

D = 2.05b [61]. Quantitative analysis

of the incoherent cross sections of Cu(NO3)2·2.5H2O, Cu(NO3)2·2.5D2O and a mixture of

both show that a deuteration level of at least 98.5% results in an incoherent cross section

comparable within a factor of 1.5 to the incoherent cross section of 100% deuterated copper

nitrate (see Appendix E).

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72 4 NRSE line shape analysis

The substitution of the two isotopes was performed in a distillation process similar to the

processes outlined by Xu [60] and Notbohm [62]. To achieve the desired deuteration ratio,

several distillation runs were necessary. In order to reduce the amount of heavy water used,

more distillation steps using less D2O compared to [62] were performed. In order to lower

the boiling point of the solution the distillation was performed under lower pressure, which

could be adjusted using a vacuum pump. The flask containing the solution was kept at

T=65C in a temperature controlled oil bath using a LakeShore temperature controller (see

Fig. 4.3).

vacuum

on on

argon

cooling water

vacuum

argon

vacuum

cooling water

Liebig condenser

oil bath

solution

Fig. 4.3: Sketch of the distillation setup. The solution (dark blue) iskept in a temperature controlled oil bath. During the vacuum pumpingprocess a mixture of normal and heavy water is removed from the solution,condensed in the Liebig cooler and collected in a second flask. To avoidair contact during the refill of heavy water the whole setup is flooded withargon.

140.99g of Cu(NO3)2·2.5H2O powder were dissolved in D2O of high purity (>99.9%). Ac-

cording to the literature [64], 140.99g of Cu(NO3)2·2.5H2O can be dissolved in 29.52g of

H2O and 32.68g D2O, respectively. Using 47.56g of D2O for the starting solution, ensured

the compound to be in solution throughout the whole distillation process. During the vac-

uum pumping process, a mixture of normal and heavy water was removed from the solution

and the condensate was weighed. The deuteration ratio of the solution should not change

significantly during this process, if at all, a small shift to a higher ratio is possible, since the

boiling point of H2O is 1.4C lower compared to D2O. After the pumping additional D2O

was added to the solution a to increase the D2O:H2O-ratio. During the refill, the setup was

flooded with argon to avoid air contact of the solution and thus, an exchange of deuterium

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4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O 73

Fig. 4.4: Highly deuterated copper nitrate single crystals grown usingenrichment and solution growth method. Large high quality crystals can begrown using this method. Left: 2×2×1cm3, m ≈ 4.1g. Right: 2×1×1cm3,m ≈ 2g.

and hydrogen. These steps were repeated several times (see Appendix E), resulting in a

calculated deuteration ratio of >99.38%.

Single crystals were then grown by cooling a saturated solution of Cu(NO3)2·2.5D2O. The

highly hygroscopic nature of copper nitrate stabilizes a second phase (Cu(NO3)2·6H2O)

below T = 26C makes the growth process rather complicated. To limit any substitution

with hydrogen the whole growth process was performed under argon atmosphere. The final

growth products were kept sealed under argon atmosphere. A slow cooling rate is essential

for the homogeneous growth of single crystals. The best parameters found for the growth

process here, were reducing the temperature from T = 85C to T = 40C in discrete steps

with a cooling rate dTdt = 0.05C

12min. During the cooling process needle shaped seed crystals

crystallize in the solution, providing the starting point of the single crystal growth. Apply-

ing this sequence, seven large single crystals with a mass of up to 4.1g each were grown (see

Fig. 4.4). The crystals tend to grow along their b-axis as has been previously found [60].

Subsequently all single crystals were orientated with X-ray Laue in the (h 0 l)-plane within

less than 1 deviation.

4.1.3 Inelastic NRSE measurements

Experiments were performed on Cu(NO3)2·2.5D2O using the thermal NRSE-TAS spectrom-

eter TRISP [66] at the FRM-II, Garching. TRISP was operated in a configuration with

scattering senses SM = −1, SS = −1 and SA = +1 and a fixed incident ki = 1.7Å−1. The

decay of the echo amplitude as a function of the spin echo time τ was investigated at the

minimum of the dispersion of the one-magnon mode at Q=(1 0 1) r.l.u., E = 0.385meV.

Here the intensity is highest and the slope of the dispersion is zero. Since the energy of

the excitation is small compared to the background TAS resolution, TAS energy scans were

performed at the base temperature T = 0.5K to determine the energy of the excitation. Due

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74 4 NRSE line shape analysis

to the fact that the polarization in spin-echo mode (precession fields of the spectrometer

arms are oriented antiparallel) was experimentally found to be higher compared to Larmor

mode (parallel orientation), TRISP was operated in spin-echo mode.

−0.2 −0.1 0 0.1 0.20

0.1

0.2

0.3

E [meV]

Inte

nsity

Scattering function T=0.5K

0 20 40 60 80 100 120

0.2

0.4

τ [ps]

echo

am

plitu

de

Spin echo signal T=0.5K

−0.2 −0.1 0 0.1 0.20

0.1

0.2

0.3

E [meV]

Inte

nsity

Scattering function T=2K

0 20 40 60 80 100 120

0.2

0.4

0.6

τ [ps]

echo

am

plitu

de

Spin echo signal T=2K

−0.2 −0.1 0 0.1 0.20

0.1

0.2

0.3

E [meV]

Inte

nsity

Scattering function T=2.5K

0 20 40 60 80 100 120

0.2

0.4

0.6

τ [ps]

echo

am

plitu

de

Spin echo signal T=2.5K

−0.2 −0.1 0 0.1 0.20

0.1

0.2

0.3

E [meV]

Inte

nsity

Scattering function T=3K

0 20 40 60 80

0.2

0.4

0.6

τ [ps]

echo

am

plitu

de

Spin echo signal T=3K

Fig. 4.5: Left: Echo amplitude as a function of spin echo time τ fordifferent temperatures (from the top: T = 0.5K, T = 2K, T = 2.5K andT = 3K) fitted with the phenomenological model (see equation (4.1)).Right: Line shape calculated from the fit parameters obtained from thespin echo data.

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4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O 75

The decay of the echo amplitude as a function of correlation time τ was measured for 4

temperatures (T = 0.5K, 2K, 2.5K and 3K) in the τ -range of 14.5ps to 112.6ps. The data

were corrected using a calibration obtained in the direct beam. Resolution effects arising

from the sample were corrected for. The results are shown in Fig. 4.5.

In order to fit the data, the phenomenological function described by equation (4.1) was

used. Note that here it is assumed that the scattering function does not vary with Q within

the TAS resolution function. While ToF methods need to take the convolution with the

resolution function explicitly into account, NRSE gives direct access to the line shape in the

time domain. Since the asymmetry vanishes for T → 0K [53], i.e. α, γ → 0, the line shape

becomes Lorentzian again. In this case the echo amplitude decays exponentially with τ .

Within the measured τ -range at the base temperature T = 0.5K no noticeable decay of

the echo amplitude was measured. This suggests, that the lifetime of the mode is at least

6.5± 52.6ns corresponding to a linewidth of Γ0.5K = 0.1± 0.8µeV. The sharp excitation at

base temperature indicates that no asymmetry is introduced by the shape of the dispersion

and resolution effects.

P0 0.404± 0.033

Γ0.5K 0.1± 0.8µeV

α0.5K 0

γ0.5K 0

Γ2K 0.009± 0.003meV

α2K 0.119± 0.087

γ2K 0.007± 0.008

Γ2.5K 0.006± 0.002meV

α2.5K 0.121± 0.026

γ2.5K 0.005± 0.002

Γ3K 0.022± 0.009meV

α3K 0.420± 0.166

γ3K 0.071± 0.064

Tab. 4.1: Fit results of the phenomenological model (see equation (4.1))applied to the spin echo data for different temperatures.

The numerical Fourier transform of the applied phenomenological model was fitted for

T = 2K, T = 2.5K and T = 3K and the results are shown in Fig. 4.5. For all 4 tem-

peratures, the parameter P (τ = 0) = P0 was used as a shared fit parameter, since P0

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76 4 NRSE line shape analysis

should not change with temperature for τ = 0. The fits are in good agreement with the

data and a clear deviation from an exponential decay can be seen. The fit parameters (see

Tab. 4.1) obtained from the spin echo data and equation (4.1) were used to determine

the temperature dependent asymmetric broadening of the excitation in energy space (see

Fig. 4.5). The value obtained for P0 of approximately 0.4 (see Tab. 4.1) is significantly

smaller than 1. For the chosen scattering plane an equal contribution of spin-flip (SF) and

non-spin-flip (NSF) scattering is expected. This can be seen by determining the scattering

amplitudes for SF and NSF scattering using the treatment of Moon, Riste and Koehler

[63], taking into account that the neutron polarization at the sample adopts all orientations

in the scattering plane for NRSE. Since the instrument was operated in the NSF sensitive

spin echo mode, the whole apparatus was detuned for the SF scattering case. This results

in a depolarized background, yielding a P0 below 1. Since SF and NSF are expected to

contribute by the same amount, an ideal P0 = 0.5 is expected. The value of P0 = 0.40(3)

obtained from the fit is lower. This is in agreement with an unpolarized background rate

of 1 countmin

.

The result of this experiment unambiguously proves an increasing asymmetry with increas-

ing temperature. A comparison of the fit parameters obtained from the spin echo data and

data from time-of-flight measurements at OSIRIS [53] is shown in Fig. 4.6. The results are

in good agreement, except the result for T = 2.5K.

0 1 2 3 4 5−5

0

5

10

15

20

25

30

35

T [K]

Γ [µ

eV]

0 1 2 3 4 5−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

T [K]

α

0 1 2 3 4 5−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

T [K]

γ

Fig. 4.6: Comparison of the fit parameters of the phenomenological modelobtained from the spin echo data (black) and the fit parameters from [53](red).

The NRSE data suggests a double peaked line shape rather than a continuous asymme-

try. The statistical error of the data is a limiting effect of the NRSE method. However,

the method does not depend on a systematic error arising from the convolution with the

resolution function as is present in time-of-flight measurements. The errors of the NRSE

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4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O 77

results increase drastically with increasing temperature since the statistical error of the spin

echo data worsens with increasing temperature and τ . This is due to the lower intensity of

the excitation at higher temperatures and a faster decay of the spin echo signal due to a

broadening of the linewidth. In addition, less data points are available for the fit compared

to the ToF data. In order to reduce the errors and to get a better fit of the profile of the

line shape, better statistics of the spin echo data are needed.

As a proof of principle it could be shown that temperature dependent asymmetric line

broadening can be determined using high resolution NRSE. This is the first time this effect

was measured with NRSE. The performed measurements explored new territory for the

NRSE method and enhanced the potential of the NRSE method beyond standard linewidth

measurements. The particular advantage of the NRSE method is its direct access to the

line shape, since there is no convolution of the signal with the resolution function of the

background spectrometer.

The results presented above, show that NRSE can be used to determine the asymmetric

broadening of an excitation with increasing temperature by measuring the decay of the echo

amplitude as a function of the spin echo time τ . Using such measurements, an asymmetry

can be determined. However, it cannot be extracted if the asymmetry develops towards

lower or higher energies relative to the excitation energy. If the mean energy

Em =

∫ωS(ω)dω∫S(ω)dω

(4.2)

of an excitation is shifted due to an asymmetric broadening of the line shape, the phase of the

spin echo signal for a fixed τ is shifted accordingly. In order to determine the orientation

of the asymmetry, phase sensitive measurements were performed at the minimum of the

dispersion Q =(1 0 1) r.l.u., E = 0.385meV for a temperature range of T = 0.5 − 3K.

The accuracy of the energy shift extracted from the phase sensitive measurements increases

when they are performed at a larger τ . Simultaneously the echo amplitude decreases with

increasing τ increasing the error of the phase. Therefore, τ = 47.23ps was chosen as a

good compromise between the accuracy of the energy shift and the magnitude of the echo

amplitude. The phase shifts relative to the phase at base temperature were converted to

energy shifts using

∆E(T ) = ~φ(T )− φ(T = 0.5K)

τ. (4.3)

Here the phase φ is expressed in radians. The results are shown in Fig. 4.7. Instead of the

expected shift to higher energies according to the results from [53] the data clearly show

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78 4 NRSE line shape analysis

a shift to smaller energies. The separation in energy between the magnetic (1 0 1) Bragg

peak and the excitation is large and a contribution of the nuclear (2 0 2) Bragg peak is not

expected due to a velocity selector installed suppressing second order scattering efficiently

at TRISP. Hence, a priori no contamination arising from the Bragg peak is expected since

the intensity of the Bragg peak is reduced by the instrument resolution. Furthermore, the

instrument is tuned to the excitation and therefore detuned for the Bragg peak. Thus,

the remaining contribution from the Bragg peak is expected to be completely depolarized.

However, the results of a more detailed quantitative analysis show, that it is necessary to

consider a contamination from the polarized fraction of the magnetic (1 0 1) Bragg peak in

the present case. As the contribution of this Bragg peak is temperature independent in the

investigated range and the amplitude of the one-magnon excitation decreases, the center of

mass is shifted to smaller energies. This is in very good agreement with the experimental

results.

0.5 1 1.5 2 2.5 3−14

−12

−10

−8

−6

−4

−2

0

2

T[K]

∆ E

[µ e

V]

Fig. 4.7: Temperature dependent data from phase sensitive measure-ments. Equation (4.5) considering a contribution of the Bragg peak isfitted to the data. A polarized fraction of 1.9(2)% of the Bragg peak con-taminating the signal is obtained.

The contribution of the Bragg peak can be estimated by the resolution function of the TAS

at the excitation energy and the intensity of the Bragg peak IBragg,calib from calibration

measurements:

IBragg = e−(

EBragg−Eexc)2

2σ2TAS · IBragg,calib = e

− (0−0.385meV)2

2(0.126meV)2 · IBragg,calib = 538.68counts

sec(4.4)

with σTAS = 0.126meV obtained from TAS calibration measurements. The function Iexc (T )

giving the intensity of the one-magnon excitation as a function of temperature T was ob-

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4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O 79

tained for τ = 47.23ps from the temperature dependent measurements described above.

Using (4.1) and the results in Tab. 4.1 gives the function C (T ), which describes the mean

energy of the asymmetric excitation.

Using these functions and considering the contribution of the Bragg peak, the polarized

fraction f of the Bragg peak, contaminating the phase sensitive measurements, the energy

shift, with respect to ∆E (T = 0.5K) at base temperature, can be fitted by

∆E (T ) =

(1− IBragg · f

IBragg · f + Iexc (T )

)· (C (T ) + 0.385meV)−∆E (T = 0.5K) . (4.5)

The fit of equation (4.5) to the data is shown in Fig. 4.7 and gives a polarized fraction of

f = 1.9± 0.2% of the Bragg peak.

An independent approach to estimate the polarized fraction uses the second order expansion

of the spin echo phase (see equation (2.42). ∆kf is substituted by ∆kf + δkf where ∆kf

is still the distribution of the kf and δkf is the change of kf due to the detuning. An

excitation energy of E = 0.385meV and an incident ki = 1.7Å−1 results in kf = 1.644Å.

Therefore, to change to kf,Bragg = 1.7Å−1 of the elastic case a δkf = 0.056Å−1 is needed.

Substituting ∆kf → ∆kf + δkf into equation (2.42) the last term, A2

(kF ·nf)3 (∆kf · nf )

2, is

replaced by 3 new terms. The term proportional to δk2f results in a constant phase, which

can be neglected. The term proportional to ∆k2f stays the same as before and the only

relevant new term is A2

|kF |3 (2∆kf · δkf )2. Here it is taken into account that the slope of the

dispersion at the minimum is zero and therefore kF ·nf = |kF |. Assuming ∆kf

kf= 1% leads

to an

FWHM∆kf ≈ 0.01 · kF ≈ 0.01644Å−1 and σ∆kf ≈ 0.00698Å−1. (4.6)

The polarization at the Bragg peak is then estimated by

P =

∣∣∣∣∣∣∣

∫e−

∆k2f

2σ2∆kf e

iA22k3

F

2δkf∆kfd∆kf

∣∣∣∣∣∣∣, (4.7)

resulting in a remaining polarization of 1.75%, which is in good agreement with the result

from the fit shown in Fig. 4.7.

Thus, it can be convincingly concluded that for Q=(1 0 1) r.l.u. a relative shift of the mean

energy of the excitation due to an asymmetric broadening cannot be measured with NRSE,

given the energy resolution of TRISP at ki = 1.7Å−1. The shift is masked due to a polarized

fraction of the (1 0 1) Bragg peak contaminating the signal.

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80 4 NRSE line shape analysis

In order to investigate the shift of the mean energy of the excitation as a function of tem-

perature, supplementary phase sensitive measurements were performed at the minimum of

the dispersion Q =(1.11 0 0.855) r.l.u., E = 0.385meV, avoiding a contamination by the

(1 0 1) Bragg peak. This was experimentally confirmed by TAS energy scans. The phase

sensitive measurements were performed in a temperature range from T = 0.5k to T = 3K

at two different spin echo times τ = 24.01ps and τ = 47.57ps each. The phase shift ∆φ

relative to the phase at base temperature T = 0.5K extracted from the data was converted

using equation (4.3).

0 1 2 3 4−10

0

10

20

30

40

50

60

70

T [K]

Ene

rgy

shift

∆ E

[µeV

]

0 1 2 3 4−30

−20

−10

0

10

20

30

T [K]

Ene

rgy

shift

∆ E

[µeV

]

Fig. 4.8: Temperature dependent data (black) from phase sensitive mea-surements for spin echo time τ = 24.01ps (left) and τ = 47.57ps (right).The data is compared to calculated mean energies using equation (4.1)and (4.2) and the fit parameters displayed in Tab. 4.1 (blue). Energyshifts calculated from the cosine Fourier transforms of the correspondingasymmetric scattering functions (see Fig. 4.10) are displayed in red.

The results for both data sets are shown in Fig. 4.8. The experimental energy shift (black

data points) is compared to calculated mean energies using equations (4.1) and (4.2) and

the fit parameters displayed in Tab. 4.1 (blue data points). Both data sets display a clear

deviation from the calculated values. Due to the absence of a second strong mode, e.g. a

Bragg peak, the deviations cannot be described by a contamination. Therefore, the data

suggest that the phase shift ∆φ is not directly proportional to the energy shift ∆E in the

case of an asymmetric line shape. Due to the difference between both data sets, the mea-

surements indicate that the measured phase shift is a non-linear function of the spin echo

time τ .

As a numerical example Fig. 4.9 displays the cosine Fourier transform (right) of four scat-

tering functions with a Lorentzian shape (left). The functions are successively shifted by

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4.1 Asymmetric line shape of excitations in Cu(NO3)2·2.5D2O 81

0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

E [meV]

Inte

nsity

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

τ [ps]

echo

am

plitu

de

Fig. 4.9: Left: Lorentzian shaped scattering function successively shiftedby 10µeV. The functions are centered at: Ec = 0.385meV (black), Ec =0.395meV (blue), Ec = 0.405meV (green) and Ec = 0.415meV (red.).Right: Corresponding numerical calculation of the cosine Fourier trans-form.

10µeV. A comparison of the phases of the different cosine Fourier transforms shows that the

phase shift for a fixed τ is a linear function of the energy shift, i.e. the phase is a linear

function of the spin echo time τ .

Results for the fit parameters of the temperature dependent asymmetric scattering func-

tions are shown in Tab. 4.1 and the corresponding numerically calculated cosine Fourier

transforms are shown in Fig. 4.10. For the minimum close to τ = 24.01ps, a phase shift

between the signals is clearly visible. The phase shift increases with increasing temperature

as expected from the calculations of the mean energy of the asymmetric line shapes. The

energy shift calculated from the cosine Fourier transform follows the trend of the shift of

the mean energy as displayed in Fig. 4.8 (left). The experimental energy shifts disagree

with the theoretical predictions from both the calculated mean energy and the energy shift

extracted from the cosine Fourier transforms.

0.2 0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

0.8

1

E [meV]

Inte

nsity

T=0.5KT=2KT=2.5KT=3K

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

τ [ps]

echo

am

plitu

de

T=0.5KT=2KT=2.5KT=3K

Fig. 4.10: Left: asymmetric scattering function calculated from the fitparameters given in Tab. 4.1. Right: Corresponding numerical calculationof the cosine Fourier transform.

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82 4 NRSE line shape analysis

In contrast for the minimum close to τ = 47.57ps almost no phase shift between the signals

is visible. The energy shift calculated from the signal according to equation (4.3) is shown

in Fig. 4.8 (right). This is in agreement with the experimental results for all temperatures

except at T = 3K, where the error is already quite large. However, it strongly disagrees with

the predictions of the calculations of the mean energy (see equation (4.2)). The difference

between the two data sets clearly demonstrates that the phase is a non-linear function of the

spin echo time τ for a line shape, differing from a Lorentzian shape. In such cases the cosine

Fourier transform no longer follows a simple exponential decay. Thus, additional terms,

which depend strongly on the line shape of the excitation, need to be taken into account

carefully for the data analysis.

As a result it could be shown that the shift of the mean energy of an excitation due to an

asymmetric line broadening cannot be measured in a straight forward manner with phase

sensitive NRSE measurements, since the phase seems to become a non-linear function of

the spin echo time τ . These results are a counter example to the assertions of a linear

dependence between phase shift and spin echo time τ . Thus, a careful treatment of results

is necessary, since these effects could easily lead to wrong data interpretation. This result

can be used to determine a deviation of the scattering function from Lorentzian shape.

4.2 Asymmetric line shape of excitations in Sr3Cr2O8

The 3-D gapped quantum spin dimer Sr3Cr2O8 has been extensively characterized recently

[65, 67, 68]. Standard TAS experiments performed at V2/FLEX, HZB, Berlin, and at the

cold triple axis spectrometer TASP, Paul-Scheerer Institut (PSI), Switzerland, showed that

the effect of temperature dependent asymmetric line broadening is also present in this system

[65, 67]. Using the high resolution NRSE technique, the TAS results were confirmed and it

was shown that the temperature development of the line shape in Sr3Cr2O8 does not have

a Lorentzian line shape.

4.2.1 Properties of Sr3Cr2O8

DC susceptibility, high field magnetization and inelastic neutron scattering measurements

have been performed by Quintero-Castro et al. [65, 67, 68] and provide a consistent picture

of Sr3Cr2O8 as a dimerized spin-12 antiferromagnet. Sr3Cr2O8 consists of a lattice of spin-12Cr5+ ions, which form hexagonal bilayers at room temperature and are paired into dimers

by the dominant antiferromagnetic intrabilayer coupling J0 [70] (see Fig.4.11). The dimers

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4.2 Asymmetric line shape of excitations in Sr3Cr2O8 83

J0

J2'

J2''

J2'''

J4'

m

m

m

J1'

J1''

J1''

J3' J

3''

J3'''

J0 5.55(1) meV

J1' -0.04(1) meV

J1'' 0.25(1) meV

J2'- J3' 0.75(1) meV

J2''- J3'' -0.54(1) meV

J2'''- J3''' -0.12(1) meV

J4' 0.06(2) meV

J4'' -0.05(1) meV

Fig. 4.11: Lowtemperature (monoclinic) crystal structure of Sr3Cr2O8

showing the magnetic Cr5+ ions only. The exchange interactions are labeledon the diagram and listed in the table [67].

are coupled three-dimensionally by frustrated interdimer interactions. Sr3Cr2O8 undergoes

a structural Jahn-Teller distortion below TJT = 285K [71]. This lowers the symmetry from

hexagonal (R3m) to monoclinic (C2/c), which is stable for T < 120K [72, 73]. The orbital or-

der and lifting of the frustration gives rise to spatially anisotropic exchange interactions. The

hexagonal lattice parameters at room temperature are a = b = 5.57Å and c = 20.17Å, while

the monoclinic lattice parameters at T = 1.6K are a = 9.66Å, b = 5.5437Å, c = 13.7882Å

and β = 103.66.

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0 M

(-1/2,1/2,-1)h

(1/2,1/2,0)h

(1,1,-3/2)h (-1/2,1/2,0)

h(1,1,0)

h (0,0,0)h

Ener

gy (m

eV)

Twin 1 Twin 2 Twin 3Flex Data

M'

Fig. 4.12: Dispersion relation of Sr3Cr2O8 (from [67]). The blue, ma-genta and black lines correspond to the fitted dispersion relations of threemonoclinic twins. Green points are fitted centers of peaks measured atV2/FLEX at HZB.

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84 4 NRSE line shape analysis

The neutron scattering experiments [67] reveal three gapped and dispersive singlet to triplet

modes arising from the three twinned domains formed below the transition (see Fig. 4.12).

The exchange constants were extracted using Random-Phase-Approximation [67]. The in-

tradimer exchange constant is identified as the intrabilayer interaction J0, which has a value

of 5.55meV as found from both, susceptibility and inelastic neutron scattering measurements

[67]. This dimerization gives rise to a singlet ground state and gapped one-magnon exci-

tations. Additionally there are significant interdimer interactions, which allow the dimer

excitations to hop and develop dispersion. The dispersion of the one-magnon mode pro-

duces a bandwidth, extending between the gap energy of 3.5meV and the maximum value

of 7.0meV, as found by magnetization and inelastic neutron scattering measurements [67].

4.2.2 Inelastic NRSE measurements

The experiments on Sr3Cr2O8 were performed at the thermal NRSE-TAS spectrometer

TRISP [66] at the FRM-II, Garching, using a single crystal grown at the HZB, Berlin [69].

TRISP was operated in a configuration with a fixed kf = 2.51Å−1 and scattering senses

SM = −1, SS = −1 and SA = +1. The investigations were done on the lowest energy

mode at the center of the Brillouin zone at Q=(0.5 0.5 3) r.l.u. where the most favorable

conditions are met. The intensity is highest, the slope of the dispersion is zero and the sepa-

ration between the modes of the three existing domains is largest (see Fig. 4.12). Since the

energy of the excitation shifts to higher values with increasing temperature [65], the back-

ground TAS parameters and the NRSE parameters were adjusted to their nominal values for

the excitation energy, appropriate for each temperature. Therefore, TAS energy scans were

performed at T = 0.5K and 15K. For intermediate temperatures TAS-data available from

measurements at V2/FLEX, HZB, Berlin, [65] was interpolated. Since the experimentally

determined polarization in Larmor mode was higher than in spin-echo mode TRISP was

operated in Larmor mode.

The decay of the echo amplitude as a function of correlation time τ was measured for 4

temperatures (T = 0.5K, 10K, 15K and 20K). The data were corrected, using a direct beam

calibration and resolution effects arising from the sample were corrected for. However, the

corrections, which are mainly due to curvature effects, are small (≈ 3%). The results are

shown in Fig. 4.13. With increasing temperature the depolarization increases for higher

values of τ . As in the previous section the phenomenological model given in equation 4.1

was fitted to the data. Again it is assumed that the scattering function does not vary in Q

within the TAS resolution ellipsoid. The asymmetry vanishes for T → 0K [53], i.e. α, γ → 0,

and the line shape becomes Lorentzian again. For this case the spin echo signal is an expo-

nential decay of the echo amplitude. Due to the statistical error of the data at T = 0.5K a

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4.2 Asymmetric line shape of excitations in Sr3Cr2O8 85

clear discrimination between an exponential decay and a signal corresponding to an already

developed asymmetry of the line shape is not possible. However, the TAS measurements

performed by Quintero-Castro et al. [65] suggest a symmetric Lorentzian shape of the scat-

tering function for T ≤ 1.6K. Thus, the fit parameters α and γ were fixed to zero for the

base temperature T = 0.5K. A clear determination of the line shape at T = 0.5K would

require further measurements and better statistics.

−0.5 0 0.50

0.2

0.4

E [meV]

Inte

nsity

Scattering function T=0.5K

0 10 20 30 40 500

0.2

0.4

τ [ps]

echo

am

plitu

de

Spin echo signal T=0.5K

−0.5 0 0.50

0.2

0.4

E [meV]

Inte

nsity

Scattering function T=10K

0 10 20 30 40 500

0.2

0.4

0.6

τ [ps]

echo

am

plitu

de

Spin echo signal T=10K

−0.5 0 0.50

0.2

0.4

E [meV]

Inte

nsity

Scattering function T=15K

0 10 20 30 40 500

0.2

0.4

0.6

τ [ps]

echo

am

plitu

de

Spin echo signal T=15K

−0.5 0 0.50

0.2

0.4

E [meV]

Inte

nsity

Scattering function T=20K

0 10 20 30 40 500

0.2

0.4

0.6

τ [ps]

echo

am

plitu

de

Spin echo signal T=20K

Fig. 4.13: Left: Echo amplitude as a function of spin echo time τ fordifferent temperatures (from the top: T = 0.5K, T = 10K, T = 15K andT = 20K) fitted with the phenomenological model (see equation (4.1)).Right: Line shape of the excitation in the energy domain calculated fromthe fit parameters obtained from the spin echo data.

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86 4 NRSE line shape analysis

The results fitting the Fourier transform of the model function given by equation 4.1 for the

temperatures T = 10K, T = 15K and T = 20K are shown in Fig. 4.13. The fits are in good

agreement with the data. The parameter P (τ = 0) = P0 was used as a shared fit parameter,

assuming that the polarization at τ = 0ps does not depend on sample temperature. The

obtained value for P0 ≈ 0.4 (see Tab. 4.2) is significantly smaller than 1. For the chosen

experiment parameters equal amounts of spin-flip (SF) and non-spin-flip (NSF) scattering

are expected. Since the instrument has been operated in the SF sensitive Larmor mode,

the whole apparatus has been detuned for any signal arising from NSF scattering. This

results in a depolarized background yielding a P0 below 1. Since SF and NSF scattering

are expected to contribute by the same amount, an ideal P0 = 0.5 is expected. The value

of P0 = 0.426(45) obtained from the fit is lower and is in agreement with an unpolarized

background rate of 1 countmin

.

The fit parameters are listed in Tab. 4.2. Using the fit results and the phenomenological

model, the scattering function and thus, the temperature dependent asymmetric line broad-

ening of the excitation in the energy domain can be calculated (see Fig. 4.13). The results

of this experiment suggest an increasing asymmetry of the line shape of the excitation with

increasing temperature.

P0 0.426± 0.045

Γ0.5K 0.027± 0.006meV

α0.5K 0

γ0.5K 0

Γ10K 0.028± 0.011meV

α10K 0.266± 0.089

γ10K 0.028± 0.020

Γ15K 0.038± 0.014meV

α15K 0.265± 0.116

γ15K 0.030± 0.025

Γ20K 0.058± 0.023meV

α20K 0.290± 0.14

γ20K 0.032± 0.032

Tab. 4.2: Fit results of the phenomenological model (4.1) applied to thespin echo data for different temperatures.

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4.2 Asymmetric line shape of excitations in Sr3Cr2O8 87

A comparison of the fit parameters from the spin echo data and TAS data [65] is shown

in Fig. 4.14. The increase of the linewidth Γ of the excitation for increasing temperature

obtained from the spin echo data is less steep compared to the TAS data. The errors of the

linewidths for the different temperatures are of the same order of magnitude. The difference

in the results for the asymmetry parameters α and γ is larger. The larger errors for the

NRSE asymmetry parameters arise from the fact that there are less data points available

for the fit compared to the TAS data. To reduce the errors and to get a better fit of the

profile of the line shape, better statistics of the spin echo data would be crucial.

0 10 20 300

50

100

150

T [K]

Γ [µ

eV]

0 10 20 300

0.1

0.2

0.3

0.4

0.5

T [K]

α

0 10 20 30

0

0.02

0.04

0.06

0.08

T [K]

γ

Fig. 4.14: Comparison of the fit parameters of the phenomenologicalmodel obtained from the spin echo data (black) and the fit parametersfrom [65] (red).

In contrast to copper nitrate the energy of the low energy mode of Sr3Cr2O8 shifts towards

higher energies with increasing temperature [68]. Since this effect masks the small shift

of the center of mass of the scattering function due to an asymmetric line broadening,

phase sensitive measurements can be used to determine the shift of the mode. In order to

check and extend previous measurements on the temperature dependent energy shift of the

mode, additional phase sensitive measurements were performed with NRSE spectroscopy

at Q=(0.5 0.5 3) r.l.u. at an energy of E = 3.6meV and τ = 9.5ps for a temperature

range of T = 0.5 − 20K. The results are shown in Fig. 4.15 and compared with TAS data

taken from [65]. Both data sets are in good agreement. With increasing temperature the

excitation energy is shifted towards higher energies. The dimers become thermally excited

as the temperature is increased. This interferes with the intersite interactions resulting in

a shift of the dispersion towards the intradimer exchange energy J0 = 5.55meV [65]. The

results of the phase sensitive NRSE measurements confirm the TAS measurements and both

data sets are in good agreement with the predictions of the Random Phase Approximation

calculations carried out in [68].

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88 4 NRSE line shape analysis

0 10 20 30 40 50 60−0.2

0

0.2

0.4

0.6

0.8

1

1.2

T [K]

∆ E

[meV

]

NRSETAS

Fig. 4.15: Energy shift ∆E of the low energy excitation at (0.5 0.5 3) fordifferent temperatures. The NRSE data and the TAS data [68] are in goodagreement.

4.3 Summary

Two systems, the 1-D Cu(NO3)2·2.5D2O and the 3-D Sr3Cr2O8, in which the effect of tem-

perature dependent asymmetric line broadening has been observed were investigated using

the NRSE technique on the thermal triple axis spectrometer TRISP. For the measurements

on Cu(NO3)2·2.5D2O large single crystals were grown from D2O-enriched solution. For the

experiment with Sr3Cr2O8 a single crystal grown at the HZB was used.

As a proof of principle it could be shown that the high resolution method of NRSE allows to

resolve line shapes, which are not Lorentzian type. Such line shape analysis is relevant for

the phenomenon of temperature dependent asymmetric line broadening as has been first ob-

served in gapped quantum magnets. In this thesis, this effect was measured for the first time

with NRSE in the time-domain. The method of NRSE gives direct access to the scattering

function without a convolution with the resolution function of the background spectrometer.

The statistical error of the NRSE data is a limiting effect, however the method does not

depend on systematic errors arising from the convolution with the resolution function as

it is the case for time-of-flight measurements. As an important result it could be shown,

that the phase shift due to an asymmetric line broadening is a non-linear function of the

spin echo time τ . This effect can also be used to determine a line shape deviation from

Lorentzian shape.

From the methodological point of view the challenge of these experiments was to explore

new territory for NRSE beyond standard linewidth measurements. It could be shown suc-

cessfully that NRSE spectroscopy can be used to reveal the effect of temperature dependent

asymmetric line broadening.

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Chapter 5

Upgrade of the NRSE option at

FLEXX

The cold triple axis spectrometer at the BER II reactor at HZB, Berlin, [74] has recently

been upgraded and rechristened FLEXX [75]. In order to benefit from the enhanced TAS

parameters the NRSE option of the spectrometer [76, 77, 78] was upgraded, which was a

major part of this thesis.

The TAS spectrometer was moved to the end of the rebuilt NL1B guide, which extends the

accessible range of wavevector transfer. The guide section has been upgraded with m = 3

neutron supermirrors increasing the neutron flux at the instrument. An elliptical guide

section focuses neutrons onto a virtual source, which is imaged on the monochromator. A

wavelength band is selected by a velocity selector obviating the need for a second order

filter and reducing the background signal. Due to the change of the beam geometry, the

former vertically focusing monochromator has been replaced by a larger double focusing

PG002 monochromator, increasing the neutron flux at the sample position. A major goal

of the upgrade was to improve the polarized neutron capabilities [75] of the spectrometer.

A vertical guide changer has been incorporated and allows to replace a part of the regular

supermirror guide with an S-bender polarizer. Subsequently a guide field is provided for the

remaining neutron flight path.

This chapter deals with the upgrade of the NRSE option of FLEXX. The major goal was

to increase the performance and the accessible parameter range of the combined NRSE-

TAS instrument. In order to feed the larger beam cross section without losses from the

monochromator to the sample and from there to the analyzer, basic components of the

NRSE option were rebuilt. New NRSE bootstrap coils, allowing for larger beam cross

sections and larger coil tilt angles, were manufactured and tested in collaboration with the

89

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90 5 Upgrade of the NRSE option at FLEXX

Max Planck Institute for Solid State Research, Stuttgart. The spectrometer arms were

redesigned making the instrument more compact and extending the accessible range in

scattering angle. New, more compact mu-metal shielding provides a better shielding of the

field free region between the NRSE coils. In contrast to the previously used spectrometer

arms, all mechanical and electrical components (e.g. goniometers) are relocated beneath

the shielding boxes to reduce scattering fields inside the field free zone. The coupling coils,

providing the non-adiabatic transition in and out of the field free regions, were redesigned

in order to increase the beam cross section. Finally, the results of successful first calibration

measurements of the NRSE option available at FLEXX are presented at the end of this

chapter.

5.1 Bootstrap coils

The centerpiece of a NRSE spectrometer are the π-coils described in section 2.2.1. Each

π-coils consist of two coils, a B0 coil providing a static magnetic field perpendicular to the

scattering plane and an RF coil creating an oscillating field within the scattering plane. For

the spin echo option of FLEXX new π-coils were manufactured at the FRM-II reactor in

collaboration with the Max Planck Institute For Solid State Research, Stuttgart. The π-coils

were assembled in pairs to bootstrap coils (see Fig. 5.1), as introduced in section 2.2.2. In a

bootstrap coil the static fields of the two B0 coils are oriented in opposite directions, which

allows to minimize stray fields outside of the coils in the field free regions. The manufacturing

of the components of the bootstrap coils is described in the following subsections.

Fig. 5.1: Design of one bootstrap coil. The two B0 coils (light blue) aremounted between cooling plates (light green) [85].

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5.1 Bootstrap coils 91

5.1.1 B0 coils

The static B0 fields are created by using a vertical air coil wound on an aluminum body (see

Fig. 5.2). The required precision of the coils is very high. Following the estimation in [12]

the required accuracy of the coils can be calculated. The Larmor precession angle after one

coil is given by

Φ = γB0lmλ

2π~(5.1)

where B0 is the applied magnetic field, l is the mean coil thickness and λ is the neutron

wavelength. If ∆Φ1 is the mean deviation from the Larmor precession angle after one coil,

then the mean deviation after M coils is

∆ΦM =√M ·∆Φ1. (5.2)

Therefore, for a polarization of P = cos(∆Φ8) > 95% after 8 π-coils, it follows that

∆Φ8 < 18 and ∆Φ1 < 6.5. Assuming a maximum magnetic field of B0,max = 165G,

a wavelength of λ = 4Å and 8 π-coils with a thickness of l = 42mm each, the relative

deviation of the phase angle should not exceed

∆Φ1

Φ1= 1 · 10−3 =

√(∆B0

B0

)2

+

(∆l

l

)2

. (5.3)

The relative deviations of the magnetic field (∆B0B0

) and of the coil thickness (∆ll ) should

contribute equally with 1·10−3√2

= 7.1 · 10−4. Assuming B0 = 165G and l = 42mm yields for

the tolerable deviations of the magnetic field and the coil thickness:

∆B0 < 115mG ∆l < 30µm (5.4)

A coil wound with wire would not satisfy the required surface flatness. Therefore, aluminum

band is used as lead. The band has a purity of about 99.9% and for the purpose of electrical

insulation it has an anodized layer of about 5µm. Aluminum has a high neutron transmis-

sion and a high conductivity. During the anodization of the aluminum band crystal water is

embedded in the surface. Since this effect causes small angle scattering [12, 79] the crystal

water is replaced with D2O. This is realized by heating the band in a D2O atmosphere in a

pressure oven at a pressure of p = 5bar. Such treatment reduces the small angle scattering

and improves the transmission of the coils [12, 79].

The aluminum body(253× 204× 42mm3

)(see Fig. 5.2) of the coil consists of two anodized

parts with windows (inner part: 64× 52mm2, outer part: 140× 52mm2) at the beam area.

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92 5 Upgrade of the NRSE option at FLEXX

Fig. 5.2: Left: Sketch of the aluminum coil body of the B0 coil [85]. Right:B0 coil wound with two layers of anodized aluminum band.

The accessible beam cross section of 56 × 52mm2 is defined by the inner cooling plate

between two B0-coils (see section 5.1.2). Note that the accessible beam cross section of

the previously used coils was 50 × 32mm2 and thus the maximum beam cross section is

enhanced by a factor of ≈ 1.6. Using a beam cross section of 56 × 52mm2 the openings

of the new B0-coils allow for a maximum coil tilt angle for the whole bootstrap coil of 45

before reducing the beam cross section by more than a factor of 0.5 (see section 5.3.4).

To improve the insulation between the coil body and the aluminum band, the whole coil

body is covered with Kapton foil. The winding process has to be done very careful. Since

the aluminum band is not very flexible, it has to be wound under very high tension. Forces

parallel to the surface could bend the band and lead to gaps between the windings resulting

in field inhomogeneities.

For the winding the body was mounted on a lathe using dedicated adapter parts. The

tension of the aluminum band was realized by a guide consisting of two Teflon blocks pressed

together with several springs. The whole guidance system was mounted on a rotary table

to avoid forces, which could twist the band. During the winding one needs to take care that

the edges of the band do not yaw against each other to avoid short circuits between the

windings. At the same time, neighboring windings should lie closely to each other to get a

homogeneous magnetic field. Due to its width, the band has a slope of about 1 with respect

to the coil axis. To avoid cross components of the magnetic field, each coil has two layers

of windings with opposite slopes. Therefore, the cross components of the different layers

compensate. Each layer is fixed with a thin layer of two-component adhesive (Araldit 2015).

The adhesive was hardened for one hour in an oven at T = 120C, while the coil was pressed

between two steel plates with a very level surface. In the beam area the flatness of the coil

surface is of highest importance. Since the aluminum band does not lie on the coil body

in this region, the band was supported from the inside during the winding process with

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5.1 Bootstrap coils 93

even plates, which fit into the coil windows. The support parts were removed afterwards.

Following this process the required flatness of the surface was reached as measured using a

straight-edge.

Static fields

Measurements of the static magnetic fields B0 yielded:

B0[G] = 3.08 · I0[A]. (5.5)

The vertical component of the B0 field along the coil axis is shown in Fig. 5.3. If both

B0 coils of a bootstrap pair are powered (blue line) with the fields oriented in opposite

directions, the return field of one coil is fed back through the other coil. Therefore, the field

profile approaches the profile of a long coil. The total field becomes more homogeneous and

slightly stronger compared to a single B0 coil (black line). The magnitude of the magnetic

field at the edges of the beam area (dashed red line) is slightly smaller (∆B0 = −30mG)

compared to the center of the beam area (red line).

−20 −18 −16 −14 −12 −10 −8 −6 −40

2

4

6

8

10

12

14

16

distance [cm]

B0 [G

]

Fig. 5.3: Vertical component of the B0 field along the coil axis in a boot-strap pair. If two coils are powered (blue line) the field is more homoge-neous and slightly enhanced as compared to a single powered coil (blackline). The center of the beam area is marked red while the edges of thebeam area are marked dashed red.

In order to reduce the stray fields of the bootstrap coil mu-metal plates with a thickness

of 3mm are screwed to the bottom and to the top of the bootstrap coil to improve the closure

of the magnetic circuit. At the maximum field strength of B0,max = 165G the magnetic flux

density within the mu-metal plates is approximately 42mm3mm

·165G = 2.3kG, which is far from

saturation. To decrease the stray fields further a second layer of mu-metal is mounted on

the bootstrap coil. This layer consists of 0.5mm thick mu-metal foil attached to the front

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94 5 Upgrade of the NRSE option at FLEXX

faces and u-shaped, 1mm thick covers on top and bottom of the coil. The second layer is

about 1cm separated from the strong magnetic flux of the coils within the inner mu-metal

plates. It has been shown that a second layer reduces the stray fields significantly by a

factor of ≈ 14 [12].

5.1.2 Cooling circuit

The heat generated by the coils at higher fields needs to be dissipated. Therefore, cooling

plates with flowing water and air are installed in front, in between and behind the coils.

The cooling plates consist of two aluminum parts (outer plates: 10mm and 2mm thick,

inner plates: 4mm and 2mm thick), in which channels for the cooling water and the air

flow are milled into. The two parts are anodized and bonded together with heat sealable

film (Collano 22.100 [80]). The cooling plates are glued to the coils with a heat conducting

adhesive (Gap Filler 1000 [81]) with a heat conductivity of λ = 1 WKm

.

The resistance of one B0 coil is R = 0.12Ω. Using P = I2R and equation (5.5) the thermal

dissipation is:

P [W] = 1.26 · 10−2B20 [G], (5.6)

which is P = 343W for B0,max = 165G. The thermal dissipation losses are transferred by

the cooling water. The cooling plates cover the coil with an area of 800cm2 where the heat

flow of about q = 0.43 Wcm2 at B0,max needs to be transferred. With an adhesive thickness of

d = 0.2mm this results in a temperature difference between coil and plates of

∆T =qd

λ≈ 1K. (5.7)

The cooling water passes 3 cooling plates one after another. A flow rate of 2− 3 lmin

is more

than sufficient and even higher dissipation losses could be easily transferred.

The beam area of the coils is not covered by the cooling plates. Without additional cooling

this area would become excessively hot. This could result in a liquefaction of the glue and a

deformation of the band. Therefore, the beam area is additionally cooled with compressed

air using a similar design as in [12]. A sketch of the cooling system of the bootstrap coils is

shown in Fig. 5.4.

The brass components of the previously used cooling circuit reacted electrochemically with

the aluminum parts of the coils. Due to the formation of aluminum oxide, the cooling plates

of the coils became blocked resulting in a cooling failure. In order to avoid electrochemical

reactions to a large extend, all brass parts were replaced by stainless steel and plastic parts

throughout.

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5.1 Bootstrap coils 95

Air

AirH O2

Fig. 5.4: Cooling system of the bootstrap coils. Cooling plates flowedthrough by water are glued to the B0 coils and the whole stack is pressedtogether. The beam areas not covered by the cooling plates are cooled withpressurized air.

5.1.3 RF coils

The tolerance of the mechanical accuracy of the RF coils is lower since deviations from

the π-flip of the magnetic moment of the neutron within the coil only contribute in second

order [12]. Following the estimate in [12], the polarization after N RF-coils with an applied

field BRF and a thickness of l can be approximated by

Pn = 1−Nπ2

4

[(∆BRF

BRF

)2

+

(∆l

l

)2]. (5.8)

Assuming a tolerable reduction of the polarization of 5% after N = 8 coils and equal relative

deviations of the magnetic field and the coil thickness yields

∆BRF

BRF=

∆l

l< 4%. (5.9)

According to equation (2.17) the amplitude of the HF field is:

BRF = 4~

m

π2

γ

1

lλ. (5.10)

The linear oscillating field BRF is divided into two counter rotating parts B+1 and B−

1 with

an amplitude of BRF

2 (see section 2.2.1). Only the part rotating in the same sense as the

precession of magnetic moment of the neutron contributes to the π-flip. For a wavelength

of λ = 2.4Å and l = 35mm the amplitude of the RF field is BRF = 15.7G. Therefore, the

accuracy needed is:

∆BRF = 0.63G ∆l = 1.4mm. (5.11)

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96 5 Upgrade of the NRSE option at FLEXX

Manufacturing

The design of the RF coils follows closely the designs discussed in [12, 79, 82]. The RF coils

consist of a rectangular coil L1 wound with one layer of round aluminum wire, which covers

the beam area and two u-shaped rectangular coils L2 wound with one layer of copper wire

and litz wire (see Fig. 5.5). The coils L2 are not in the neutron beam and feed back the

magnetic field of coil L1. This design simultaneously reduces the stray fields in the field

free region and the eddy currents induced in the cooling plates and thus the power losses

by dissipation. Since the new RF coils are thicker than the previous coils used at FLEXX,

smaller magnetic fields and therefore less current is needed for a π-flip of the magnetic mo-

ment of the neutron.

L1

L2

L2

B1

B1

B1

Fig. 5.5: Left: Sketch of the RF coil as an assembly of three coils. Themagnetic field of the inner coil L1 is actively fed back by the outer coils L2.Right: Side view of the complete RF coil.

The whole coil body consists of a glass-reinforced plastic frame for L1 and two massive PVC

bodies for L2. To feed back the field BRF of L1, the coils L2 need to produce the same field

BRF . The wiring diagram is shown in Fig. 5.6. In order to reduce the total inductance, the

two coils L2 are connected parallel to each other and in series with L1. Since the current

passing through each L2 coil is half of the current passing through L1, the winding density

of L2 needs to be twice as high as for L1 to produce the same magnetic field BRF .

The inner coil L1 is wound with four aluminum wires in parallel with a cross section of 1mm2

each. The straight part of the outer coils L2 is wound with three copper wires in parallel

with a cross section of 0.75mm2 each. The curved parts of L2 are wound with litz wire such

that the winding density is the same as for the straight section. The total inductance of the

RF coils was measured to be L = 36.6µH and the DC resistance is R = 0.35Ω.

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5.2 Spin echo instrument arms 97

L1

L2

L2

I

I/2

I/2

Fig. 5.6: RF coil wiring diagram.

Since the inductance of the new RF coils is different compared to the previously used

coils it was necessary to adapt the parameters for the impedance matching. Details of the

impedance matching procedure are described in Appendix F.

5.2 Spin echo instrument arms

The base frames of the new spin echo spectrometer arms were completely rebuilt in a more

compact design. They are assembled using an aluminum ITEM profile [83]. The NRSE

spectrometer arms are attached to the TAS components by coupling devices and are mov-

able by four air pads. The frames are designed such that every part, which potentially needs

maintenance, is easily accessible. Beside the valves for the air pads and the air cooling of

the bootstrap coils, the bottom part of the frames hosts the valves and the flow meter of the

cooling water circuit. If the flow of the cooling water is reduced below a certain level, the

flow meters activate the interlock signal of the power supplies to ensure that overheating of

the bootstrap coils is avoided. At the top part of the base frames, all rotary and translation

stages (see section 5.2.3) are mounted on an aluminum baseplate. On top of the base frame

the magnetic shielding boxes for the bootstrap coils are installed. The shielding performance

is described in section 5.2.1. The separation of the coils and the dimensions of the magnetic

shielding are chosen such that the coils can be rotated by 360. Before the upgrade, the hard

limit of the coil tilt angles was ≈ 45 while intensity losses due to geometrical restrictions

of the available beam cross section were encountered from Θ1 = Θ2 ≈ 35. Combined with

the new design of the coils the instrument now allows for much larger coil tilt angles (see

section 5.3.4). The new coils enhance this value to Θ ≈ 45. The housing of the impedance

matching capacitors (see Appendix F) is mounted on the back side of the base frame.

The new base frames are slimmer compared to the previously used frames and allow for

larger scattering angles. Here the maximum value for the scattering angle A4 is enhanced

from 110 to 120. The previous spin echo spectrometer could not be positioned in di-

rect beam geometry, i.e. A4 = 0 and A6 = 0 (detector angle), in the ki-range from

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98 5 Upgrade of the NRSE option at FLEXX

ki = 1.3Å−1 to ki = 1.7Å−1 since the maximum radius of the instrument was too large

for the old instrument floor. Therefore, direct beam calibration measurements (see section

5.3.4) were not possible in this ki-range. The new, more compact design shortens the spin

echo spectrometer arms, decreasing the maximum length of the instrument. As FLEXX is

installed at a new instrument position in the neutron guide hall of BER II, which provides a

larger instrument floor area, direct beam calibration measurements are now possible for the

whole ki-range of the instrument. Hence, the base frames hosting the NRSE components

increase the measurement range for the spin echo option of FLEXX. The accessible ki-range

is between 1.37Å−1 and 2.37Å−1 corresponding to a maximum λ = 4.57Å and a minimum

λ = 2.77Å, respectively. This range is determined by the monochromator shielding and was

limited by mechanical constraints at the time of the experiment.

The combination of more compact spectrometer arms and larger accessible coil tilt angles

(see section 5.3.4) allows for a larger range of scattering angles in Larmor diffraction geome-

try (see section 3.3.3). The previous NRSE option allowed for a |Q|-range in Larmor diffrac-

tion geometry from 2.69Å−1 to 3.11Å−1. These values are calculated for ki = kf = 1.9Å−1.

Given the new geometry of the spectrometer arms, the enhanced |Q|-range is now from

2.44Å−1 to 3.29Å−1 for ki = kf = 1.9Å−1.

5.2.1 Magnetic shielding

Between the monochromator and the analyzer no depolarization of the neutron beam due

to uncontrolled magnetic fields should occur. The earth’s magnetic field (Be ≈ 0.3G) gives

rise to a field integral of about 1Gm over the whole instrument and would lead to four

360-precessions of the polarization vector for a wavelength of λ = 6Å and about three

360-precessions for a wavelength of λ = 4Å. A tolerable variation in precession angle of

Φ = 25 reduces the polarization by 10% and corresponds to a residual mean magnetic field

of Br ≈ 5mG for λ = 6Å and Br ≈ 8mG for λ = 4Å. The shielding factor is defined by:

S =Ba

Br(5.12)

where Ba is the mean magnetic field without any shielding. Therefore, a shielding factor of

at least 60 is required.

The design of the NRSE option previously used at FLEXX provided the “field free” region

by shielding the instrument with a single layer of standard mu-metal with a high relative

permeability of 50000 [84]. The new shielding design, which has been basically adopted

from the previously used NRSE components, consists of three parts: Each spectrometer

arm has a square tube with a rectangular cross section (780 × 380 × 380mm3, thickness

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5.2 Spin echo instrument arms 99

d = 2mm) to shield a pair of bootstrap coils. The covers are removable and the end caps are

properly connected by screws ensuring a good magnetic contact. A split vertical cylinder

(∅500mm × 500mm each, thickness d = 2mm) shields the sample position (see Fig. 5.7).

The rectangular tubes of the spectrometer arms are closed with openings for the neutron

beam and have a removable cover. Compensation coils at the outer faces allow for an addi-

tional active shielding.

Fig. 5.7: Sketch of the magnetic shielding design of the NRSE option ofFLEXX.

Smaller mu-metal cylinders (∅100mm × 300mm each) shield the region between the spec-

trometer arms and the sample position. The mu-metal cylinder towards the sample region

is flange-mounted to the caps. At the entrance face of the first and the exit of the second

spectrometer arm a mu-metal square tube (80 × 80 × 100mm3) is flange-mounted to the

caps as a shielding housing for the coupling coils (see section 5.2.2) of the spectrometer.

Since the thickness of the shielding of the bootstrap coils is increased from d = 1.5mm to

d = 2mm, the shielding factor is enhanced. The shielding factor of a box with a quadratic

cross section can be calculated to good approximation by the ballistic demagnetization fac-

tor N and the relative flux concentration in the walls of the shielding [86]. The ballistic

demagnetization factor for this geometry is given by:

N =1

1 + p

(1−

43

q + 203

)1− 3

40

(q − 9

5

)(q − 20)

(q + 13

2

) 135

, (5.13)

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100 5 Upgrade of the NRSE option at FLEXX

where p = ab is the ratio of the edge lengths a and b of the box and q = 1

p+p. Approximations

for the perpendicular and longitudinal shielding factor are [86]:

Sp = 0.7µd

a(5.14)

Sl =2.52N

1 + 0.56 ba

µd

b. (5.15)

For the new dimensions of the shielding the approximations yield Nl ≈ 0.58 and Sl ≈ 87 for

the longitudinal case and Np ≈ 0.42 and Sp ≈ 184 for the perpendicular case. The calculated

shielding factors for the old dimensions (1000 × 300 × 300mm3, thickness d = 1.5mm) are

Sl ≈ 40 and Sp ≈ 175. The longitudinal shielding factor for the spectrometer arms is

therefore substantially increased by a factor of 2 while the perpendicular shielding factor

is slightly enhanced. Measurements showed that the residual fields in the horizontal and

vertical directions within the new shielding do not exceed the critical value of Br = 5mG in

both arms in the magnetic field of the earth.

The vertical field component at the sample position needs to be actively shielded since the

two cylinders are separated. Therefore, compensation coils at the outer ends are installed

to reduce this field component. The vertical field at the gap between the cylinders is

higher since the field components are bundled within the shielding walls [12]. This region

is bypassed using small, flange-mounted mu-metal cylinders extending into the vertical

cylinder. The horizontal field component at the sample region is compensated using the

coils mounted to the caps of the shielding of the spectrometer arms.

Important for the operation of the NRSE option are stray fields arising from a possible

operation of magnets, vertical or horizontal, at the neighboring V4/SANS instrument, which

may give rise to unwanted spin precessions. The sample position of the V4/SANS instrument

is about 11m away from the beginning of the field free region of the first spectrometer arm.

Stray field measurements for the strongest vertical magnet available at the HZB, the VM-1B

with a maximum field of 17.5T, show that the remaining vertical field at the position of the

NRSE option is about 200mG. Since Sp ≈ 184, such a field is easily shielded by the actual

mu-metal shielding. However, stray field measurements for the strongest horizontal magnet

available at the HZB, the HM-1 with a maximum field of 6T, show that the stray field

components parallel to the neutron flight path are an issue. Depending on the orientation

and on the chosen geometry of the NRSE instrument the stray field components parallel

and perpendicular to the neutron flight path are in the range of 0.5−1G. The perpendicular

component is sufficiently shielded. Since Sl ≈ 87 the remaining field within the field free

zone exceeds 5mG and will cause significant depolarization. Therefore, a parallel operation

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5.2 Spin echo instrument arms 101

of the horizontal magnet at V4/SANS and the NRSE option at FLEXX should be avoided

and be considered for the scheduling of the experiments of both instruments.

5.2.2 Coupling coils

Fig. 5.8: Side view of the new coupling coils.

The neutron beam is polarized and analyzed in the vertical z-direction perpendicular to

the scattering plane. Thus, the polarization vector needs to be turned by 90 into the

scattering plane before entering the first field free region. After the second field free region

the process has to be reversed by turning the polarization vector into the vertical direction.

This is realized by coupling coils. The coupling coils are positioned at the entrance of the

first shielding and the exit of the second shielding box. In standard operation the first

coupling coil turns the incoming neutron spin (polarized in the vertical z-direction) into the

horizontal plane by an adiabatic transition and transports the neutron spin into the screened

area by a non-adiabatic transition (see Appendix G). The second coupling coil at the exit

of the second shielding provides the transition from a field free region to a region with guide

field and rotates the neutron polarization vector again by 90 to be perpendicular to the

scattering plane. In order to tune the single RF-coils of the NRSE option the coupling coils

are turned by 90 around the beam axis. The neutron polarization vector is then left aligned

in the vertical direction providing only the non-adiabatic transition into the screened area.

The coupling coils manufactured at HZB have a new design. Each coil (see Fig. 5.8) is

wound on an anodized aluminum body with an Al-wire (∅0.5mm). In order to have an

adiabatic transition at the side outside the magnetic shielding the wire is bent out of the

neutron beam. For the non-adiabatic transition the neutrons pass through the wire within

the magnetic shielding. To avoid a leakage of the return field into the screened area, the coil

is covered with mu-metal to guide the return field. Compared to the former design [12] the

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102 5 Upgrade of the NRSE option at FLEXX

adiabatic transition is realized by a diagonal wound opening of the coil. Simultaneously the

wire is everywhere properly attached to the coil body. This makes the design more compact

and robust at the same time.

5.2.3 Motors and encoders

Each bootstrap coil is mounted on its own rotary stage allowing to rotate the coil around its

vertical axis for measuring dispersive excitations or for measurements in Larmor diffraction

geometry. The rotary stages of the first and fourth coil are mounted on a linear stage. This

allows for translating the coils along the beam direction to fine tune or to scan the field

integral L ·B. Due to the upgrade it is now possible to translate both the first and the last

NRSE coils. With the new, more compact magnetic shielding the rotary and translation

stages were positioned outside of the shielding and placed below. Hence, the motors and

encoders do not produce any stray fields within the magnetic shielding. The new position

of the mechanical devices also facilitates maintenance work.

5.3 Calibration of the new NRSE option at FLEXX

Prior to performing any measurements on samples with a completely new NRSE spectrom-

eter, several calibrations measurements have to be done. Since the parameters of all coils of

the instrument changed, the optimum current parameters will also change. Therefore, these

parameters have to be determined first. For the calibration measurements the coils in each

of the spectrometer arms were aligned parallel using a theodolite. The S-bender was moved

into the FLEXX neutron guide system. The S-bender has a magnetization field perpendic-

ular to the scattering plane and the vertical directions of the polarization is maintained up

to the coupling coils. A bender was mounted in front of the detector. In future for polarized

neutron work a Heussler analyzer will be installed replacing the PG analyzer. However, the

device was not available at the time of the calibration measurements described here.

Conventional NRSE spectrometers are run in two modes. In the so called 4π-mode only the

inner π-coils of each spectrometer arm are used for the measurement (4 π-coils in total are

in use). This mode gives access to smaller spin echo times. The so called 8π-mode uses all

π-coils to benefit from the bootstrap enhancement factor and to give access to higher spin

echo times.

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5.3 Calibration of the new NRSE option at FLEXX 103

5.3.1 Calibration of currents and HF voltage

New NRSE look-up table

The optimum parameters to perform a π-flip within a π-coil need to be determined experi-

mentally. The look-up table for experimentally optimized values of the currents I0 applied

to the B0-coils and the pick-up Voltage Upu of the RF coils was measured for a wavelength

of λi = 3.3Å. Since the neutron velocity and therefore the time of the neutron spent in

the magnetic field is linear in λ, the optimum values can easily be extrapolated for other

wavelengths. The same argument holds for a change in the coil tilt angles.

8 10 120

2

4

6

8

10x 10

4

I0 [A]

Inte

nsity

[Cou

nts

/ 5s]

Coil #1Coil #2Coil #3Coil #4

8 10 120

2

4

6

8

10x 10

4

I0 [A]

Inte

nsity

[Cou

nts

/ 5s]

Coil #5Coil #6Coil #7Coil #8

Fig. 5.9: Detector counts as a function of the current of the B0-coils spec-trometer arm 1 (left) and 2 (right). If the Larmor frequency of the staticfield is equal to the frequency f of the RF coils the detector counts reacha minimum. The results show very little variation between the individualcoils manufactured.

First, the optimum currents for the B0-coils as a function of the frequency applied to the

RF-coils were determined. For these measurements the incoming polarization vector is ori-

ented parallel to the axis of the B0-coil. This is realized by operating the coupling coils in

a mode to provide a field perpendicular to the scattering plane. Hence, the initially verti-

cally oriented polarization vector is not turned into the horizontal plane and the coupling

coil only provides the non-adiabatic transition into the screened region. The current of a

single static field coil is then scanned for a fixed frequency f of the RF coil. If the Larmor

frequency corresponding to B0 is the same as f , the magnetic moment of the neutron will

perform the largest possible rotation around the RF field and therefore causes a minimum

in the detector counts. The RF amplitude may differ from its optimum value and the mini-

mum of the polarization becomes more pronounced with the RF amplitude approaching the

optimum value. Fig. 5.9 shows the intensity at the detector as a function of the B0-coil

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104 5 Upgrade of the NRSE option at FLEXX

current I0 for all 4 coils of each spectrometer arm for f = 100kHz. The plot shows very

little variation between the individual coils manufactured. This is very important since all

B0-coils of a spectrometer arm are operated in series using a single power supply. Due to

the negligible variation in the optimum I0 currents no additional resistors were needed to

adjust the static magnetic field amplitudes.

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

f [kHz]

B0 c

urre

nt [A

]

data 1st armdata 2nd armlinear fit

0 50 100 150 200 250 300 350 400−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

f [kHz]

Diff

eren

ce D

ata

− F

it [A

]

data 1st armdata 2nd arm

Fig. 5.10: Left: Optimum currents for the first B0-coils of each spectrom-eter arm as function of RF frequency. A linear function is fitted to thedata. Right: Difference between data and linear fit.

The calibration measurements for I0 as a function of frequency were performed using the

first B0-coil of each spectrometer arm. The results for the optimum current I0 over the

frequency range probed are shown in Fig. 5.10, displaying the expected linear behavior.

They are almost identical for both arms. A linear function is fitted to the data and the

deviations from the fit are shown. The deviations are larger for small frequencies but since

the minimum in intensity is quite broad, depolarization effects arising from variations in B0

can be neglected.

Similar measurements were performed to obtain the optimum values for the amplitude of

the RF signal. The signal from a small pick-up coil located at one of the U-shaped parts of

the RF coil, directly probing th RF field by the induced voltage Upu, is used as a control

variable (see Appendix F). The RF amplitude is scanned for a fixed frequency f at the op-

timum value of B0(f). At the optimum the magnetic moment of the neutron will perform a

π-flip resulting in a minimum of the detector counts. The measurements for each frequency

were performed using the first π-coil of each spectrometer arm. The results are shown in

Fig. 5.11 and are almost identical for both arms. A linear function is fitted to the data

and the deviations from the fit are shown. The deviations are larger for small frequencies

but since the minimum of the detector counts is quite broad, depolarization effects can be

neglected.

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5.3 Calibration of the new NRSE option at FLEXX 105

0 50 100 150 200 250 300 350 4000

5

10

15

20

f [kHz]

Pic

k−U

p V

olta

ge [V

]

data 1st armdata 2nd armlinear fit

0 50 100 150 200 250 300 350 400−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

f [kHz]D

iffer

ence

Dat

a −

Fit

[V]

data 1st armdata 2nd arm

Fig. 5.11: Left: Optimum pick-up voltages Upu for the first RF coils ofeach spectrometer arm as function of RF frequency. A linear function isfitted to the data. Right: Difference between data and linear fit.

Coupling coils

The coupling coils provide the transition of the polarization vector into and out of the

field free region. In the limit of large coupling coil currents the adiabatic transition is best

realized. However, the stray fields increase with increasing coupling coil current. An exper-

imental calibration of the currents is therefore needed. For the calibration measurements

the coupling coils were operated in their standard orientation, i.e. the incoming vertical po-

larization vector is guided into the scattering plane by the first coupling coil and is guided

back to the vertical direction by the second coupling coil at the end of the field free region.

Depending on the relative orientation of the magnetic fields, i.e. the relative sign of the

currents, of the coupling coils the detector counts will either reach a maximum plateau (the

polarization vector is restored in vertical direction) or a minimum plateau (the incoming

polarization vector is rotated by 180). The current of one coil was scanned while the cur-

rent of the other coil was fixed at +0.9A.

The results for both coupling coils are shown in Fig. 5.12. The current scans for both cou-

pling coils clearly show the two plateaus. With increasing coil current the detector counts

decrease slightly because of depolarization effects of the increasing stray field of the coupling

coils. Components of the polarization vector perpendicular to the nominal direction cause a

slight oscillation of the signal. This effect is due to a rotation of perpendicular components

around the effective magnetic field. Stray field effects decrease the intensity with increasing

coil current. Apparently this effect is stronger for the second coupling coil (right) due to the

different transition geometry between coupling coil and guide field. For further calibration

measurements Ic1 and Ic2 were set to +1.5A.

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106 5 Upgrade of the NRSE option at FLEXX

−5 −4 −3 −2 −1 0 1 2 3 4 50

2

4

6

8

10x 10

4

Ic1

[A]

Inte

nsity

[Cou

nts

/ 10s

]

−5 −4 −3 −2 −1 0 1 2 3 4 50

2

4

6

8

10x 10

4

Ic2

[A]

Inte

nsity

[Cou

nts

/ 10s

]Fig. 5.12: Intensity at the detector as a function of the current of thecoupling coils of spectrometer arm 1 (left) and 2 (right). While scanning thecurrent Ic1 (Ic2) the current Ic2 (Ic1) was set to Ic2 = +0.9A (Ic1 = +0.9A).

5.3.2 Spin echo curve and echo point

According to [2] the envelope of the spin echo curve is the Fourier transform of the incoming

wavelength distribution

AE = 〈cosΦNSE〉 =∫

f (λ) cos(ωeffm

2π~∆Lλ

)dλ, (5.16)

where ωeff = γBeff = 2NγB0 is the effective Larmor frequency, N the number of active

π-coils within one bootstrap coil (see section 2.2.2) and ∆L = L1 −L2. Approximating the

wavelength distribution with a Gaussian shape

f(λ) =1

σλ√2π

e− (λ−λ0)

2

2σ2λ (5.17)

with a mean wavelength λ0, the Fourier transform yields

AE = e− 1

8

(ωeffm

2π~∆Lσλ

)2

cos(ωeffm

2π~∆Lλ0

)(5.18)

According to equation (2.11) the beam intensity after the analyzer is

I =I02(1 +AE) (5.19)

Fig. 5.13 shows a typical spin echo curve obtained by scanning the position of the fourth

bootstrap coil using all π-coils (N = 2) with a frequency of 400kHz. The accessible range

of the spin echo curve is determined by the range of the translation table.

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5.3 Calibration of the new NRSE option at FLEXX 107

−25 −20 −15 −10 −5 0 5 10 15 20 250

1

2

3

4

5

6

7

8

x 104

Position of fourth bootstrap coil [mm]

Inte

nsity

[Cou

nts

/ 10s

]

Fig. 5.13: Spin echo curve obtained by direct beam measurements. Thefit yields a wavelength of λ = 3.3498(1)Å and ∆λ

λ≈ 1.3%.

The cosine term in equation (5.18) causes the rapid oscillations. By fitting the period length

of one oscillation

∆L =2π~

ωeffmλ0(5.20)

the mean wavelength λ0 can be determined. In this case for a frequency of 400kHz, N = 2

and a period length ∆L = 0.73812(3)mm the obtained wavelength is λ = 3.3498(1)Å. The

spectral width is fitted to ∆λλ ≈ 1.3%. This is in good agreement with the properties of the

focusing monochromator used during the experiment.

Supplementary spin echo curves were measured in the direct beam for nominal wavevectors

ki = 1.37Å−1 and ki = 2.37Å−1. Fitting the data results in ki,fit = 1.3762(1)Å−1 (λ =

4.8686(2)Å) with a corresponding ∆λλ ≈ 0.8% and ki,fit = 2.3666(3)Å−1 (λ = 2.2666(3)Å)

with a corresponding ∆λλ ≈ 1.4%. These two wavevectors are the limits of the measur-

able ki-range due to mechanical constraints of the monochromator shielding at the time of

the experiment. The value of ∆λλ decreases from ∆λ

λ ≈ 1.4% (λ = 2.27Å) to ∆λλ ≈ 0.8%

(λ = 4.87Å) due to the fact, that the same angular variation of neutron trajectories emanat-

ing from the monochromator corresponds to a larger wavelength spread at lower wavelengths.

The condition of the echo point is given by equation (2.8). In the case of an unscattered

neutron beam, there is no energy transfer and therefore no change in the neutron velocity.

Hence, the field integrals B1L1 and B2L2 of the two spectrometer arms need to be the same.

In the present experimental setup the effective precession fields B1 and B2 are the same in

both spectrometer arms and therefore the lengths L1 and L2 need to be equal too. It is

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108 5 Upgrade of the NRSE option at FLEXX

best to determine the distances between the coils using neutron experiments. Hence, the

echo amplitude for a fixed frequency f was measured as a function of the position of the

last bootstrap coil by varying the length L2 in direct beam geometry. The maximum of this

spin echo curve is the echo point where L1 = L2. For conventional spin echo spectrometers

with a broad wavelength distribution (∆λλ ≈ 10%), where ∆λ is a FWHM, the spin echo

point can easily be identified. However, if f(λ) becomes small the spin echo curve becomes

very broad and the echo point cannot be determined with high precision.

In the case of a broad spin echo curve, the spin echo point can be determined by measuring

the spin echo curve for several frequencies, which should be coprime. In the case of ∆L =

0 all curves should have a common maximum or a minimum depending on the relative

orientation of the fields provided by the coupling coils (see equation (5.18)). During the

measurements the coupling coils were operated in an antiparallel configuration. This leads to

a phase shift of π since the incoming polarization vector oriented in the positive z-direction

is turned by 180 so that the outgoing polarization vector points in the negative z-direction.

Thus, in this case all curves should have a minimum in the spin echo point. The fitted

curves of such a measurement are shown in Fig. 5.14. The data were collected in 8π-mode

with a wave length of λ = 3.35Å for the frequencies 271kHz, 399kHz and 437.5kHz. The

minima of the curves are at x = −3.26± 0.01mm. The small deviations of the curves from

the common minimum arise from the slightly different static fields of the coils leading to a

small phase shift of the spin echo curve.

−10 −8 −6 −4 −2 0 2 4 60

1

2

3

4

5

6

7

8

x 104

Position of fourth coil [mm]

Inte

nsity

[Cou

nts

/ 10s

]

437.5 kHz399 kHz271 kHz

Fig. 5.14: At the spin echo point x = −3.26 ± 0.01mm (∆L = 0) spinecho curves for different frequencies have a common minimum. The mea-surements were performed in 8π-mode using a wavelength of λ = 3.35Å forthe frequencies 271kHz, 399kHz and 437.5kHz.

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5.3 Calibration of the new NRSE option at FLEXX 109

The position of the spin echo point can be used to determine the absolute distance L1 = L2

between the coils, i.e. the length of the effective magnetic field, at the spin echo point. In

the first measurement the same frequency was applied to the coils in both spectrometer

arms. By decreasing the frequency of the second arm by a factor of 0.97 the length L′2

should increase to

L′2 =

L

0.97= L+∆x (5.21)

to satisfy the spin echo condition. Here ∆x is the shift of the spin echo point. The distance

between the coils can then be calculated from

L =0.97

0.03∆x. (5.22)

2 4 6 8 10 12 14 160

1

2

3

4

5

6

7

8

x 104

Position of fourth coil [mm]

Inte

nsity

[Cou

nts

/ 10s

]

f1=437.5 kHz, f

2=424.375 kHz

f1=399 kHz, f

2=387.03 kHz

f1=271 kHz, f

2=262.87 kHz

Fig. 5.15: At the spin echo point x = 8.86 ± 0.01mm (∆L = 0) all spinecho curves have a minimum. The measurements were performed in 8π-mode with a wavelength of λ = 3.35Å. The frequencies applied to thesecond spectrometer arm were lowered by a factor of 0.97 compared to thefirst arm.

The results for the measurements of the shifted spin echo point are shown in Fig. 5.15. The

frequency of the second arm was lowered by a factor of 0.97. The spin echo point is now at

x = 8.86 ± 0.01mm leading to ∆x = 12.12 ± 0.01mm, which yields L8π = 391.9 ± 0.3mm,

which is in very good agreement with the distance between coils measured mechanically.

The same measurement was performed in 4π-mode to determine the distance between the

inner π-coils at the spin echo point. The data measured at λ = 3.35Å with the frequencies

122kHz, 225kHz and 297kHz applied to the RF-coils in both spectrometer arms is shown in

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110 5 Upgrade of the NRSE option at FLEXX

Fig. 5.16 (top). The spin echo point is at x = −2.78 ± 0.01mm. The signal corresponding

to f = 122kHz is slightly shifted. This can caused by residual magnetic fields. However, the

shifted minimum is still consistent with the correct common minimum of the other signals.

−20 −15 −10 −5 0 5 10 150

1

2

3

4

5

6

7

8

9x 10

4

Position of fourth coil [mm]

Inte

nsity

[Cou

nts

/ 10s

]

122 kHz225 kHz297 kHz

−10 −5 0 5 10 15 20 250

1

2

3

4

5

6

7

8

9x 10

4

Position of fourth coil [mm]

Inte

nsity

[Cou

nts

/ 10s

]

f1=122 kHz, f

2=118.34

f1=225 kHz, f

2=218

f1=297 kHz, f

2=288.09

Fig. 5.16: Top: At the spin echo point x = −2.78± 0.01mm all spin echocurves have a common minimum. The measurements were performed in 4π-mode using a wavelength of λ = 3.35Å for the frequencies 122kHz, 255kHzand 297kHz. Bottom: The frequencies applied to the second spectrometerarm were lowered by a factor of 0.97 compared to the first arm and thespin echo point is shifted to x = 7.63± 0.01mm.

To determine the shift of the spin echo point the frequencies applied to the second arm

were again lowered by a factor of 0.97 as for the measurement in 8π-mode. In 4π-mode the

RF-coils of the outer π-coils of each spectrometer arm are not connected though the B0-

coils are powered and the magnetic moments precess in these static fields with the Larmor

frequency. If the same frequencies are applied in all coils, i.e. the same current is applied to

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5.3 Calibration of the new NRSE option at FLEXX 111

the B0-coils, the additional Larmor phase collected in the static fields cancels out after the

second arms. However, if the frequency in the second arm and thus the B0-coil current is

lower a difference in the total Larmor phase ∆Φ collected in the static fields remains. Due

to the additional phase the resulting spin echo curve is shifted and the measured curves

need to be corrected for each frequency separately. Since the phases corresponding to the

inner π-coils cancel at the spin echo point, ∆Φ at the spin echo point can be calculated as:

∆Φ = 2πf12Lcoil

v− 2πf2

2Lcoil

v= 4π

Lcoil

v(f1 − f2) (5.23)

=2Lcoilmλ

~(f1 − 0.97f1) = 0.03

2Lcoilmλ

~f1. (5.24)

(5.25)

Here f1 and f2 are the frequencies applied to the spectrometer arms and Lcoil is the thickness

of one π-coil. Using the period of one oscillation

L0(f) =2π~

2Nfmλ(5.26)

the correction in distance can be calculated for each frequency by

∆Lcorr(f) = L0∆Φ

2π. (5.27)

Fig. 5.16 (bottom) shows the corrected data using the coil thickness of 52mm. The spin

echo point is at x = 7.63 ± 0.01mm leading to ∆x = 10.41 ± 0.01mm, which yields L4π =

336.6± 0.3mm. This is in good agreement with direct distance measurements.

5.3.3 Phase stability

The phase stability of the NRSE spectrometer was tested by repeating a spin echo measure-

ment several times and comparing the phases of the individual scans. The measurements

were done in 8π-mode using λ = 3.35Å and a frequency of 300kHz. Each spin echo run

took about 10 minutes. Fig. 5.17 (left) shows the fits of the single scans. The results for

the phases are plotted in Fig. 5.17 (right). During the measuring time of one hour the

phase shifted by ∆Φ = 0.058rad. If the phase shifts while recording a spin echo signal, the

contrast of the resulting spin echo signal is decreased, i.e. the echo amplitude decreases.

For a long spin echo scan with a typical total measurement time (here assumed to be 3

hours), assuming a perfectly polarized beam and neglecting all other depolarizing effects,

the remaining polarization would be 1 − cos(0.058 radh

· 3h) = 98.5%. Thus, this effect is

negligible compared to the statistical error in most cases.

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112 5 Upgrade of the NRSE option at FLEXX

The phase shift of the signal is due to thermal drifts. During a measurement the coils heat

up, which changes the resistance slightly, i.e. the currents, the magnetic fields and the

Larmor frequencies change slightly, too. The estimations for the depolarization made here

present a conservative limit since for longer measurements the phase reaches a plateau when

the coils and the cooling plates reach thermal equilibrium. For more precise estimations of

systematic errors longer test measurements would be needed, which could not be performed

due to the limited beam time at FLEXX.

−0.5 0 0.50.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

5

Position of fourth coil [mm]

Inte

nsity

[cou

nts

/ 20s

]

Run 1Run 2Run 3Run 4Run 5Run 6

0 2 4 6 81.74

1.75

1.76

1.77

1.78

1.79

1.8

1.81

1.82

Run number

Pha

se [r

ad]

Fig. 5.17: Left: Fits of the spin echo scans performed at 300kHz in 8π-mode with λ = 3.35Å. Right: Phase as a function of run number. Within1 hour the phase shifted by ∆Φ = 0.058rad.

5.3.4 Calibration of coil tilt angles

Mezei pointed out [2] that the correction for instrumental depolarization effects can be

performed by dividing the measured echo amplitude from inelastic scattering

AE,meas(τ) = AE,meas(f1, f2,Θ1,Θ2, ki, kf ) (5.28)

by the echo amplitude AE0 from direct beam measurements corresponding to the same spin

echo parameters:

AE(τ) = AE(f1, f2,Θ1,Θ2, ki, kf ) =AE,meas(f1, f2,Θ1,Θ2, ki, kf )

AE0(f1, f2,Θ1,Θ2, ki, kf ). (5.29)

Since a convolution of the signal and the instrumental resolution in real space correspond

to a multiplication in Fourier space and vice versa, this simplifies the correction compared

to classical neutron spectroscopy where a deconvolution is performed.

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5.3 Calibration of the new NRSE option at FLEXX 113

For quasielastic scattering experiments a purely elastic scattering sample can be used to

obtain AE0. However, for inelastic scattering no standard samples exist since there is in

principle no dispersive excitation with zero linewidth. Therefore, AE0 is obtained by direct

beam measurements with f1 = f2, Θ1 = Θ2 and ki = kf . For direct beam measurements

the echo amplitude AE,calib can be written as a product of the echo amplitudes AE1 and

AE2 after each spectrometer arm:

AE,calib(f1,Θ1, ki) = AE1(f1,Θ1, ki) ·AE2(f1,Θ1, ki). (5.30)

Assuming that the depolarizing effects are the same in both arms yields

AE1 = AE2 =√AE,calib(f1,Θ1, ki). (5.31)

In inelastic spin echo measurements the spin echo parameters of both spectrometer arms

are different, i.e. AE1(f1,Θ1, ki) and AE2(f2,Θ2, kf ). The corresponding echo amplitude

AE0 is then obtained by:

AE0(f1, f2,Θ1,Θ2, ki, kf ) = AE1(f1,Θ1, ki) ·AE2(f2,Θ2, kf ) (5.32)

=√

AE,calib(f1,Θ1, ki) ·√AE,calib(f2,Θ2, kf ). (5.33)

AE,calib is measured for a discrete set of frequencies and coil tilt angles according to the

chosen ki and kf of the experiment. If the actual spin echo parameters of the experiment

differ from the parameters of the calibration measurements, the corresponding AE,calib is

obtained by interpolation between the two closest calibration sets.

Fig. 5.18 shows calibration data for 4π-mode (left) and 8π-mode (right). The set was

recorded for coil tilt angles of 0, 10, 20, 30, 40 and 50 for ki = 1.8757Å−1. The effec-

tive frequency feff is the actual frequency f applied to the π-coils times the enhancement

factor 2N (N = 1 for 4π-mode and N = 2 for 8π-mode). As the frequency increases, the

magnetic moment of the neutron performs more precessions within the magnetic field and

the stray fields increase. The depolarization increases with increasing frequency. This can

be seen for both modes in Fig. 5.18 where the echo amplitude decreases with increasing feff

for each coil tilt angle. Since the flight path of the neutrons within the coil and therefore

the time spent in the magnetic field increases with increasing coil tilt angle, the influence

of depolarizing effects, e.g. due to field inhomogeneities, increases. As seen in Fig. 5.18

especially for high frequencies in 8π-mode the echo amplitude decreases with increasing coil

tilt angle. However, the similar values for the echo amplitude for all coil tilt angles in the low

and middle regime of feff display the accurate alignment of the coils. If the field boundaries

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114 5 Upgrade of the NRSE option at FLEXX

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

feff

[kHz]

echo

am

plitu

de

θ = 0°θ = 10°θ = 20°θ = 30°θ =40°θ = 50°

0 500 1000 15000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

feff

[kHz]ec

ho a

mpl

itude

θ = 0°θ = 10°θ = 20°θ = 30°θ =40°θ = 50°

Fig. 5.18: NRSE calibration data for ki = 1.8757Å−1

. The data set wasmeasured for coil tilt angles of 0, 10, 20, 30, 40 and 50 for both NRSEmodes, 4π (left) and 8π (right).

of the coils are not parallel to each other, the flight paths in the coils would differ and the

depolarization would increase more strongly with increasing frequency and increasing coil

tilt angle. The comparably low echo amplitude at feff = 100kHz in 4π-mode corresponds

to a frequency of f = 50kHz and is a result of the Bloch-Siegert-Shift (see section 2.2.1).

Similarly the effect is noticeable in the 8π data at feff = 400kHz. Supplementary calibra-

tion sets were measured for ki = 2.3666Å−1 for both NRSE modes, 4π and 8π. The results

are shown in Fig. 5.19.

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

feff

[kHz]

echo

am

plitu

de

θ = 0°θ = 20°θ = 30°θ =40°θ = 50°

0 500 1000 15000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

feff

[kHz]

echo

am

plitu

de

θ = 0°θ = 20°θ = 30°θ =40°θ = 50°

Fig. 5.19: NRSE calibration set for ki = 2.3666AA−1. The set wasmeasured for coil tilt angles of 0, 20, 30, 40 and 50 for both NRSEmodes, 4π (left) and 8π (right).

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5.3 Calibration of the new NRSE option at FLEXX 115

Fig. 5.20 shows the fitted intensities I0 for ki = 1.8757Å−1 according to equation (5.19)

for all measured spin echo signals of the calibration set. The variation in intensity within

one coil tilt angle set is due to variation in reactor power since the data was taken for fixed

count time and is shown not normalized to monitor counts.

0 200 400 600 8000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

feff

[kHz]

Inte

nsity

[Cou

nts

/ s]

θ = 0°θ = 10°θ = 20°θ = 30°θ =40°θ = 50°

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

feff

[kHz]

Inte

nsity

[Cou

nts

/ s]

θ = 0°θ = 10°θ = 20°θ = 30°θ =40°θ = 50°

Fig. 5.20: Fitted intensities I0 according to equation (5.19) for all mea-sured spin echo signals of the calibration set shown in Fig. 5.18 separatedaccording to the mode used (left: 4π, right: 8π).

With increasing coil tilt angle the intensity decreases since the neutron beam cross section is

gradually reduced by the finite window width of the NRSE coils. Fitting an average intensity

for every coil tilt angle set and normalizing the average intensity to the value for Θ = 0

the effect of reduced effective beam cross section on intensity at the detector is shown in

Fig. 5.21. Here, the mean intensities for both modes obtained from the calibration set as

a function of the coil tilt angle are displayed. The black curve is the calculated normalized

beam cross section as calculated from the geometric dimensions of the bootstrap coils. The

data is in very good agreement with the calculations, however, the steep reduction of the

neutron beam seems to set in at smaller tilt angles than predicted. This effect can arise

from a dislocation of the coils to each other perpendicular to the neutron beam. Such

a dislocation would reduce the accessible beam cross section, but would not change the

length of the flight path within the coil. The green curve displays the beam cross section

as calculated from the geometric dimensions of the previously used bootstrap coils. Note

that the enhancement of the coil window height from 32mm to 52mm is taken into account

for the calculations. The previously used bootstrap coils had a hard limit for the coil tilt

angle of Θ ≈ 45 and at Θ0.5 ≈ 35 the beam cross section was reduced by a factor of 0.5.

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116 5 Upgrade of the NRSE option at FLEXX

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

coil tilt angle [°]

norm

aliz

ed b

eam

cro

ss s

ectio

n

I0mean

I0mean

Fig. 5.21: Normalized mean intensities fitted to the calibration set forboth modes as a function of coil tilt angle. This displays the cropping ofthe neutron beam. The black (green) line is a rough estimation accordingto the geometric dimensions of the current (previously used) bootstrap coiland is in good agreement with the fit results.

The new coils have no hard limit for the coil tilt angle and the value for Θ0.5 was enhanced

to ≈ 45. The measurements show that the new bootstrap coils allow for much higher tilt

angles compared to the previously used bootstrap coils.

A comparison between calibration data from experiments using the previous NRSE option

and the calibration measurements presented here show, that the polarized neutron flux at

the sample position is increased by a factor of 5. This was measured for ki = 1.9Å−1 in

direct beam geometry. The enhancement is due to the larger beam cross section available

at FLEXX now using a double focusing monochromator, the more compact instrument and

the larger accessible beam cross section of the coupling and the bootstrap coils.

5.4 Summary

The upgrade of the NRSE option of the cold TAS FLEXX was a major part of this thesis.

Redesigned NRSE bootstrap coils were manufactured in collaboration with the Max Planck

Institute For Solid State Research, Stuttgart. The new coils allow for larger beam cross sec-

tions and larger coil tilt angles. Hence, steeper dispersions are accessible. The spectrometer

arms were redesigned, making the instrument more compact, leading to an extended range

in scattering angle. In combination with larger coil tilt angles, the range of scattering an-

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5.4 Summary 117

gles in Larmor diffraction geometry is increased. A new, more compact mu-metal shielding

guarantees a better shielding of the field free area between the NRSE coils. In contrast to

the previously used spectrometer arms, all mechanical and electrical components (e.g. go-

niometers) were relocated beneath the shielding boxes to reduce scattering fields inside the

field free zone. In order to benefit from the larger beam cross section available at FLEXX,

now using a double focusing monochromator, the coupling coils were redesigned with a

larger cross section and a more compact design. The larger beam cross section available at

FLEXX in combination with the new geometries of the new NRSE result in an increase of

the polarized neutron flux at the sample position by a factor of 5.

Extensive calibration measurements were performed in order to demonstrate the functional-

ity of the instrument and to obtain the parameters necessary to make the TAS-NRSE spec-

trometer available for future experiments. In particular, look-up tables for the impedance

matching and coil currents were measured. The echo point and the distance between the

coils were determined. The measurements prove the phase stability and calibration sets for

different coil tilt angles show that the properties of the individually produced bootstrap

coils are practically identical.

The calibration measurements clearly demonstrate that the NRSE option works without

any major technical problems and demonstrated its reliable performance.

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118 5 Upgrade of the NRSE option at FLEXX

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Chapter 6

Conclusion and perspectives

Two new kinds of NRSE experiments were investigated in this thesis: Resolving mode dou-

blets with an energy separation smaller than the typical energy resolution of a standard

triple axis spectrometer and line shape analysis of temperature dependent asymmetric line

broadening. The data analysis of these experiments requires a detailed model for the echo

amplitude as a function of correlation time as introduced by Habicht et al. [6]. This model

includes depolarization effects due to sample imperfections, the curvature of the dispersion

surface and the TAS resolution function. This is essential for a thorough understanding of

the results. Up to now the existent model was limited to cubic systems.

In this thesis, major generalizations of the existing formalism were developed. As a result,

the model can now be applied to any crystal symmetry class. Furthermore, the extended

model allows for violated spin echo conditions, arbitrary local gradient components of the

dispersion surface and detuned TAS parameters, giving rise to important additional depo-

larizing effects, which have been neglected before. The model was successfully tested by

experiments on phonons in a high quality single crystal of Pb. For a detuned background

TAS, realized by rotating the sample, the model predicts additional phase terms and a

non-linear behavior of the spin echo phase. The results of the experiment are in very good

agreement with the extended resolution model and demonstrate the stringent necessity to

consider second order effects.

Subsequently, the formalism was extended to analyze mode doublets. As a major general-

ization, detuning effects for both modes are taken into account in contrast to the previous

treatment [23]. Nb was chosen for experimental verification as it is a well understood model

system. The realization of a unique tunable double dispersion setup allowed to generate

artificially split modes. Echo amplitudes were measured in elastic Larmor diffraction ge-

ometry for two neighboring Bragg peaks. The results demonstrate the echo modulation in

119

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120 Conclusion and perspectives

spin echo length space to be in good agreement with a simplified phenomenological model

describing the signature of split modes [21]. Inelastic NRSE spectroscopy on an effectively

split dispersion clearly shows the modulation of the echo amplitude. The results agree with

the developed extended model predictions, indicating persistence of the modulation over

the entire spin echo time range, probed by the experiment. It was proven once more that

inelastic NRSE measurements on a single crystal require the consideration of second order

effects. Since the dispersion of Nb is steep for the chosen experimental parameters, strong

depolarizing effects occur. Hence, neglecting these effects would result in a significant over-

estimation of the linewidths. It was shown that the phenomenological model is sufficient for

the determination of the splitting. However, for a detailed analysis the use of the generalized

resolution function, developed in this thesis, is inevitable.

The results prove the potential of NRSE spectroscopy to resolve mode doublets with an

energy separation smaller than the typical energy resolution of a standard TAS. Hence,

the NRSE method can give valuable input to investigations on effects like the splitting of

magnon excitations as observed by Náfrádi et al. [24], hybridized magnon-phonon modes

in multiferroics [25, 26], and excitations with small energy separations, which are found in

orbital Peierls systems [27].

The second class of new NRSE experiments treated in this thesis were dedicated to line

shape analysis of temperature dependent asymmetric line broadening. This effect has been

observed in two systems: Cu(NO3)2·2.5D2O, a model system for a 1-D bond alternating

Heisenberg chain [53], and Sr3Cr2O8, a 3-dimensional gapped quantum spin dimer [65],

with standard ToF and TAS techniques. In order to explore the potential of NRSE to

investigate the effect, inelastic spin echo measurements were performed. For this purpose

high quality single crystals of Cu(NO3)2·2.5D2O were grown in the course of this thesis.

Temperature dependent inelastic NRSE measurements were performed and analyzed using

a phenomenological model for asymmetric line shapes for both systems. For the first time

this effect was measured with NRSE. As a proof of principle the results clearly show that the

NRSE method can be used to detect temperature dependent asymmetric line broadening.

Since there is no convolution of the signal with the resolution function of the spectrometer,

the NRSE method gives direct access to the line shape in the time domain. This is an

important advantage compared to other high resolution methods, such as ToF. It was shown,

that for a line shape differing from Lorentzian shape, the phase of the spin echo signal

becomes a non-linear function of the spin echo time τ . The results are a counter example

to the assertion of a general linear dependence between phase shift and spin echo time τ .

Thus, phase sensitive measurements can be used to determine a deviation of the scattering

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Conclusion and perspectives 121

function from a Lorentzian shape. Exceptions are temperature dependent excitation energies

as present in Sr3Cr2O8.

Applications of NRSE are very interesting and might stimulate further experiments on

different systems, since the results for Cu(NO3)2·2.5D2O and Sr3Cr2O8 are suggested to be

applicable to a broad range of quantum systems [53, 65].

The second major part of this thesis was the upgrade of the NRSE option of FLEXX at

the BER II neutron source at HZB, Berlin. In the course of this project, redesigned NRSE

bootstrap coils were manufactured in collaboration with the Max Planck Institute For Solid

State Research, Stuttgart. The new coils allow for larger beam cross sections and higher coil

tilt angles. Hence, steeper dispersions are accessible. A major advantage is the exploitation

of the larger beam cross section, now available at FLEXX. The newly designed spectrometer

arms resulted in a more compact instrument. Thus, direct beam calibration measurements

are now feasible for the entire accessible wavevector range. In combination with higher coil

tilt angles the accessible Q-range in Larmor diffraction geometry is enlarged. A new, more

compact mu-metal shielding guarantees a better shielding of the field free area between the

NRSE coils. All mechanical and electrical components (e.g. rotary tables) were relocated

beneath the shielding boxes to reduce stray fields inside the field free zone. In order to

benefit from the larger beam cross section, the coupling coils were redesigned.

Extensive calibration measurements were performed. The parameters, necessary for user op-

eration of the instrument, were obtained and look-up tables, important for the impedance

matching and coil currents, are available now. The echo point and the distance between the

coils were determined. Measurements were successfully done to prove the phase stability.

Calibration sets for different coil tilt angles show that the properties of the individually

manufactured bootstrap coils are practically identical. The measurements clearly demon-

strate that the NRSE option works without any major technical problems and prove its

reliable performance.

The upgrade of the NRSE option of FLEXX was performed during the scheduled shut down

and upgrade of the BER II neutron source at HZB, Berlin. Since originally only a six months

shut down period until April 2011 was planned, NRSE measurements at the cold triple axis

spectrometer FLEXX on samples were foreseen in the course of this thesis. Due to the

unexpected prolongation of the shut down of BER II until April 2012, the available beam

time was severely limited. The shortage of beam time restricted the NRSE experiments

to commissioning of the NRSE option and calibration measurements. Hence, the thermal

NRSE-TAS TRISP at FRM-II, Garching, was chosen to perform NRSE line shape analysis

on Cu(NO3)2·2.5D2O.

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122 Conclusion and perspectives

A feasible future advancement could be the upgrade of the NRSE-TAS instrument to be

able to perform MIEZE I&II measurements (Modulated IntEnsity by Zero Effort) [8].

MIEZE is a quasi-elastic variant of the NRSE technique generating a high frequent, time-

dependent sinusoidal signal at the detector and can be regarded as a high resolution ToF

spectrometer [8]. MIEZE I provides the advantage that the signal modulation is achieved

before the sample. Hence, the measurement becomes insensitive to depolarizing effects at

the sample region. Thus, the sample region does not need to be shielded magnetically

and measurements including magnetic fields and ferromagnetic samples are feasible. Such

an upgrade would require a detector with a high time resolution suitable for resolving the

time-dependent sinusoidal signal and minor changes of the instrument control software.

A MIEZE spectrometer is a promising instrument candidate [91, 92] for the future European

Spallation Source (ESS), Lund, Sweden. Applying the method to pulsed sources is techni-

cally challenging and requires major efforts and experimental verification. Such tests could

be performed at the ESS test beam line currently being built at the BER II, Berlin. Here,

the existing upgraded NRSE option of FLEXX can be used. The highly flexible spectrom-

eter arms could be adapted to other instrument components without major modifications

and provide ready to use NRSE components. A cost intensive and time consuming design

and manufacturing process would thus be redundant.

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Appendix

A UB matrix formalism

The UB matrix formalism as discussed in [17] extends the original formalism introduced for

four-circle diffractometers [16] to TAS and ToF spectrometers. The UB matrix formalism

allows to calculate the instrument parameters (ν, µ, ω, φ) for a given (Q, E) and vice versa.

Here µ and ν are the sample goniometer angles, φ is the angle between kI and kF and ω is

related to the sample angle A3 by

ω = A3− θ (A.1)

with

θ =kI − kF cosφ

kF sinφ. (A.2)

A.1 The B matrix

The B matrix transforms any wavevector Q expressed in the basis of the reciprocal lattice

vectors b1,2,3 into a Cartesian system attached in a defined way to the reciprocal lattice.

The definition of Q reads:

Q =

h

k

l

= hb1 + kb2 + lb3. (A.3)

Follow Busing and Levy [16] here the x-axis is chosen to be parallel to b1, the y-axis to be in

the plane of b1 and b2 and the z-axis to be perpendicular to this plane. The transformation

of Q is then described by

Qc = BQ. (A.4)

123

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124 Appendix

This equation follows the crystallographic convention, i.e. the factor 2π is omitted. The

magnitude of the wavevector is given by QM = 2π |Qc| and B is given by

B =

(B1 B2 B3

)=

a∗ b∗ cos γ∗ B13

0 b∗ sin γ∗ B23

0 0 B33

(A.5)

with

B13 = c∗ cosβ∗ (A.6)

B23 = c∗(cosα∗ − cos γ∗ cosβ∗)

sin γ∗= −c∗ sinβ∗ cosα∗ (A.7)

B33 =√c∗2 −B2

3x −B23y = c∗ sinβ∗ sinα∗. (A.8)

Note that Busing and Levy give B33 = 1/c. The usual crystallographic relations apply [47]

for the reciprocal lattice

a∗ =bc

Vsinα, b∗ =

ca

Vsinβ, c∗ =

ab

Vsin γ (A.9)

using

V = abc√1− cos2 α− cos2 β − cos2 γ + 2 cosα cosβ cos γ (A.10)

and

cosα∗ =cosβ cos γ − cosα

sinβ sin γ(A.11)

cosβ∗ =cos γ cosα− cosβ

sin γ sinα(A.12)

cos γ∗ =cosα cosβ − cos γ

sinα sinβ. (A.13)

A.2 The U matrix

The procedure introduced by Busing and Levy uses two non-collinear reflections Q1 and Q2

to calculate the orientation matrix U.

U = TνT−1c , (A.14)

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A UB matrix formalism 125

where Tc is defined by a triple of right-handed orthogonal unit vectors t1c, t2c, t3c and Tν

is defined by a triple of right-handed orthogonal unit vectors t1ν , t2ν , t3ν . The matrix Tc

then reads

Tc =

(BQ1

|BQ1|,(BQ1 ×BQ2)×BQ1

|(BQ1 ×BQ2)×BQ1|,BQ1 ×BQ2

|BQ1 ×BQ2|

)(A.15)

where t1c is parallel to BQ1, t2c is in the plane of BQ1 and BQ2 and t3c is perpendicular to

this plane. Note that the matrix T−1c transforms from the crystal lattice Cartesian system

into the Cartesian system defined by the scattering system where Q1 points along the x-axis.

The matrix Tν is calculated in a similar way using the vectors u1ν and u2ν , given by

uiν =

cosωi cosµi

− sinωi cos νi + cosωi sinµi sin νi

sinωi sin νi + cosωi sinµi cos νi

. (A.16)

The vectors u1ν and u2ν represent the two non-collinear reflections expressed in the labora-

tory system by the experimental values (νi, µi, ωi) (i = 1, 2) of each reflection. The matrix

Tν therefore transforms from the Cartesian system defined by the scattering system into

the laboratory system and reads

Tν =

(u1ν

|u1ν |,(u1ν × u2ν)× u1ν

|(u1ν × u2ν)× u1ν |,u1ν × u2ν

|u1ν × u2ν |

). (A.17)

With the matrices U and B determined, the vector Qn for any given set of instrumental

positioning parameters (ν, µ ,ω, ϕ) can be calculated using

Qn =QM

2π(UB)−1

unν . (A.18)

A.3 The R matrix

As pointed out in [17] there are ambiguities for calculating the instrument parameters for

a given parameter set (Qn, E), since the Q-vector can be rotated around its own axis via

the sample goniometers without changing the scattering condition. Here, the condition

“keep a reference plane as horizontal as possible” is adopted from [17]. The matrix R then

transforms from the ν-coordinate system, where the rotation axis of the upper goniometer

points along the x-axis, into the θ-coordinate system, with Q pointing along the x-axis. The

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126 Appendix

transformation reads

RQν = Qθ =1

QM

0

0

. (A.19)

The matrix R is related to the elementary rotation matrices

R = ΩMN, (A.20)

where

Ω =

cosω − sinω 0

sinω cosω 0

0 0 1

(A.21)

M =

cosµ 0 sinµ

0 1 0

− sinµ 0 cosµ

(A.22)

N =

1 0 0

0 cos ν − sin ν

0 sin ν cos ν

(A.23)

represent the rotations of the sample table and the sample goniometers, respectively. The

matrix elements of R are explicitly

R =

R11 R12 R13

R21 R22 R23

R31 R32 R33

=

ωcµc ωcµsνs − ωsνc ωcµsνc + ωsνs

ωsµc ωsµsνs + ωcνc ωsµsνc − ωcνs

−µs µcνs µcνc

, (A.24)

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A UB matrix formalism 127

where the instrumental positioning parameters can be obtained from

µs = sinµ = −R31 (A.25)

µc = cosµ =(R2

11 +R221

)1/2(A.26)

νs = sin ν = R32/(R2

11 +R221

)1/2(A.27)

νc = cos ν = R33/(R2

11 +R221

)1/2(A.28)

ωs = sinω = R21/(R2

11 +R221

)1/2(A.29)

ωc = cosω = R11/(R2

11 +R221

)1/2. (A.30)

Note that the proper quadrants are obtained from

µ = arcsin (−R31) (A.31)

ν = arcsin(R32/

(R2

11 +R221

)1/2)(A.32)

and

ω =

arcsin(R21/

(R2

11 +R221

)1/2) if R11 > 0

π − arcsin(R21/

(R2

11 +R221

)1/2) if R11 < 0.(A.33)

In order to calculate the matrix elements of R and following Lumsden et al. [17], the unit

vector perpendicular to the reference plane is defined by

uν⊥ =

− sinµplane

cosµplane sin νplane

cosµplane cos νplane

. (A.34)

Using the unit vectors Qν

|Qν | and uν⊥, a right-handed orthogonal set of vectors is constructed

by

t1ν = u1ν =Qν

|Qν |=

UBQ

|UBQ| (A.35)

t2ν = uν⊥ × u1ν (A.36)

t3ν = t1ν × t2ν . (A.37)

With

T1ν =

(t1ν t2ν t3ν

)(A.38)

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128 Appendix

the result for R is:

R = T−11ν . (A.39)

Now the matrix elements of R are known and equations (A.31 – A.33) together with equa-

tions (A.1) and (A.2) allow to calculate the instrument positioning parameters for a given

Q.

B Matrix elements

B.1 The Ψ-matrix

The non-zero elements of the symmetric (6× 6)-matrix Ψ in the resolution function given

by equation (2.81) in section 2.3 are:

Ψ11 = − 2D2

C2fN

2F τ

′′2

(B.1)

Ψ22 = − ~

m− ~

mtan2 θ1 − 2

τ1τ ′′2

NI

kI ·ni− 2

D2

τ ′′2

C2i N

2I

C2fN

2F

(B.2)

Ψ33 = − ~

m

1

cos2 θ1− 2

D2

τ ′′2

e2i2N2I

C2fN

2F

1

cos2 θ1(B.3)

Ψ44 =~

m

1

cos2 θ2+ 2

~

m

tan θ2cos2 θ2

1

Cfef2 − 2

D2

τ ′′2

e2f2C2f

1

cos2 θ2(B.4)

Ψ55 = − ~

m− 2

D2

τ ′′2

e2i3N2I

C2fN

2F

(B.5)

Ψ66 =~

m− 2

D2

τ ′′2

e2f3C2f

(B.6)

Ψ12 = 2D2

τ ′′2

CiNI

C2fN

2F

(B.7)

Ψ13 = 2D2

τ ′′2

ei2NI

C2fN

2F

1

cos θ1(B.8)

Ψ14 = −2D2

τ ′′2

ef2C2fNF

1

cos θ2+

~

m

tan θ2cos θ2

1

CfNF(B.9)

Ψ15 = 2D2

τ ′′2

ei3NI

C2fN

2F

(B.10)

Ψ16 = −2D2

τ ′′2

ef3NFC2

f

(B.11)

Ψ23 =~

m

tan θ1cos θ1

− 2D2

τ ′′2

CiN2I

C2fN

2F

ei21

cos θ1(B.12)

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B Matrix elements 129

Ψ24 = − ~

m

tan θ2cos θ2

CiNI

CfNF+ 2

D2

τ ′′2

CiNI

C2fNF

ef21

cos θ2(B.13)

Ψ25 = −2D2

τ ′′2

CiN2I

C2fN

2F

ei3 (B.14)

Ψ26 = 2D2

τ ′′2

CiNI

C2fNF

ef3 (B.15)

Ψ34 = 2D2

τ ′′2

ei2ef2NI

C2fNF

1

cos θ1 cos θ2− ~

m

tan θ2cos θ2 cos θ1

NI

CfNFei2 (B.16)

Ψ35 = 0 (B.17)

Ψ36 = −2D2

τ ′′2

ei2ei3N2I

C2fN

2F

1

cos θ1+ 2

D2

τ ′′2

ei2ef3NI

C2fNF

1

cos θ1(B.18)

Ψ45 = 2D2

τ ′′2

ef2ei3NI

C2fNF

1

cos θ2− ~

m

tan θ2cos θ2

1

Cf

NI

NFei3 (B.19)

Ψ46 = −2D2

τ ′′2

ef2ef3C2f

1

cos θ2+

~

m

tan θ2cos θ2

1

Cfef3 (B.20)

Ψ56 = 2D2

τ ′′2

ei3ef3NI

C2fNF

(B.21)

with

D2 =

(−τ2

NF

kF ·nf− ~

2mτ ′′2 − ~

2mτ ′′2 tan2 θ2

). (B.22)

B.2 The I−1C -matrix

The non-zero elements of the (6× 6)-matrix I−1C in the resolution function given by equation

(2.116) in section 2.3.4 are:

I−1C,11 = NI (ei1 cos θ1 − ei2 sin θ1) (B.23)

I−1C,12 = NI (ei1 sin θ1 + ei2 cos θ1) (B.24)

I−1C,13 = ei3NI (B.25)

I−1C,14 = −NF (ef1 cos θ2 + ef2 sin θ2) (B.26)

I−1C,15 = −NF (ef1 sin θ2 − ef2 cos θ2) (B.27)

I−1C,16 = −ef3NF (B.28)

I−1C,21 = cos θ1 (B.29)

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130 Appendix

I−1C,22 = sin θ1 (B.30)

I−1C,32 = I−1

C,45 = I−1C,53 = I−1

C,66 = 1. (B.31)

B.3 The IC-matrix

The non-zero elements of the (6× 6)-matrix IC in the resolution function given by equation

(2.116) in section 2.3.4 are:

IC,12 =1

cos θ1(B.32)

IC,13 = − tan θ1 (B.33)

IC,41 = − 1

NF ef2 sin θ2 +NF ef1 cos θ2(B.34)

IC,42 = − ei2NI sin θ1 − ei1NI cos θ1ef1NF cos θ1 cos θ2 + ef2NF cos θ1 sin θ2

(B.35)

IC,43 =ei2NI cos

2 θ1 + ei2NI sin2 θ1

ef1NF cos θ1 cos θ2 + ef2NF cos θ1 sin θ2(B.36)

IC,44 = −ef1 sin θ2 − ef2 cos θ2ef1 cos θ2 + ef2 sin θ2

(B.37)

IC,45 =NIei3

ef2NF sin θ2 + ef1NF cos θ2(B.38)

IC,46 =ef3

ef1 cos θ2 + ef2 sin θ2(B.39)

IC,23 = IC,35 = IC,54 = IC,66 = 1. (B.40)

C Triple axis transmission function in matrix notation

The standard approximation of the resolution function for a conventional TAS has been

derived by Cooper and Nathans [48] (corrections by Dorner [49]) and reformulated in a

covariant matrix formalism by Stoica [50]. The formalism has been extended by Popovici

[51] to include spatial effects, such as monochromator and analyzer focusing and finite spatial

dimensions of the beam optical elements and the sample. Explicitly the TAS resolution

function reads [51]:

RTAS (X) =R0

(2π)2

√detMTASe

− 12XTMTASX (C.1)

where X is the four component column vector (X1, X2, X3, X4) = (Q−Q0, ω(q)− ω0(q0)),

i.e. the resolution function is defined in (Q, ω)-space. Note that X4 is the fourth com-

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D Dispersion relation properties of RbMnF3 131

ponent of the vector X and the quantity ∆ω = ω(q) − ω0(q) used in the calculations is

different and refers to energy deviations from the dispersion surface for a given wavevec-

tor q. R0

(2π)2

√detMTAS is a normalization factor and R0 is given in equation (16) in [51].

Following Popovici the resolution matrix is given by:

M−1TAS = BA

[D(S+TTFT

)−1DT]−1

+G

−1

ATBT (C.2)

where G−1 is the covariance matrix of the distribution of the angular variables and F−1

is the covariance matrix of the reflectivity function. The matrices A, D, S, T, and G are

defined as in Appendix I and II in [51]. The linearized relation

X = BY (C.3)

holds, where the six component column vector Y = (∆ki,∆kf ) (see section (2.3.4). The

matrix B transforms from the coordinate space of Y into the coordinate space of X. In

order to express the exponent of the resolution function as a function of the six component

column vector J = (∆ω,∆kin, y1, y2, z1, z2), equation (2.124) is used

Y = ICJ. (C.4)

The (6× 6)-matrix IC is defined in Appendix B.3. Therefore, equation (C.2) can be refor-

mulated yielding for the TAS resolution matrix in the frame of J:

LTAS =

I−1C A

[D(S+TTFT

)−1DT−1

+G

]−1

AT(I−1C

)T−1

− (C.5)

D Dispersion relation properties of RbMnF3

The dispersion relation of RbMnF3 can be obtained from linear spin wave theory [52]:

E(q) =[gβHA + 2S (6J1 − 12J2 + 8J3) + 4SJ2B2 − 4SJ1A+ 4SJ3C2

] 12, (D.1)

where

A = cos qha0 + cos qka0 + cos qla0 (D.2)

B = cos (qh + qk) a0 + cos (qh − qk) a0 + cos (qk + ql) a0

+cos (qk − ql) a0 + cos (ql + qh) a0 + cos (ql − qh) a0 (D.3)

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132 Appendix

C = cos (qh + qk + ql) a0 + cos (qh − qk − ql) a0

+cos (qh + qk − ql) a0 + cos (qh − qk − ql) a0. (D.4)

J1, J2 and J3 are the exchange interactions between first, second and third neighbors and

HA is the anisotropy field. Here qh,k,l = (2π/a0 ·Qh,k,l) and Qh,k,l are in r.l.u.. Magnetic

zone centers are located at (±0.5,±0.5,±0.5). According to [52] it is assumed that HA = 0,

J2 = 0, J3 = 0, S = 5/2, a0 = 4.204Å and J1 = 0.293meV. Hence, the dispersion relation

reduces to

E(q) = ~ω(q) = 4SJ1

[9− A(q)2

] 12. (D.5)

Since RbMnF3 has cubic symmetry, there is no need for additional transformations into a

Cartesian coordinate system and the components of the gradient of the dispersion read

∂E

∂qh,k,l= 4SJ1a0 sin a0qh,k,l

A√9−A2

. (D.6)

The elements of the curvature matrix can be calculated from the analytical expressions

∂2E

∂qh∂qh= − 4Sa20J1

(9−A2)32

[9 sin2 qha0 − 9A cos qha0 +A3 cos qha0

](D.7)

and∂2E

∂qh∂qk= −36SJ1a

20 sin qha0 sin qka0

(9−A2)32

. (D.8)

Other second derivatives are obtained from cyclic permutation of h, k and l.

E Growth of copper nitrate single crystals

Required deuteration ratio

Using the incoherent cross sections listed in Tab. 6.1, the incoherent cross section σxinc of

Cu(NO3)2·2.5H2O, Cu(NO3)2·2.5D2O and a mixture of both can be calculated. Here x

is defined as the percentage fraction of Cu(NO3)2·2.5D2O in the mixture. Therefore, the

incoherent cross section σxinc of pure Cu(NO3)2·2.5D2O and pure Cu(NO3)2·2.5H2O yields:

σ100inc =

1 · 0.55 + 2 · 0.5 + 8.5 · 0 + 5 · 2.0516.5

b = 0.72 b (E.1)

σ0inc =

1 · 0.55 + 2 · 0.5 + 8.5 · 0 + 5 · 80.2616.5

b = 24.42 b. (E.2)

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E Growth of copper nitrate single crystals 133

Element σinc [b]

H 80.26

D 2.05

O 0

N 0.5

Cu 0.55

Tab. 6.1: Incoherent cross sections of the elements present in the com-pounds Cu(NO3)2·2.5H2O and Cu(NO3)2·2.5D2O [61].

A satisfactory deuteration ration is achieved, if the incoherent cross section of the mixture

σxinc is comparable to the incoherent cross section σ100

inc of pure Cu(NO3)2·2.5D2O. Thus, a

deuteration ratio of 98.5% resulting in

σ98.5inc =

1 · 0.55 + 2 · 0.5 + 8.5 · 0 + 5 · (80.26 · 0.015 + 2.05 · 0.985)16.5

b

= 1.0707 b = 1.487 σ100inc , (E.3)

would fulfill the requirement.

Distillation process

In order to calculate the remaining amount of H2O after the distillation process the molar

masses of the different compounds are needed. The molar masses of Cu(NO3)2·2.5H2O,

D2O and H2O are 232.59 gmol

, 20 gmol

and 18 gmol

, respectively.

For the distillation process 140.99g Cu(NO3)2·2.5H2O powder material, containing 27.04g

H2O, was dissolved in 47.56g high purity D2O (>99.9%). Therefore, using the molar masses,

the resulting hydrogen mass ratio and the atomic ratio are 36.25% and 38.71%, respectively.

In the first destillation run 12.80g liquid, containing 4.64g H2O and 8.16g D2O, were distilled

from the mixture. 22.29g D2O were added in the next step and decreased the hydrogen molar

ratio and the atomic ratio to 26.64% and 28.75%, respectively. By repeating these two steps

17 times, the deuteration level of the solution was successively increased, resulting in a

calculated deuteration ratio of 99.38%. The results of the distillation runs are listed in Tab.

6.2. Note that the actual deuteration ratio may be slightly higher than the calculated one

due to a small amount of liquid, remaining in the Liebig condenser of the setup, which was

not weighed after each distillation run. In the last run liquid was distilled off to ensure a

saturated solution for the crystal growth.

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134 Appendix

addedD2O[g]

distilledliquid

[g]

thereofH2O[g]

thereofD2O[g]

H2O insolution

[g]

D2O insolution

[g]

D2Ofraction[g-%]

D2Ofraction[mol-%]

47.56 0.00 0.00 0.00 27.04 47.56 63.75 61.29

22.29 12.80 4.64 8.16 22.40 61.69 73.36 71.25

22.71 30.50 8.12 22.38 14.28 62.02 81.29 79.63

16.62 23.20 4.34 18.86 9.93 59.79 85.75 84.41

18.04 21.20 3.02 18.18 6.91 59.65 89.61 88.59

30.29 30.90 3.21 27.69 3.70 62.25 94.38 93.80

12.05 8.90 0.50 8.40 3.20 65.90 95.36 94.87

11.50 13.70 0.64 13.06 2.57 64.33 96.16 95.75

16.45 15.00 0.58 14.42 1.99 66.36 97.08 96.77

10.92 9.30 0.27 9.03 1.72 68.25 97.54 97.27

13.11 13.10 0.32 12.78 1.40 68.58 98.00 97.78

8.47 3.70 0.07 3.63 1.33 73.42 98.23 98.03

11.32 10.90 0.19 10.71 1.13 74.04 98.49 98.33

11.48 14.40 0.22 14.18 0.92 71.33 98.73 98.59

17.31 19.80 0.25 19.55 0.66 69.10 99.05 98.94

18.12 13.80 0.13 13.67 0.53 73.55 99.28 99.20

17.70 13.90 0.10 13.80 0.43 77.45 99.44 99.38

0.00 15.80 0.09 15.71 0.35 61.73 99.44 99.38

Tab. 6.2: List of distillation steps performed to obtain a molar deuterationlevel of 99.38% of the solution.

F Impedance matching of the RF coils

In order to measure at different spin echo times τ during an NRSE experiment, different

frequencies need to be applied to the RF coils. Thus, the optimum parameters of the res-

onant circuit of the RF coils need to be adapted, i.e. the impedance needs to be matched

to the required frequency. The spin echo option of FLEXX uses an automatic impedance

matching device implemented in the circuit shown in Fig F.1. All four RF coils of each spec-

trometer arm are connected in parallel and the RF coils of each NRSE arm are connected

to a waveform generator via a power amplifier. The phases of both waveform generators

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F Impedance matching of the RF coils 135

are locked. In order to have a direct control over the magnetic field amplitude in the RF

coils, a pick-up coil is attached to one of the U-shaped parts of the RF coil. The oscillating

magnetic field induces a voltage Upu, which is read by an RF voltmeter and transferred to

the control program.

C2

C1

L

R

Z

Fig. F.1: Impedance matching circuit using two capacities [90].

The impedance matching proceeds as follows: The required frequency is applied to the RF

coils using a small signal amplitude. Optimum values for the capacitances C1 and C2 are

obtained from a look-up table for the specific frequency. The capacitance C2 is kept fix

while C1 is adjusted until the pick-up voltage Upu reaches a maximum. The amplitude of

the waveform generator is then increased until Upu has reached its target value obtained

from the look-up table (see section 5.3.1).

The impedance of one RF coil is:

ZRF = iωL+R, (F.1)

where R is the DC resistance of the RF coil (≈ 0.35Ω) and L is the inductance (≈ 36.6µH).

The output resistance of the power amplifiers is 50Ω and frequency independent. In order

to maximize the forward power of the amplifier, the real part of the load impedance needs to

match the 50Ω. Otherwise the reflected power could overheat the amplifier. The impedance

matching device uses a simple matching circuit with a discrete set of capacitances for each

of the two capacitances C1 and C2 [90] (see Fig. F.1). The impedance of the circuit is:

1

Z= iωC2 +

11

iωC1+ iωL+R

. (F.2)

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136 Appendix

The real part of equation (F.2) should be 50Ω while the imaginary part should be zero:

Re

(1

Z

)=

R

R2 +(ωL− 1

ωC1

)2 = 50Ω (F.3)

Im

(1

Z

)= ωC2 −

ωL− 1ωC1

R2 +(ωL− 1

ωC1

)2 = 0 (F.4)

Therefore, the resulting capacitances are:

C1(ω) =1

ω2L− ω√ZR−R2

(F.5)

C2(ω) =L− 1

ω2C1

R2 +(ωL− 1

ωC1

)2 . (F.6)

The existing range of discrete capacitors (C1: 25pF to 3.19µF, C2: 800pF to 1.63µF) is

sufficient for a frequency range from 50kHz to 500kHz in 4π-mode (2 parallel RF coils)

and 8π-mode (4 parallel RF coils). For calibration purposes, to provide a parameter set

C1(ω) and C2(ω) for NRSE measurements, the capacitance C2 was calculated according to

equation (F.6) and C1 was then optimized. By varying C1 stepwise the read out induced

voltage was maximized for a constant output amplitude of the waveform generator.

C1 [nF]

C2 [n

F]

f = 100kHz

700 750 800

300

310

320

330

340

350

360

0.9

1

1.1

1.2

1.3

C1 [nF]

C2 [n

F]

f = 200kHz

80 85 90 95145

150

155

160

165

170

175

1.2

1.4

1.6

1.8

2

2.2

2.4

C1 [nF]

C2 [n

F]

f = 300kHz

28 30 32 34

115

120

125

130

135

140

1

1.5

2

2.5

3

Fig. F.2: Resonance of the induced voltage as a function of the capacitiesC1 and C2 for frequencies of 100kHz, 200kHz and 300kHz. The measure-ments were done in 8π-mode (4 parallel RF coils). The black line showsthe optimum values according to equation (F.6).

Fig. F.2 shows the induced voltage for different sets of capacitances for frequencies of

100kHz, 200kHz and 300kHz. For the measurement all four parallel RF coils were con-

nected, resulting in Rtot ≈ 0.25Ω and Ltot ≈ 11µH. The maximum value of the resonance

shows little sensitivity to C2. For f = 100kHz the resonance is quite broad, while it gets

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G Adiabatic and non-adiabatic transitions of the magnetic moment of the neutron 137

smaller with increasing frequency. The discrete steps in the voltage values as seen in Fig.

F.2 arise from the fact that the effective capacitances C1 and C2 are each a combination

of parallel capacitances. If the effective capacitance is increased stepwise, the number of

parallel capacitances may change by a factor of two or more, resulting in a step change

of the impedance of the effective capacitance. This effect does not affect the automatic

impedance matching since for every fixed C2 there is still a pronounced maximum of the

resonance, which is found by varying C1. The black line in Fig. F.2 displays the theoretical

optimum capacitance sets (C1, C2) according to equation (F.6). The experimental results

agree completely with the calculations from the model.

In order to generate a new look-up table containing the frequency dependent optimum start-

ing values of the capacitances, the resonances were scanned for different frequencies for both

spectrometer arms. The measurements were performed in 4π-mode and 8π-mode.

G Adiabatic and non-adiabatic transitions of the magnetic

moment of the neutron

If the magnetic moment of the neutron oriented parallel to the guide field enters a temporally

and/or spatially varying magnetic field, two different limiting cases are possible [2, 89]:

• Adiabatic transition: The magnetic moment follows the change of a smoothly varying

magnetic field direction

• Non-adiabatic transition: The orientation of the magnetic moment does not change if

the magnetic field direction changes abruptly

The condition for the adiabatic case can be formulated as

ωL ≫ ωB, (G.1)

where ωL = |γ| ·B is the Larmor frequency and ωB =d( ~B/| ~B|)

dt is the frequency of the change

of the magnetic field direction. If ωL is much larger than ωB, the magnetic moment of the

neutron keeps its orientation along B and follows the change of the magnetic field direction.

If the magnetic field changes by | ~B| within the distance L and the velocity of the neutron

is v, equation (G.1) can be rewritten as

ωLL

v≫ 1. (G.2)

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138 Appendix

For the non-adiabatic transition the analogous condition reads:

ωL ≪ ωB (G.3)

and can be rewritten as

ωLL

v≪ 1. (G.4)

Adiabatic transition: The guide field from the monochromator to the coupling coil is ori-

ented in the vertical direction and is of the order of 30G. The change of the magnetic field

into the horizontal plane occurs along a length of 18cm. For a wavelength of λ = 2.3Å and

thus, a neutron velocity of v = 1675ms

equation (G.2) yields:

ωLL

v= 2916

HzG

· 30G 0.18m1675m

s

≈ 9.4 ≫ 1, (G.5)

which satisfies, to a good approximation, the adiabaticity condition. For larger wavelengths

the adiabaticity parameter is > 10.

Non-adiabatic transition: The maximum field in horizontal direction in the coupling coils is

30G. The maximum change of field from the inner part of the coupling coil to the shielded

region with a remaining field of Br = 5mG is therefore approximately 30G. The transmission

occurs within a distance equal to the wire thickness of 0.5mm. For a wavelength of λ = 6Å

and therefore a neutron velocity of v = 659ms

equation (G.4) yields:

ωLL

v= 2916

HzG

· 30G0.0005m659m

s

≈ 0.066 ≪ 1 (G.6)

Thus, the transition at the current sheet is non-adiabatic.

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Acknowledgments

Since a PhD thesis is no single combat, I want to thank all my friends and colleagues who

gave advice and support and made this time so special:

First of all, I want to thank my direct supervisor Dr. Klaus Habicht who welcomed me to

the HZB and gave me the opportunity to work on the FLEXX instrument. He shared all

his experience and took his time for all the invaluable discussions. I am especially grateful

for the opportunity to upgrade the NRSE option. Thanks for all the ideas, the support and

that you stayed even when the polarization already went home.

Special thanks to my supervisor Prof. Dr. Alan Tennant for giving me the opportunity to

perform this work. Thanks for the discussions and all the ideas.

Thanks to Prof. Dr. Peter Böni, who kindly accepted to be the second reviewer of my

thesis.

Many thanks to Dr. Katharina Rolfs for all the discussions, a lot of encouraged help during

the crystal growth, the motivation and the fun. I am really going to miss the coffee rein-

forced discussions.

Thanks to the whole FLEXX-Team, former and present members, for the nice and enjoyable

time I had as a part of this group. Especially I would like to thank Dr. Manh Duc Le for

sharing his Matlab knowledge, Dr. Kirrily Rule for all the explanations regarding triple

axis spectrometer and Dr. Markos Skoulatos for a really warm welcome to the group, nice

discussions and of course for the oregano.

Thanks to the coffee round, Dr. Katharina Rolfs, Dr. Daniil Nekrassov, Morten Sales, Dr.

Rasmus Toft-Peterson, Dr. Manh Duc Le, Dr. Mirko Boin, Dr. Robert Wimpory and Jen-

nifer Schulz, for all the delicate, sometimes unsettling, but always funny discussions during

the coffee breaks.

Thanks to Kathrin Buchner for all the patience and enthusiasm while manufacturing the

new, really nice bootstrap coils. Standing beside a lathe for 2 months, wearing a gas mask

and watching a coil slowly rotating can actually be very funny. It is not possible to com-

pensate your help with Haloren-Kugeln and Knusperflocken, although I tried.

Thanks to Dr. Thomas Keller for all the support during the beam time at TRISP. Thanks

145

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146 Acknowledgments

for all the helpful discussions and patiently sharing your expertise and knowledge about

NRSE.

Thanks to Bernd Urban for his tremendous support designing and manufacturing compo-

nents as well as setting up the NRSE instrument.

Thanks to Norbert Beul for designing and manufacturing the coupling coils and the com-

pensation coils.

Thanks to Dr. Diana Quintero-Castro for the Sr3Cr2O8 crystal and a lot of fruitful discus-

sions.

Thanks to the team of sample environment, especially Dr. Klaus Kiefer and Martin Petsche

for setting up the Attocube setup.

Thanks to Dr. Wolfgang Jauch for the γ-ray diffraction experiments on the Nb crystals.

Thanks to the staff of the HZB workshops, solving all the urgent problems and orders.

Thanks to Dr. Klaus Habicht, Dr. Katharina Rolfs, Dr. Manh Duc Le and Dr. Diana

Quintero-Castro for proof reading this thesis.

Many thanks to my good friend Marijke Haffke for supporting me with your funny and

calming calls throughout all the busy times.

Thanks to my friends Christine Walch, Steffen Walch and Hannelore Fiebig, who made

Berlin a home to me.

Thanks to my parents and my family for their support throughout my studies and for the

interest in my work.

Many thanks to my girlfriend Jule, who had to go without me during the beam time, the

coil winding and all the other busy times. Thanks for the love, the care and being the door

to the world outside the institute.

Last but not least, thanks to my coffee machine for three years of continuous and reliable

work keeping the level up.


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