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

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 forward, 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(NO ) 2.5D O and Sr Cr O . For this purpose high quality single crystals 3 2· 2 3 2 8 of Cu(NO ) 2.5D O were grown in the course of this thesis. As a proof of principle the 3 2· 2 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 enhanced. Extensive calibration measurements were performed and the results clearly demonstrate the reliable performance of the new NRSE option, now available for the broader user community at FLEXX.

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 dieser 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, werden 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(NO ) 2.5D O sowie an Sr Cr O durchgeführt. Hierfür wurden eigens 3 2· 2 3 2 8 hochwertige Cu(NO ) 2.5D O-Einkristalle gezüchtet. Die Ergebnisse zeigen deutlich, dass 3 2· 2 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 entwickelten Spektrometerarme kompaktere Instrument ermöglicht Kalibrationsmessungen im direkten Strahl für den gesamten zugänglichen Wellenvektor-Bereich. In Kombination mit 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. 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 Spinechophononfocusing...... 12 2.3 ExtendedNRSEresolutionfunction ...... 16 2.3.1 Generalized spin echo phase - violated spin echo conditions . . . . . 17 2.3.2 The τ dependenceofthepolarization...... 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- persionsurface ...... 31 2.3.5 Quantitative description of depolarization due to sample imperfections andcurvatureofthedispersionsurface ...... 35 2.3.6 Dispersion surface not coinciding with the center of the TAS resolution ellipsoid...... 36 2.3.7 Numericalexamples ...... 40 2.4 Experimentaltest...... 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 Experimentalsetup...... 54 3.3.2 Niobium dispersion models ...... 55

i 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(NO ) 2.5D O...... 70 3 2· 2 4.1.1 Properties of Cu(NO ) 2.5D O...... 70 3 2· 2 4.1.2 Sample deuteration and growth of single crystals ...... 71 4.1.3 InelasticNRSEmeasurements...... 73

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

4.2.1 Properties of Sr3Cr2O8 ...... 82 4.2.2 InelasticNRSEmeasurements...... 84 4.3 Summary ...... 88

5 Upgrade of the NRSE option at FLEXX 89 5.1 Bootstrapcoils ...... 90

5.1.1 B0 coils ...... 91 5.1.2 Cooling circuit ...... 94 5.1.3 RFcoils...... 95 5.2 Spinechoinstrumentarms...... 97 5.2.1 Magnetic shielding ...... 98 5.2.2 Coupling coils ...... 101 5.2.3 Motorsandencoders ...... 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 Spinechocurveandechopoint ...... 106 5.3.3 Phasestability ...... 111 5.3.4 Calibration of coil tilt angles ...... 112 5.4 Summary ...... 116

6 Conclusion and perspectives 119

Appendix 123

References 139 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 instruments. 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 understanding 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 spectrometers. 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 measurements 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 measurements. 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 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 developed 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 asymmetric line broadening. This effect has recently been observed in Cu(NO ) 2.5D O, a model 3 2· 2 system for a 1-D bond alternating Heisenberg chain [53], and Sr3Cr2O8, a 3-D gapped quantum 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 performed 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 signal 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 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. 4 Introduction 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 spectroscopy 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 ω = γB , where γ = 2π 2.913 103 Hz 1 1 · · G

5 6 2 NRSE resolution theory

Sample B1 B2 polarized A

P beam Detector

x z L1 L2 y 1

Polarization -1

Fig. 2.1: Top: Schematic NSE setup. The beam is polarized by a polarizer P. The average polarization decays in the ﬁrst spectrometer arm. For elastic scattering and equal ﬁeld integrals the polarization is restored in a spin echo after the second spectrometer arm (black) and analyzed by an analyzer A. For the inelastic case (red) the polarization at the echo point decreases as a function of the spin echo time τ and the phase is shifted. Bottom: The average beam polarization as a function of the instrument length for the elastic 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 ∆λ B = 300G, L = 1m, λ = 6Å and a wavelength distribution of λ = 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 = cos Φ , it is zero after the first field region. By scattering at h i 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 φ = ω t , resulting in a total angle of 2 − 2 1

γB1L1 γB2L2 ΦNSE = (2.1) v1 − v2 after the second spectrometer arm. For elastic scattering (v = v ), a symmetric instrument configuration ( B L = B L ) 1 2 | 1 1| | 2 2| and B~ = B~ , the phase φ will be zero for all velocities after the second field region. 1 − 2 NSE 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 2.1 Principle of inelastic neutron spin echo spectroscopy 7

neutron spin echo is the use of ΦNSE to determine the energy transfer m ~ω = v2 v2 . (2.2) 2 1 − 2 To get a quantitative relation between ΦNSE and ω, the scattering function S (ω) is considered 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 =v ¯1 + dv1 (2.3)

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

γB1L1 γB2L2 ΦNSE = (2.4) ~ v¯1 + dv1 − (¯v + dv )2 2 (ω + dω) 1 1 − m 0 q Expanding equation (2.4) to ﬁrst order in dv1 and dω yields:

γB L γB L v¯ ~ γB L Φ = Φ 1 1 2 2 1 dv 2 2 dω (2.5) NSE NSE,0 − 2 − 3 1 − m 2 (¯v1) (¯v2) (¯v2) where ΦNSE,0 corresponds to the mean velocities v¯1 and v¯2

γB1L1 γB2L2 ΦNSE,0 = (2.6) v¯1 − v¯2 and v¯2 is the mean neutron velocity after the sample

~ 2 v¯2 = (¯v1) ω0. (2.7) r − 2m If the so called spin echo condition

3 B1L1 (¯v1) = 3 (2.8) B2L2 (¯v2) holds, the phase ΦNSE becomes independent of the velocity spread dv1 and dΦNSE is proportional to dω: dΦ = Φ Φ = τ dω (2.9) NSE NSE − NSE,0 − NSE The instrumental constant ~γB1L1 ~γB2L2 τNSE = 3 = 3 (2.10) m (¯v1) m (¯v2) 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 I = 0 (1 + P ) (2.11) 2 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)

A = cos Φ = S (dω) cos (Φ dω τ ) d (dω) . (2.12) E h NSEi NSE,0 − · NSE Z As a classical example from quasi elastic scattering let the scattering function S (ω) be a Lorentzian line with a linewidth of Γ:

Γ S (ω) (2.13) ∝ Γ2 + ~2ω2

Equation (2.12) then yields:

S (ω)cos(ωτNSE) dω ΓτNSE A (τ )= = e− ~ (2.14) E NSE S (ω) dω R R 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 B~ 0 pointing in the z-direction and a rotating field B~ RF in the xy-plane with the frequency ωRF . The superposed magnetic field is described by the vector

B~ 1 (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 2.2 Principle of neutron resonance spin echo 9

B1 x

Fig. 2.2: Principle of the π-coil. The incoming neutron polarization vector (green) precesses around the effective magnetic field B1 and leaves the coil πv0 with a new orientation (blue). Setting B1 = γd the 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]: ω B~ = B , 0,B RF . (2.16) 1 RF 0 − γ 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: πv B = 0 (2.17) RF γd 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 B~ . Simultaneously B~ precesses around the z-axis by an angle of ω d . 1 1 RF v0 Thus, after leaving the coil the phase of the magnetic moment is:

d Φ1 = 2ωRF t1 + ωRF Φ0 (2.18) v0 − 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 + B1 parts B1 and B1− with an amplitude of 2 . Only the part rotating in the same direction as the magnetic moment of the neutron contributes to the π-flip. The counter rotating part 10 2 NRSE resolution theory can be neglected for B B . This causes a small shift of the resonance frequency, the so 0 ≫ 1 called “Bloch-Siegert-Shift” [11] 2 ∆ω BRF = 2 , (2.19) ω 4B0 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: π2 ∆v 2 A = 1 M (2.20) E − 4 v 0 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 B~ 1 and are driven with a frequency of ω1 = γB1 while the coils C and D after the sample region have the static

field B~ 2 with a frequency of ω2 = γB2. The two fields B~ 1 and B~ 2 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

d ΦA = 2ω1tA + ω1 . (2.21) v1

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 t = t + L1 . Therefore, B A v1 the phase angle of the magnetic moment after the second coil is:

d L1 ΦB = 2ω1tB + ω1 ΦA = 2ω1 . (2.22) v1 − v1 2.2 Principle of neutron resonance spin echo 11

zero-field regions

Spin

x z y AB CD

L1 L2 d sample region

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

Φ becomes independent from the time t and only depends on the time of ﬂight L1 between B A v1 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

d ΦC = 2ω2tC ω2 ΦB. (2.23) − − v2 −

Hence, after the last coil D the phase is

d ΦD = 2ω2tD ω2 ΦC − − v2 − L1 L2 (2.24) = 2ω1 2ω2 v1 − v2

= ΦNRSE = 2ΦNSE. 12 2 NRSE resolution theory

Therefore, the previous discussion on spin echo applies directly to NRSE if B1 and B2 ω1 ω2 are substituted by 2 γ and 2 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

L L Φ = 2N ω 1 ω 2 = NΦ (2.25) N 1 v − 2 v NRSE 1 2 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 ∆λ and a broad wavelength distribution λ = 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 2.2 Principle of neutron resonance spin echo 13

ω0

q0 q Fig. 2.4: The resolution function of the TAS selects a certain region in the (q,ω)-space. The lines of constant Larmor phase (Φ = const, pale blue) have to be parallel to the dispersion curve. This can be reached by inclining 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) ω (q)) . (2.26) − 0 In order to measure equation (2.26) the Larmor precession angle needs to be proportional to (ω (q) ω (q)): − 0 Φ Φ = τ (ω (q) ω (q)) = τ∆ω. (2.27) − 0 − − 0 − 14 2 NRSE resolution theory

precession field regions i0 B1 j1 j0 l i1 kI sample kI 0

l1 ti nI Θ B2 1 nI si ni L1 kF Q L2

kF nF Θ2 j2

nF l2 sf tf i2 nf Fig. 2.5: Schematic drawing of the tilted precession ﬁeld regions in an inelastic NSE setup. The deﬁned Cartesian coordinate systems are used throughout.

Assuming a planar dispersion and expanding the dispersion to ﬁrst order in (q q ) yields: − 0

∆ω = ω (q) ω (q)= ω (q) [ω (q )+(q q ) qω (q0)] . (2.28) − 0 − 0 0 − 0 ·∇

Using momentum and energy conservation with mean wavevectors kI and kF before and after the scattering:

ki = kI +∆ki and kf = kF +∆kf ~2 ~2 (2.29) ω(q)= k2 k2 and ω (q)= k2 k2 , 2m i − f 0 2m I − F ∆ω is expressed as a function of the deviations from the mean wavevectors ∆ki and ∆kf : ~ ∆ω = ω (q) ω (q)= [(k ∆k ) (k ∆k )] . (2.30) − 0 m I · i − F · f Using the relation for the momentum in a perfect crystal

q = k k G and q = k k G (2.31) i − f − 0 0 I − F − 0 where G0 is the reciprocal lattice vector and combining equations (2.28) and (2.30) yields: ~ ~ ∆ω =∆k k qω (q ) ∆k k qω (q ) (2.32) i m I −∇ 0 0 − f m F −∇ 0 0 2.2 Principle of neutron resonance spin echo 15

The Larmor precession angle after two tilted ﬁeld regions is

m ωL1L1 cosΘ1 ωL2L2 cosΘ2 Φ = Φ0 +∆Φ = ~ ki cosΘ1 − kf cosΘ2 (2.33) m ω L cosΘ ω L cosΘ A A = L1 1 1 L2 2 2 = 1 2 ~ k n − k n k n − k n i · i f · f i · i f · f 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

A1 A2 Φ0 = (2.34) k n − k nf I · i F · and expanding the total Larmor precession angle to ﬁrst order yields

A1 A2 Φ Φ0 = (∆ki ni)+ (∆kf nf ) . (2.35) − −(k n )2 · (k n )2 · I · i F · f For equation (2.27) to hold, the coeﬃcients in equations (2.32) and (2.35) have to be compared. The result for the normal vectors of the precession ﬁeld boundaries is:

~ ~ m kI,F qω0(q0) m kI,F qω0(q0) ni,f = ~ −∇ = −∇ (2.36) k qω (q ) NI,F m I,F −∇ 0 0 and for the spin echo time

A1,2 A1,2 τ = ~ = . (2.37) (k n ) k qω (q ) (kI,F ni,f ) NI,F I,F · i,f m I,F −∇ 0 0 ·

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

2 ~ 2 ω L cosΘ2 (kI ni) kI qω0(q0) cosΘ (k n ) N L1 1 = · m −∇ = 2 I · i I . (2.38) ω L 2 ~ 2 L2 2 cosΘ1 (kF nf ) m kF qω0(q0) cosΘ1 (kF nf ) NF · −∇ ·

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 16 2 NRSE resolution theory with a background TAS, is given by [6]

1 A = S (Q,ω) R (k , k ) eiΦ(ki,kf )d3k d3k + c.c. (2.39) E N T AS i f i f Z where RT AS (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

A S (∆ω) R (∆ω) eiΦd∆ω + c.c. (2.40) E ∝ T AS Z with ∆ω given by equation (2.28). The application of spin echo spectroscopy is only reasonable if the linewidth S (∆ω) of the excitation is much smaller than the energy resolution

T (∆ω) of the background TAS. Therefore, RT AS (∆ω) is assumed to be constant. Thus, the result for the echo amplitude is again the cosine Fourier transform of the line shape:

A S (∆ω)cos(τ∆ω) d∆ω (2.41) E ∝ Z 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(q ) may have components out of the ∇ c0 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. 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 treatment 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, depolarization 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:

A1 A2 Φ(ki, kf ) Φ0 = 2 (∆ki ni)+ 2 (∆kf nf ) − − (kI ni) · (kF nf ) · · · (2.42) A1 2 A2 2 + (∆ki ni) (∆kf nf ) . (k n )3 · − (k n )3 · I · i F · f 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)

∆k =∆k n (2.45) in i · i 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

1 T ω (q )=∆ω + ω (q )+∆q ω (q )+ ∆q H q ∆q (2.46) c c0 c0 ·∇ c0 2 c c| c0 c 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′ =∆q ∆G =∆k ∆k ∆G (2.47) c c − c i − f − c 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) ·∇ − ·∇ . (2.50) 1 T ∆G ω (q )+ ∆q′ H q ∆q′ − c ·∇ c0 2 c c| c0 c 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 (q ) = k2 k2 (2.51) c 2m i − f ~ ~ = ω (q )+ k ∆k k ∆k c0 m I · i − m F · f ~ ~ + ∆k2 ∆k2 (2.52) 2m i − 2m f 2.3 Extended NRSE resolution function 19 using q = k k (2.53) c0 I − F and ~ ω (q )= k2 k2 . (2.54) c0 2m I − F Combining equations (2.50) and (2.51) yields

~ ~ ∆ω = k ω (q ) ∆k k ω (q ) ∆k m I −∇ c0 · i − m F −∇ c0 · f ~ ~ 2 2 1 T + ∆k ∆k +∆Gc ω (qc0) ∆q′ Hc ∆q′ . (2.55) 2m i − 2m f ·∇ − 2 c |qc0 c Now unit vectors are deﬁned as

~ ~ kI ω (qc0) kI ω (qc0) ǫ = m −∇ = m −∇ (2.56) i ~ N m kI ω (qc0) I ~ −∇ ~ k ω (q ) k ω (q ) ǫ m F c0 m F c0 f = ~ −∇ = −∇ . (2.57) k ω (q ) NF m F −∇ c0

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:

A1,2 A1,2 τ1,2 = ~ = . (2.58) (k n )2 k ω (q ) (∆k n )2 N I,F · i,f m I,F −∇ c0 I,F · i,f I,F

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: 20 2 NRSE resolution theory

A2 ~ τ2∆ω = kI ω (qc0) ∆ki k n 2 m −∇ · ( F f ) NF · ~ A2 m kF ω (qc0) 2 −∇ ∆kf −(kF nf ) NF · ~ · ~ + τ ∆k2 τ ∆k2 2m 2 i − 2m 2 f 1 T +τ2∆Gc ω (qc0) τ2∆q′ Hc ∆q′ (2.59) ·∇ − 2 c |qc0 c A2 NI ǫ A2 ǫ = 2 i ∆ki 2 f ∆kf (kF nf ) NF · − (kF nf ) · ~· ~ · + τ ∆k2 τ ∆k2 2m 2 i − 2m 2 f 1 T +τ2∆Gc ω (qc0) τ2∆q′ Hc ∆q′ . (2.60) ·∇ − 2 c |qc0 c

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 s = n t . Note that the unit vectors s are identical i,f i,f × i,f i,f since they are both perpendicular to the scattering plane. The components ei,f1, ei,f2 and ei,f3 are defined by

e = ǫ n e = ǫ t e = ǫ s (2.62) i1 i · i i2 i · i i3 i · i e = ǫ n e = ǫ t e = ǫ s . (2.63) f1 f · f f2 f · f f3 f · f As a generalization compared to [6], the case where the gradient vector of the dispersion ω (q ) has a component out of the scattering plane is included. Inserting equation (2.61) ∇ c0 in equation (2.59) yields 2.3 Extended NRSE resolution function 21

A2 ∆kf nf = τ2′′∆ω + τ2′′NI ei1∆kin (k n )2 · − F · f 1 +τ ′′N e ∆k tan θ + y 2 I i2 − in 1 1 cos θ 1 +τ ′′N e z τ ′′N e z 2 I i3 1 − 2 F f3 2 1 τ2′′NF ef2 y2 − cos θ2 ~ ~ 2 2 + τ ′′∆k τ ′′∆k 2m 2 i − 2m 2 f 1 T +τ ′′∆Gc ω (qc0) τ ′′∆q′ Hc ∆q′ (2.64) 2 ·∇ − 2 2 c |qc0 c with τ2 τ ′′ = . (2.65) 2 (e e tan θ ) f1 − f2 2 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:

s ∆k = z (2.66) i · i 1 s ∆k = z (2.67) f · f 2 n ∆k = ∆k . (2.68) i · i in

∆k = (x cos θ + y sin θ ) n +( x sin θ + y cos θ ) t + z s (2.69) i 1 1 1 1 i − 1 1 1 1 i 1 i 1 = ∆k n + ∆k tan θ + y t + z s (2.70) in i − in 1 1 cos θ i 1 i 1 1 ∆k = (∆k n ) n + (∆k n )tan θ + y t + z s . (2.71) f f · f f − f · f 2 2 cos θ f 2 f 2 1 ti ∆ki = ∆kin tan θ1 + y1 (2.72) · − cos θ1 and 1 tf ∆kf = (∆kf nf )tan θ2 + y2 . (2.73) · − · cos θ2

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 required to provide the Fourier transform of the scattering function S (Q,ω): 22 2 NRSE resolution theory

φ(k , k ) φ = τ N ∆k τ ′′∆ω + τ ′′N e ∆k i f − 0 − 1 I in − 2 2 I i1 in 1 +τ ′′N e ∆k tan θ + y + τ ′′N e z 2 I i2 − in 1 1 cos θ 2 I i3 1 1 1 τ2′′NF ef2 y2 τ2′′NF ef3z2 − cos θ2 − 1 T +τ ′′∆Gc ω (qc0) τ ′′∆q′ Hc ∆q′ 2 ·∇ − 2 2 c |qc0 c ~ ~ 2 2 + τ ′′∆k τ ′′∆k 2m 2 i − 2m 2 f N N +τ I ∆k2 τ F (∆k n )2 . (2.74) 1 k n in − 2 k n f · f I · i F · f 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 (∆k n )2 is now substituted in equation (2.74) by using equation (2.64). Since f · f only second order effects are considered and higher order terms are neglected, it is sufficient to consider equation (2.64) to first order only:

1 NI Ci ∆kf nf = ∆ω + ∆kin · −Cf NF NF Cf NI 1 1 1 + ei2 y1 ef2 y2 Cf NF cos θ1 − Cf cos θ2 NI 1 1 + ei3z1 ef3z2 + ∆Gc′ ω (qc0) (2.75) Cf NF − Cf Cf NF ·∇ using

A2 τ2 = (2.76) (k n )2 N f · f F C = e e tan θ (2.77) i i1 − i2 1 C = e e tan θ . (2.78) f f1 − f2 2 2.3 Extended NRSE resolution function 23

1 The term + ∆G′ ω (qc0) considers only the ﬁrst order terms arising from the lattice Cf NF c ·∇ 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). 2 2 ∆Gc′ is deﬁned in equation (2.99). ∆ki and ∆kf are substituted in equation (2.74) using equations (2.69) and (2.71):

1 2 ∆k2 =∆k2 + ∆k tan θ + y + z2 (2.79) i in − in 1 1 cos θ 1 1 1 2 ∆k2 = (∆k n )2 + (∆k n )tan θ + y + z2. (2.80) f f · f − f · f 2 2 cos θ 2 2 Inserting equations (2.79) and (2.80) into equation (2.74) and using again equation (2.75) to substitute all ∆k n terms allows to express the total Larmor precession angle as a f · f 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]:

T 1 T φ(k , k ) φ = τ ′′T J τ ′′J ΨJ i f − 0 2 − 2 2 +τ ′′∆G ω (q ) 2 c ·∇ c0 1 T τ ′′∆q′ Hc ∆q′ + X(∆Gc) (2.81) −2 2 c |qc0 c

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:

τ N e T = 1 T = N C 1 C T = I i2 (2.82) 1 − 2 I i − τ f 3 cos θ 2 1 NF ef2 T4 = T5 = NI ei3 T6 = NF ef3. (2.83) − cos θ2 −

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) 24 2 NRSE resolution theory and thus

τ2′′ = τ2 (2.87)

τ1 = τ2. (2.88)

The matrix Ψ then reduces to the matrix given in [6]. Since only T = 1 remains non-zero, 1 − 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 (∆G = 0) and a planar dispersion (H q = 0) is c c| c0 derived. With these assumptions equation (2.81) reduces to

T 1 T φ(k , k ) φ = τ ′′T J τ ′′J ΨJ. (2.89) i f − 0 2 − 2 2 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:

1 R (k , k ) = exp JT L J . (2.90) T AS i f −2 T AS Substituting the equations (2.89) and (2.90) into the fundamental equation (2.39) yields:

1 T 1 T 6 A = S (Q,ω)exp iτ ′′T J exp J L J d J + c.c. (2.91) E N 2 −2 I n Z with

LI = LT AS + iτ2′′Ψ. (2.92)

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

1 A = S (Q,ω)exp iτ ′′∆ω d∆ω E N − 2 Z T 1 T 5 exp iτ ′′T J exp J L J d J (2.93) × 2 −2 I n Z e e e e e 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 . e e × Following [6] and using the general matrix theorem [18] e ∞ 1 exp KT J exp JT MJ dnJ −2 n Z −∞ n/2 (2π) 1 T 1 = exp K M− K (2.94) 1/2 −2 (det M) the resolution function is obtained

det LI (τ2′′ = 0) 1 2 T 1 FI τ2′′ = exp τ2′′ T LI− τ2′′ T . (2.95) s det LI (τ ) −2 2′′ e e e e For strongly violated spin echo conditionse 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. e

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 RbMnF at the zone boundary excitation Q = [0.5 0.5 1], E = 8.287meV, for 3 − 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 obtained 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 dΘ = 9 would decrease the echo amplitude by about 3%. 1,2 ± ◦ 26 2 NRSE resolution theory

1 1

0.8 0.8

0.6 0.6

0.4 0.4 echo amplitude echo amplitude

0.2 0.2 a b 0 0 −40 −20 0 20 40 −40 −20 0 20 40 dΘ [deg] dΘ [deg] 1 2

1 1

0.8 0.8

0.6 0.6

0.4 0.4 echo amplitude echo amplitude

0.2 0.2 c d 0 0 −40 −20 0 20 40 −40 −20 0 20 40 df [kHz] df [kHz] 1 2

Fig. 2.6: Depolarization eﬀects due to a detuning of spin echo parameters at 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 f1 3. For τ = 12.4ps a de- f2 ≈ tuning of df = 25kHz and df = 8kHz would decrease the echo amplitude by about 3%. 1 ± 2 ± 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 frequency. 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 dΘ = 1.5 would de- 1,2 ± ◦ 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 f and 10kHz for f ± 1 ± 2 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 f is assumed to be detuned to f . A numerical example for the zone 2 − 2 boundary excitation Q = [0.5 0.5 1], E = 8.287meV, in RbMnF is shown in Fig. 2.8. − 3 2.3 Extended NRSE resolution function 27

1 1

0.8 0.8

0.6 0.6

0.4 0.4 echo amplitude echo amplitude

0.2 0.2 a b 0 0 −40 −20 0 20 40 −40 −20 0 20 40 dΘ [deg] dΘ [deg] 1 2

1 1

0.8 0.8

0.6 0.6

0.4 0.4 echo amplitude echo amplitude

0.2 0.2 c d 0 0 −40 −20 0 20 40 −40 −20 0 20 40 df [kHz] df [kHz] 1 2

Fig. 2.7: Depolarization eﬀects due to a detuning of spin echo parameters at 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.

0.8

0.6

0.4 echo amplitude 0.2

0 0 5 10 15 20 τ [ps]

Fig. 2.8: Depolarization effects for spin-flip scattering without reversing the second field region. The example shown applies to the zone boundary excitation Q = [0.5 0.5 1],E = 8.287meV, in RbMnF3. − 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 1 | ∇ | an incident wavevector of ki = 1.7Å− 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.

1 SM=−1 SS=−1 SA=−1 SM=−1 SS=−1 SA=+1 0.8 SM=−1 SS=+1 SA=−1 SM=−1 SS=+1 SA=+1

0.6

0.4 echo amplitude

0.2

0 0 1000 2000 3000 4000 τ [ps]

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

2.3.3 Quantitative description of depolarization due to sample imperfections

In this section the additional term +τ ∆G ω (q ) in the total Larmor phase given by 2′′ c ·∇ c0 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. 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.

Δω

Δq

lattice vector G

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

Correction for sign errors and using the correct second order term rather than that stated in [6] yield:

∆G = G G (2.96) c c − c0 (Gc0 +∆Gc)cos∆ν cos∆η   = (Gc0 +∆Gc)cos∆ν sin∆η . (2.97)      (Gc0 +∆Gc)sin∆ν      Expanding small variations in the lattice vector Gc and neglecting terms higher than second order leads to

∆G 1 G ∆ν2 +∆η2 c − 2 c0   ∆Gc = Gc0∆η +∆ Gc∆η . (2.98)      Gc0∆ν +∆Gc∆ν      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 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:

∆Gc   ∆Gc′ = G0c∆η (2.99)      G0c∆ν      1 G ∆ν2 +∆η2 − 2 0c   ∆Gc′′ = ∆Gc∆η . (2.100)      ∆Gc∆ν      The complex resolution matrix can now be written as

1 LM = I− LI I + 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 (q ). The non-zero elements of the (9 9) matrix N are: ∇ c0 × 1 1 1 N77 = 2 , N88 = 2 , N99 = 2 . (2.104) ΥS ηS νS

Here ΥS is the 1σ standard deviation for the Gaussian distribution of lattice vectors. The quadratic terms in ∆η, ∆ν and ∆G are taken into account by the symmetric (9 9) matrix c × W. Its non-zero elements are:

W = W = iτ ′′C , W = W = iτ ′′C (2.105) 78 87 − 2 y 79 97 − 2 z W88 =+iτ2′′CxG0c, W99 =+iτ2′′CxG0c. (2.106)

Note that equation (2.106) is the corrected version of equation (83) in [6]. The linear terms T in ∆η, ∆ν and ∆Gc arising from ∆Gc′ can be written as +iτ2′′Tg JM , with the column 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

τ1 NI ei2 NF ef2 1,NI Ci Cf , , ,NI ei3, NF ef3, T = − − τ2 cos θ1 − cos θ2 − . (2.109) M   Cx, CyG0c, CzG0c  − −    Analog to the treatment in the previous subsection, terms in ∆ω higher than linear are neglected, leading to

1 A = S (Q,ω)exp iτ ′′∆ω d∆ω (2.110) E N − 2 Z T 1 T 8 exp iτ ′′T J exp J L J d J , (2.111) × 2 M M −2 M M M n Z e e e e e 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 . e e × Applying the general matrix theorem (2.94), leads to the resolution function in the form e

det LM (τ2′′ = 0) 1 2 T 1 FM τ2′′ = exp τ2′′ TM LM− τ2′′ TM , (2.112) s det LM (τ ) −2 2′′ e e e e which includes general lattice imperfections.e 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

1 T In this subsection the additional term τ ′′∆q′ Hc ∆q′ in the total Larmor phase given − 2 2 c |qc0 c 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 depolarization eﬀects, since neutrons corresponding to diﬀerent points on the dispersion surface 32 2 NRSE resolution theory

1 η =1’ 0.9 S η =2.5’ S 0.8 η =5’ S 0.7

0.6

0.5

0.4 echo amplitude 0.3

0.2

0.1

0 0 50 100 150 200 250 300 τ [ps]

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 expressed in the Cartesian coordinate system related to the reciprocal lattice by the B-matrix [16]. The elements of the Hessian H are deﬁned by

∂2 Hijc = ω (qc) i,j = 1, 2, 3. (2.113) ∂qic∂qjc

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

∂2 Hij = ω (q) , (2.114) ∂qi∂qj and the Hessian H needs to be transformed into Cartesian coordinates using the B matrix:

1 Hc = BHB− . (2.115)

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

∆Gc = 0, ∆qc′ reduces to ∆qc and LI and J are used. Terms arising from sample imperfections will be included in the next subsection. Following [6], the modiﬁed matrix reads:

1 1 LC = LI + iτ ′′I− Θ− Hc ΘC IC (2.116) 2 C C |qc0 2.3 Extended NRSE resolution function 33

where the matrix ΘC IC transforms the Hessian Hc into the coordinate space of LI . The matrix ΘC describes the transformation

∆qθ = ΘC Y (2.117) with the column vector Y = (x1,y1,z1,x2,y2,z2). The variation of the wavevector ∆qθ = ∆k ∆k is expressed in the basis of the Cartesian system (i , j and l ). In this coordinate i − f 0 0 0 system the total wavevector transfer Qθ is parallel to i0 (see Fig. 2.5):

QM 2π   Qθ = 0 . (2.118)      0      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

∆k =(x cos φ y sin φ) i +(x sin φ + y cos φ) j + z l (2.119) i 1 − 1 0 1 1 0 1 0 ∆k =(x cosΞ y sinΞ) i +(x sinΞ + y cosΞ) j + z l (2.120) f 2 − 2 0 2 1 0 2 0 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

cos φ sin φ 0 cosΞ sinΞ 0 − −   ΘC = sin φ cos φ 0 sinΞ cos Ξ 0 . (2.121)  − −     0 010 0 1   −    Note that here the definitions of φ and Ξ are different to [6] and the sign errors in the T definition of ΘC are corrected. In order to evaluate the expression ∆q Hc ∆qc, the c |qc0 Hessian Hc needs to be rotated into Hθ. According to [17] the transform of the vector Q is defined as

Qθ = ΩMNUBQ (2.122) 34 2 NRSE resolution theory and that of the matrix H is

1 1 1 1 1 Hθ = ΩMNUBHB− U− N− M− Ω− , (2.123)

T T since Qθ HθQθ = Q HQ 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 = IC J. (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 = k ω (q ) ∆k k ω (q ) ∆k (2.125) m I −∇ c0 · i − m F −∇ c0 · f = N ǫ ∆k N ǫ ∆k . (2.126) I i · i − F f · f

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

∆ω = x (N e cos θ N e sin θ ) 1 I i1 1 − I i2 1 +y1 (NI ei1 sin θ1 + NI ei2 cos θ1)+ ei3NI z1 x (N e cos θ N e sin θ ) − 2 F f1 2 − F f2 2 y (N e sin θ N e cos θ ) e N z (2.127) − 2 F f1 2 − F f2 2 − f3 F 2 and

∆kin = x1 cos θ1 + y1 sin θ1. (2.128)

According to equation (2.124) the transformation reads

1 J = Ic− Y. (2.129)

1 The elements of the matrices IC− 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 2.3 Extended NRSE resolution function 35 curved dispersion surface is given by

T T 1 T 1 1 ∆q Hc ∆qc = Y Θ− Hc ΘC Y = J I− Θ− Hc ΘC IC J. (2.130) c |qc0 C |qc0 C C |qc0 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

1 A = S (Q,ω)exp iτ ′′∆ω d∆ω E N − 2 Z T 1 T 5 exp iτ ′′T J exp J L J d J . (2.131) × 2 −2 C n Z e e e e e Using equation (2.94) the resolution function taking into account curvature eﬀects and neglecting contributions from sample imperfections is then given by

det LC (τ2′′ = 0) 1 2 T 1 FC τ2′′ = exp τ2′′ T LC− τ2′′ T . (2.132) s det LC (τ ) −2 2′′ e e e e e

2.3.5 Quantitative description of depolarization due to sample imperfections and curvature of the dispersion surface

The results of the previous two subsections can now be combined to give the resolution matrix T LMC = I LC I + N + W (2.133)

The resolution function can be expressed as

det LMC (τ2′′ = 0) 1 2 T 1 FMC τ2′′ = exp τ2′′ TM LMC− τ2′′ TM (2.134) s det LMC (τ ) −2 2′′ e e e e This resolution function FMC eincludes 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)

0.9 0.8

0.7

0.6

0.5

0.4 echo amplitude 0.3 only instr 0.2 instr + sample 0.1 instr + curv instr + sample + curv 0 0 20 40 60 80 100 τ [ps]

Fig. 2.12: Numerical results for different depolarizing effects for the [2 0.1 0] TA phonon in Pb. Considering only instrumental resolution (black), instrumental resolution and sample imperfections (red), instrumental resolution and curvature of the dispersion surface (blue). The depolarization due 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 resolution 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 kF 0 and an energy of ~ω0T AS. The wavevector transfer is

Q = k k (2.137) 0 I0 − F 0 2.3 Extended NRSE resolution function 37

and the TAS resolution ellipsoid is centered at the energy of (Q0,ω0T AS), where ~ ω (k , k )= k2 k2 . (2.138) 0T AS I0 F 0 2m I0 − F 0 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 ∆ΩT AS:

∆ω ∆ω ∆Ω (2.140) → − T AS where ∆Ω = ω ω (Q ) . (2.141) T AS 0T AS − 0S 0 The resolution matrix reads

T LMC = I LC I + N + W (2.142) T 1 1 = I LT AS + iτ ′′Ψ + iτ ′′I− Θ− Hc ΘC IC I + N + W. (2.143) 2 2 C C |qc0 Since only the TAS transmission function changes, the following substitution has to be made

JT IT L IJ JT IT L IJ (2.144) T AS → T AS T AS T AS with the substitution

J = J J′=(∆ω ∆Ω , ∆k ,y ,y ,z ,z ), (2.145) T AS − − T AS in 1 2 1 2 where J′ = (∆Ω, 0, 0, 0, 0, 0). Therefore, the new TAS resolution matrix is deﬁned as

T LS = I LT ASI. (2.146)

T T The Matrix LS is symmetric and therefore J′ LSJ = J LSJ′. Using this symmetry yields

T T T T J L J = J L J 2J L J′ + J′ L J′ (2.147) T AS S T AS S − S S = JT L J 2∆Ω (L ) J +(L ) ∆Ω2 . (2.148) S − T AS S 1n S 11 T AS 38 2 NRSE resolution theory

where (LS)1n deﬁnes the 6 dimensional row vector of the matrix LS and (LS)11 is the 2 (1,1) element of the matrix LS. The term proportional to (LS)11 ∆ΩT AS only produces a constant, which can be absorbed in the normalization factor N. With

1 1 JT L J = JT L J + ∆Ω (L ) J −2 T AS MC,T AS T AS −2 MC T AS S 1n 1 (L ) ∆Ω2 (2.149) −2 S 11 T AS the modiﬁed expression for the polarization reads

1 A = S (Q,ω)exp iτ ′′∆ω + ∆Ω (L ) ∆ω d∆ω E N − 2 T AS S 11 Z 1 exp TT J exp JT L J d5J . (2.150) × T AS −2 MC n Z Here the deﬁnition e e e e e

TT AS = iτ2′′TM + ∆ΩT AS LS . (2.151) 1n was used. Therefore, the ﬁnale result fore the resolution function,e including its normalization to 1 at τ = 0, is

′′ det LMC (τ2 =0) e ′′ det LMC (τ2 ) r e 1 T 1 FMC,T AS τ2′′ = exp T L− (τ ) T . (2.152) × 2 T AS MC 2′′ T AS

1 T 1 exp 2 TT ASe(0) LeMC− (τ2′′ =e 0) TT AS (0) × −

Since e e e

T 1 2 T 1 exp T L− T = exp τ ′′ T L− T T AS MC T AS − 2 M MC M T 2 1 e e e exp +∆Ωe Te AS LeS LMC− LS × 1n 1n 1 exp +i2τ2′′∆ΩT ASe TM LeMC− LeS , (2.153) × 1n e e e 2.3 Extended NRSE resolution function 39 equation (2.152) yields

′′ det LMC (τ2 =0) e ′′ det LMC (τ2 ) r e T 1 2 1 exp ∆ΩT AS LS LMC− (τ2′′) LS × 2 1n 1n T FMC,T AS τ ′′ = 1 2 1 . (2.154) 2 exp ∆ΩT AS LeS LeMC− (τ2′′ = 0)e LS × − 2 1n 1n 1 2 T 1 exp τe T Le− (τ ) T e × − 2 2′′ M MC 2′′ M

2 T 1 exp iτ2′′ ∆ΩT ASeTMeLMC− (τ2′′)e LS × 1n

Note that the phase term arising from e e e

1 T 1 exp T (0) L− τ ′′ = 0 T (0) (2.155) −2 T AS MC 2 T AS T 1 e 2 e 1 e = exp ∆ΩT AS LS LMC− τ2′′ = 0 LS (2.156) −2 1n 1n e e e is of no interest in this context, since the term is independent from τ2′′ 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 (∆Ω (L ) ∆ω) S (∆ω)exp iτ ′′∆ω d∆ω. (2.157) T AS S 11 − 2 Z This term

exp (∆ΩT AS (LS)11 ∆ω) (2.158) multiplied with the scattering function introduces a small asymmetry. However, in practical cases ∆Ω (L ) ∆ω 1 (2.159) T AS S 11 ≪ Thus, the exponential

exp (∆ΩT AS (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 ∆ΩT AS with respect to the excitation 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 40 2 NRSE resolution theory automatically to a detuning of the spin echo parameters. A detuning of the TAS effectively corresponds to a new Q0′ , which can be derived using the UB matrix formalism described in Appendix A. Q0′ 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 ∆ΩT AS. 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 calculations 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 k and k . • I F ν and µ: These parameters are the angular variables of the sample goniometers. • ∆Ω : A detuning in ∆Ω corresponds to a shift of the center of the TAS reso- • T AS T AS 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. 2.3 Extended NRSE resolution function 41

Detuning the sample angle A3 and the scattering angle A4

1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4 echo amplitude echo amplitude 0.3 0.3

0.2 dA3 = 0° 0.2 dA4 = 0° dA3 = 0.1° dA4 = 0.1° 0.1 dA3 = 0.5° 0.1 dA4 = 0.5° dA3 = 1° dA4 = 1° 0 0 0 20 40 60 80 0 20 40 60 80 τ [ps] τ [ps]

Fig. 2.13: Calculated depolarization eﬀects for RbMnF3 at the zone boundary excitation Q = [0.5 0.5 1], E = 8.457meV, arising from a detuned 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 depolarizing 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 depolarizing 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

1 1

0.8 0.8

0.6 0.6

0.4 0.4 ν µ echo amplitude echo amplitude d =0° d =0° ν µ 0.2 d =0.5° 0.2 d =0.5° dν=1° dµ=1° dν=2° dµ=2° 0 0 0 20 40 60 80 0 20 40 60 80 τ [ps] τ [ps]

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

0.9

0.8

0.7

0.6

0.5

0.4

echo amplitude ∆Ω = 0 meV 0.3 TAS ∆Ω = 0.1 meV TAS 0.2 ∆Ω = 0.5 meV TAS 0.1 ∆Ω = 1 meV TAS 0 0 20 40 60 80 τ [ps]

Fig. 2.15: Calculated depolarization eﬀects for RbMnF3 at the zone boundary excitation Q = [0.5 0.5 1], E = 8.287meV, arising from a shift of 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 ∆ΩT AS = 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 ∆ΩT AS = 0.5meV the depolarizing effects become significant. For ∆ΩT AS = 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 f = 50kHz frequency of the RF coils. If the instrument ≈ RF 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). 2.4 Experimental test 43

Comparison of diﬀerent detuned TAS parameters

0.9

0.8

0.7

0.6

0.5

0.4 echo amplitude 0.3 no detuning 0.2 dΩ=0.5meV 0.1 dA3=0.5° dA4=0.5° 0 0 20 40 60 80 τ [ps]

Fig. 2.16: Depolarization eﬀects for RbMnF3 at the zone boundary excitation Q = [0.5 0.5 1], E = 8.287meV, for diﬀerent detuned TAS parameters. −

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 (∆ΩT AS) 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 ∆ΩT AS. 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 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 II of HZB, Berlin, with its spin echo option mounted (before the upgrade in 2010-2011).

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 1 incident wavevector kI = 1.7Å− . 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]. 2.4 Experimental test 45

240

220

200

180

160

Counts per monitor 140

120

100 10 15 20 25 dl2 [mm]

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.

200

100

−100

−200

−300 phase shift [°]

−400

−500

−600 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 ∆ A3 [°]

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

Using a simplified approach of a first-order expansion of the Larmor phase, allows to extract dω the slope of the dispersion dq from the linear coefficient of the phase dependence. A linear 46 2 NRSE resolution theory

dω fit [6] gives dq = 6.42 (14) meVÅ. Though using a full force-constant parameterization of the dispersion including three-body interactions [22] gives a significantly higher value of dω dq = 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. 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 simultaneously, 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 resolution 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 48 3 NRSE investigations on split modes

plitude AE is described by the Fourier transform of two Lorentzians. Including the relevant energy oﬀsets yields:

1 A = S (∆ω ω ) R (∆ω ω )exp( i (∆ω ω ) τ) d∆ω . (3.1) | E| N − 0S1 T AS − 0T AS − − 0NRSE Z

Here S (∆ ω ω ) is the scattering function describing two Lorentzian profils, given by − 0S1 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 2 2 ~ (∆ω ω0T AS) RT AS (∆ω ω0T AS) = exp 4ln2 −2 , (3.2) − − ET AS ! where the TAS resolution ellipsoid is centered at ω0T AS with a FWHM of ET AS. 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

Γ1 Γ2 S (∆ω ω0S1)= A1 + A2 , (3.3) − Γ2 + (∆ω ω )2 Γ2 + (∆ω ω ∆Ω)2 1 − 0S1 2 − 0S1 − 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 proportional 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

1 A = S (∆ω) R (∆ω (ω ω ))exp( i∆ωτ) d∆ω, (3.4) E N T AS − 0T AS − 0S1 − Z representing the fact that the scattering function is centered at the ﬁrst mode and the center of the TAS resolution is at ω ω . The normalization factor is then given by 0T AS − 0S1

N = S (∆ω) R (∆ω (ω ω )) d∆ω. (3.5) T AS − 0T AS − 0S1 Z 3.1 Two modes within the TAS resolution ellipsoid - Simpliﬁed NRSE model 49

10 8

) [a.u.] 6 ω

∆ 4 S( 2 0 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 ∆ ω [meV]

0 10

−1

echo amplitude 10

0 5 10 15 20 25 30 τ [ps]

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 multiplied with the TAS energy resolution function assuming ETAS = 1.25meV (red solid) and scattering function convoluted with the TAS resolution function (grey dashed). Bottom: Single exponential decay corresponding to Γ (black solid), Fourier transform of S (∆ω) (blue solid), Fourier transform of 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]

τ τΓ A = A + (1 A ) cos 2π exp , (3.6) E | M | −| M | T − ~ n o where the period of the modulation is inversely proportional to the separation in energy

4π~ T = (3.7) ∆Ω 50 3 NRSE investigations on split modes and the amplitude of the modulation is determined by the intensity ratio of the modes A1 A2

A1′ A2′ AM = − (3.8) A1′ + A2′ with

A′ = A R ( ω ) (3.9) 1 1 T AS − 0T AS A′ = A R (∆Ω ω ) . (3.10) 2 2 T AS − 0T AS Another approximation is

τ τΓ A = A + (1 A )cos2 2π exp , (3.11) E | M | −| M | T − ~ n o 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 separation 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 depolarized 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 A1 , i.e. A2 the contrast, becomes τ-dependent. 3.2 Second dispersion surface within the TAS resolution ellipsoid - Generalmodel 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 10 15

10 ) [a.u.] ω

∆ −1 5 10 S( 0 echo amplitude −0.2 −0.1 0 0.1 0.2 0 20 40 60 ∆ ω [meV] τ [ps]

0 10 15

10 ) [a.u.] ω

∆ −1 5 10 S( 0 echo amplitude −0.2 −0.1 0 0.1 0.2 0 20 40 60 ∆ ω [meV] τ [ps]

0 10 20 ) [a.u.] ω

∆ −1 10 S( 0 echo amplitude −0.2 −0.1 0 0.1 0.2 0 20 40 60 ∆ ω [meV] τ [ps]

Fig. 3.2: Calculated examples for diﬀerent model scattering functions S (∆ω) (left) and the corresponding Fourier transform (right), assuming Γ=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 ellipsoid - 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 ωT AS 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

1 A = S (Q,ω) R (k , k )eiφ1(ki,kf )d3k d3k E N 1 1 i f i f Z iφ2(ki,kf ) 3 3 + S2 (Q,ω) R2(ki, kf )e d kid kf . (3.12) Z 52 3 NRSE investigations on split modes

The echo amplitude at the spin echo point will be given by A . Using the results from | E| section 2.3.6 AE is

1 A = S (∆ω) R (∆ω ∆Ω , J )eiφ1(∆ω,Jn)d∆ωd5J E N 1 T AS − 1 n e n Z + S (∆ω) R (∆ω ∆Ω , J e)eiφ2(∆ω,Jn)d∆ωd5J e , (3.13) 2 T AS − 2 n e n Z where e e

∆Ω = ω ω (q ) (3.14) 1 0T AS − 0S1 c0 ∆Ω = ω ω (q ) . (3.15) 2 0T AS − 0S2 c0 The scattering function does not contain the energy oﬀsets:

Γ1 S1 (∆ω) = A1 2 2 (3.16) Γ1 +∆ω Γ2 S2 (∆ω) = A2 2 2 . (3.17) Γ2 +∆ω

Substituting the TAS resolution function RT AS in equation (3.13), the most general case reads:

1 A = S (∆ω)exp iτ ′′ ∆ω + ∆Ω (L ) ∆ω d∆ω E N 1 − 2,1 1 S 11 Z 1 exp TT J exp JT L J d5J × T AS1 −2 MC1 n Z (3.18) + S (∆eω)expe iτ ′′ ∆ω +e ∆Ωe (Le) ∆ω d∆ω 2 − 2,2 2 S 11 Z 1 exp TT J exp JT L J d5J × T AS2 −2 MC2 n Z where e e e e e

TT AS1,2 = iτ2′′,1,2TM1,2 + ∆Ω1,2 LS . (3.19) 1n In principle this formalisme can be extendede to a multiplee 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 3.2 Second dispersion surface within the TAS resolution ellipsoid - Generalmodel 53

for one of the two modes would be violated by a sign reversal in τ2, resulting in a rapid depolarization 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]. 1 The measurement was performed at a fixed kf = 2.51Å− 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 10 echo amplitude

−1 10 0 5 10 15 20 25 30 τ [ps]

Fig. 3.3: Experimental data measured on the zone boundary magnon Q = [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 general model developed in this thesis. Calculations assuming several simpliﬁca- tions [23] are given by the dashed blue line. The general model gives a better description of the data and demonstrates the importance of taking depolarization eﬀects for both excitations into account. 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 neutron 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.

Ω Rotation stage: (ΔΩ A3) Δ

Goniometer 1: tiltμ μ Goniometer 2: tiltν Nb crystal 1: movable ν cylinder axis (100) 1 1 0 0 0 1 Nb crystal 2: fixed cylinder axis (110)

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 the sample stick. The Nb crystals are oriented in the (h h l) scattering plane. Right: Sketch of the sample rotations provided by the module. 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.

3 E [meV] 2

1 DAF CGW Baypai 0 0 0.05 0.1 0.15 0.2 q=[0 0 l] [r.l.u]

Fig. 3.5: Comparison of the dispersion data obtained from the diﬀerent implemented models for Nb. The wavevector of the excitation q is directed along the [0 0 l] direction. For small values of q the models give almost the same result. 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 Slope H(1,1) H(2,2) H(3,3) [meV] [meV Å] [meV Å2] [meV Å2] [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 models for q = 0.05r.l.u. . Elements of the curvature matrix H not listed are zero. The elements of H are expressed in the Cartesian system of the reciprocal 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. 3.3 Experimental verification 57

kF kI kF kI

k L ┴

θ θ

Fig. 3.6: Left: Larmor diffraction setup. The precession field boundaries are parallel to the lattice planes and the magnetic fields are oriented in the same direction. All neutrons fulfilling the Bragg condition have the same k⊥ and therefore undergo the same Larmor precession. This setup is sensitive to a spread of the lattice spacing. Right: Mosaicity sensitive Larmor diffraction geometry. The lattice planes of the crystal are slightly tilted with respect to each other. Hence, neutrons acquire a different total Larmor precession angle, as their paths depend on which crystallite they are scattering off. The magnetic fields of the precession regions are oriented antiparallel.

It can be shown that for parallel oriented magnetic ﬁelds the total Larmor phase reads:

2mω L Φ = L d (3.20) tot π~ 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 ∆d ∆Φ=Φ . (3.21) tot d 5 ∆d 6 Since Φtot can be up to 10 rad, a relative resolution of d ∼= 10− 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 measurements, the setup becomes sensitive to mosaic spread, rather than to the distribution of lattice constants [45] (see Fig. 3.6 right). 58 3 NRSE investigations on split modes

Experimental results

The elastic part of the measurements using the tunable double dispersion setup was performed 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 1 an incident wavevector of kI = 1.9Å− . 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 diffraction measurements performed at HZB, Berlin.

0.8

0.6

0.4 echo amplitude

0.2

0 0 200 400 600 800 1000 δ [Å]

Fig. 3.7: Gaussian distribution ﬁtted to the elastic NRSE data measured at the [1 1 0] Bragg peak of the individual crystals. The black and blue line 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 approximated with a Gaussian, modulated by a cosine term [21]:

δ δ2 A = A + (1 A )cos 2π exp 4ln2 , (3.22) E | M | −| M | ∆ − w2 s S

3.3 Experimental veriﬁcation 59

0.8

0.6

0.4 echo amplitude

0.2

0 0 200 400 600 800 δ [Å]

Fig. 3.8: Elastic NRSE data measured at the [1 1 0] Bragg peaks of two Nb crystals. The red (green) data points were measured at a nominal relative angle of 0.3◦ (0.39◦) between the [1 1 0] Bragg peaks. The model described by equation (3.22) is ﬁtted to the data. where δ is the spin echo length. The modulation of the echo amplitude is inversely proportional to the diﬀerence in sample rotation angle ∆A3 and the magnitude of the reciprocal lattice vector QB 4π ∆s = . (3.23) QB∆A3

The FWHM wS of the distribution function in correlation length may be obtained from the

FWHM of the sample mosaicity ηS.

2π wS = . (3.24) QBηS

In a mosaicity sensitive Larmor diﬀraction setup using NRSE, the correlation length δ reads:

m LtanΘ 1 δ = 4π feff , (3.25) ~ kI QB 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 separations of 0.30◦ (red) and 0.39◦ (green). As expected, the modulation period of the signal 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 A1 A1 A1 of A2 = 0.63, A2 = 0.16 and A2 = 0.04 are obtained from the Gaussian fits. Thus, the contrast of the modulation should decrease with increasing A3.

1200

1000

800

600 counts / 1 s 400

200

0 44 44.5 45 45.5 46 rocking angle A3 [°]

Fig. 3.9: TAS rocking scan at an angular separation of 0.39◦ between the two Bragg peaks. Two Gaussians (dashed blue lines) are ﬁtted too the data. 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- 3.3 Experimental veriﬁcation 61

0.8

0.6

0.4 echo amplitude

0.2

0 0 200 400 600 800 1000 δ [Å]

Fig. 3.10: Elastic NRSE data measured at the [1 1 0] Bragg peaks of two Nb crystals with an angular separation of 0.39◦. The measurements were performed 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 the amplitude ratio, i.e. the contrast of the modulation, decreases.

A1 ing amplitude ratios A2 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 approximation. 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- A1 tion (3.22) is suitable to determine the splitting of the Bragg peaks only if A2 > 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 scattering senses SM = 1 (monochromator), SS = 1 (sample) and SA = +1 (analyzer) − − 1 with an experimental transverse Q-resolution of about 0.006Å− FWHM at fixed incident 1 ki = 1.9Å− [43]. The two Nb crystals were aligned with [h h l] plane as the scattering 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 centered around zero. The required tilt angles of the precession field boundaries, Θ = 31.60 1 − ◦ and Θ2 = +27.27◦, were calculated using SERESCAL [15].

1600

1400

1200

1000

800

600

400 Intensity [Counts / 15min]

200

0 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 E [meV]

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 the focused TAS conﬁguration leading to higher intensities. The energy of the excitation is E = 1.144(8)meV. Due to resolution eﬀects of the background TAS 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

2000 1500

1500 1000

1000

500 500 Intensity [Counts / 1s] Intensity [Counts / 12.3min]

0 0 44 44.5 45 45.5 46 46.5 47 −0.5 0 0.5 1 1.5 2 A3 [°] E [meV]

Fig. 3.12: Left: A3 rocking scan giving a split in A3 of 0.84◦. The movable peak (A3 = 45.72◦) is defined as the [1 1 0] Bragg− peak. Right: The corresponding TAS energy scan at [1 1 -0.05] gives an energy of 1.607meV for the shifted excitation resulting in an energy split of 0.463meV. 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].

1 exponential decay 0.9 extended model

0.8

0.7

0.6

0.5

echo amplitude 0.4

0.3

0.2

0.1 0 5 10 15 20 25 30 τ [ps]

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. The extended resolution model gives a linewidth, which is zero within error. In contrast, the linewidth obtained from the exponential decay would suggest an unphysical linewidth of Γ = 40(3)µeV.

In order to determine the linewidth of the excitation at T = 65K, inelastic NRSE measurements 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 dispersion 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. 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 example 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 ± P = (0.71 0.11). Since the data is corrected using the calibration scans, a P significantly 0 ± 0 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 P = (0.94 0.17). The results for Γ of both · − ± 0 ± 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.

800

700

600

500

400

300

200 Intensity [Counts / 1s]

100

0 44 44.5 45 45.5 46 46.5 A3 [°]

Fig. 3.14: TAS rocking scan to determine the splitting in A3 of the two modes.

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 parameters, 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 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

1 1

0.8 0.8

0.6 0.6

0.4 0.4 echo amplitude echo amplitude 0.2 0.2

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 τ [ps] τ [ps]

Fig. 3.15: The inelastic data at [1 1 0.05], E = 1.28meV, at T = 68K shows a 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

τ τΓ A = A + (1 A )cos 2π exp (3.26) E | M | −| M | T − ~ 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 excitation 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 A1 below 0.5, which A2 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 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 A1 . A2

A1 2 Model (equation) Amplitude ratio A2 Γ [µeV] ∆Ω [meV] χ (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 model corresponding 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 A1 = 0.35 0.86. A2 ± 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 modulation of the echo amplitude. However, the value obtained for the amplitude ratio is in agreement with the TAS calibration scans within the error. 3.4 Summary 67

1 1 single dispersion double dispersion 0.8 0.8

0.6 0.6

0.4 0.4 echo amplitude echo amplitude

0.2 0.2

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 τ [ps] τ [ps]

Fig. 3.16: Left: The general model discussed in section 3.2 ﬁtted to the inelastic data. The ﬁt is in very good agreement with the data and gives a separation in energy of the modes of ∆Ω = 0.274 0.026meV. Right: Fit results of the single and the double dispersion using± the extended model. The maxima of the modulation follow the decay of the single dispersion signal.

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 conditions for inelastic scattering and is appropriate for high resolution spin echo measurements 68 3 NRSE investigations on split modes on mode doublets. A simplified model describing the signature of split modes, a modulation 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. 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]

1 1 S (ω)= (4.1) π 2 3 2 1+ ω α ω + γ ω Γ − Γ Γ takes the form of a modified Lorentzian and describes an asymmetric line shape. Here, the ω 2 usual argument is replaced by a polynomial, that includes an asymmetry term α Γ and ω 3 a damping term γ Γ . 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 convolution 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(NO ) 2.5D O (copper nitrate), a model system for a 1-D 3 2· 2 bond alternating Heisenberg chain, have established thermal development of an asymmetric

69 70 4 NRSE line shape analysis

1 0.35 0.9

0.3 0.8

0.25 0.7 0.6 0.2 0.5

0.15 0.4

0.1 echo amplitude 0.3 Scattering function 0.2 0.05 0.1 0 −0.2 −0.1 0 0.1 0.2 0.3 0 20 40 60 80 ∆ω [meV] τ [ps]

Fig. 4.1: Left: Model scattering function with an asymmetric line shape (black) using the phenomenological function (see equation (4.1)) and a normal Lorentzian scattering function (dashed blue). Right: Calculated echo 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 broadening [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(NO ) 2.5D O and Sr Cr O , at TRISP at the FRM-II, Garching. For the first time, 3 2· 2 3 2 8 this effect was measured with NRSE.

4.1 Asymmetric line shape of excitations in Cu(NO ) 2.5D O 3 2· 2 4.1.1 Properties of Cu(NO ) 2.5D O 3 2· 2 High ﬁeld magnetization and inelastic neutron scattering measurements have been performed in the past [56, 57, 58, 60, 55] and provide a consistent picture of Cu(NO3)2 2.5D2O 1 · as a 1-D dimerized spin- 2 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- 4.1 Asymmetric line shape of excitations in Cu(NO ) 2.5D O 71 3 2· 2

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 [ 1 , 1 , 1 ] and [ 1 , 1 , 1 ] directions and project onto the same 2 2 2 2 − 2 2 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 0.227, J ≈ 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

ρ J´ J

d Fig. 4.2: Alternating chain layout with the dimer coupling strength provided by J and the intradimer coupling provided by J ′. ρ donates the separation 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(NO ) 2.5H O. In its original composition 3 2· 2 Cu(NO ) 2.5H O copper nitrate produces a high incoherent cross section due to the large 3 2· 2 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 inc inc of deuterium is comparably low: σH = 80.26b and σD = 2.05b [61]. Quantitative analysis of the incoherent cross sections of Cu(NO ) 2.5H O, Cu(NO ) 2.5D O and a mixture of 3 2· 2 3 2· 2 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). 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=65◦C in a temperature controlled oil bath using a LakeShore temperature controller (see Fig. 4.3).

argonargon

solution Liebig condenser

vacuumvacuum oil bath vacuum coolingcooling water water on on

Fig. 4.3: Sketch of the distillation setup. The solution (dark blue) is kept in a temperature controlled oil bath. During the vacuum pumping process a mixture of normal and heavy water is removed from the solution, condensed in the Liebig cooler and collected in a second flask. To avoid air contact during the refill of heavy water the whole setup is flooded with argon.

140.99g of Cu(NO ) 2.5H O powder were dissolved in D O of high purity (>99.9%). Ac- 3 2· 2 2 cording to the literature [64], 140.99g of Cu(NO ) 2.5H O can be dissolved in 29.52g of 3 2· 2 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 vacuum 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 1.4◦C 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 4.1 Asymmetric line shape of excitations in Cu(NO ) 2.5D O 73 3 2· 2

Fig. 4.4: Highly deuterated copper nitrate single crystals grown using enrichment and solution growth method. Large high quality crystals can be grown 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(NO ) 2.5D O. The 3 2· 2 highly hygroscopic nature of copper nitrate stabilizes a second phase (Cu(NO ) 6H O) 3 2· 2 below T = 26◦C 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 = 85◦C to T = 40◦C in discrete steps dT 0.05◦C with a cooling rate dt = 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(NO ) 2.5D O using the thermal NRSE-TAS spectrom- 3 2· 2 eter TRISP [66] at the FRM-II, Garching. TRISP was operated in a configuration with 1 scattering senses SM = 1, SS = 1 and SA = +1 and a fixed incident k = 1.7Å− . The − − i 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 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.

Spin echo signal T=0.5K Scattering function T=0.5K

0.3 0.4 0.2

0.2

Intensity 0.1

echo amplitude 0 0 20 40 60 80 100 120 −0.2 −0.1 0 0.1 0.2 τ [ps] E [meV] Spin echo signal T=2K Scattering function T=2K 0.6 0.3 0.4 0.2

0.2

Intensity 0.1

echo amplitude 0 0 20 40 60 80 100 120 −0.2 −0.1 0 0.1 0.2 τ [ps] E [meV] Spin echo signal T=2.5K Scattering function T=2.5K 0.6 0.3 0.4 0.2

0.2

Intensity 0.1

echo amplitude 0 0 20 40 60 80 100 120 −0.2 −0.1 0 0.1 0.2 τ [ps] E [meV] Spin echo signal T=3K Scattering function T=3K 0.6 0.3 0.4 0.2

0.2

Intensity 0.1

echo amplitude 0 0 20 40 60 80 −0.2 −0.1 0 0.1 0.2 τ [ps] E [meV]

Fig. 4.5: Left: Echo amplitude as a function of spin echo time τ for different temperatures (from the top: T = 0.5K, T = 2K, T = 2.5K and T = 3K) fitted with the phenomenological model (see equation (4.1)). Right: Line shape calculated from the fit parameters obtained from the spin echo data. 4.1 Asymmetric line shape of excitations in Cu(NO ) 2.5D O 75 3 2· 2

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 Γ K = 0.1 0.8µeV. The sharp excitation at ± 0.5 ± base temperature indicates that no asymmetry is introduced by the shape of the dispersion and resolution eﬀects.

P 0.404 0.033 0 ± Γ K 0.1 0.8µeV 0.5 ± α0.5K 0

γ0.5K 0

Γ K 0.009 0.003meV 2 ± α K 0.119 0.087 2 ± γ K 0.007 0.008 2 ± Γ K 0.006 0.002meV 2.5 ± α K 0.121 0.026 2.5 ± γ K 0.005 0.002 2.5 ± Γ K 0.022 0.009meV 3 ± α K 0.420 0.166 3 ± γ K 0.071 0.064 3 ± 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 temperatures, the parameter P (τ =0) = P0 was used as a shared fit parameter, since P0 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 count of 1 min . The result of this experiment unambiguously proves an increasing asymmetry with increasing 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.

35 0.14

30 0.6 0.12

25 0.5 0.1

20 0.4 0.08 eV] γ α

µ 15 0.3 0.06 [ Γ 10 0.2 0.04

5 0.1 0.02

0 0 0

−5 −0.1 −0.02 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 T [K] T [K] T [K]

Fig. 4.6: Comparison of the ﬁt parameters of the phenomenological model obtained 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 asymmetry. 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 4.1 Asymmetric line shape of excitations in Cu(NO ) 2.5D O 77 3 2· 2 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

ωS(ω)dω E = (4.2) m S(ω)dω R of an excitation is shifted due to an asymmetricR 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 φ(T ) φ(T = 0.5K) ∆E(T )= ~ − . (4.3) τ 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 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.

−2

−4 eV] µ −6 E[ ∆ −8

−10

−12

−14 0.5 1 1.5 2 2.5 3 T[K]

Fig. 4.7: Temperature dependent data from phase sensitive measurements. Equation (4.5) considering a contribution of the Bragg peak is ﬁtted to the data. A polarized fraction of 1.9(2)% of the Bragg peak contaminating 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:

2 E −Eexc ( Bragg ) (0−0.385meV)2 σ2 counts I = e− 2 TAS I = e− 2(0.126meV)2 I = 538.68 (4.4) Bragg · Bragg,calib · Bragg,calib sec with σT AS = 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- 4.1 Asymmetric line shape of excitations in Cu(NO ) 2.5D O 79 3 2· 2 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

I f ∆E (T )= 1 Bragg · (C (T ) + 0.385meV) ∆E (T = 0.5K) . (4.5) − I f + I (T ) · − Bragg · exc 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 1 excitation energy of E = 0.385meV and an incident ki = 1.7Å− results in kf = 1.644Å. 1 1 Therefore, to change to kf,Bragg = 1.7Å− of the elastic case a δkf = 0.056Å− is needed. A2 2 Substituting ∆kf ∆kf + δkf into equation (2.42) the last term, 3 (∆kf nf ) , is → (kF nf ) · 2 · replaced by 3 new terms. The term proportional to δkf results in a constant phase, which 2 can be neglected. The term proportional to ∆kf stays the same as before and the only A2 2 relevant new term is 3 (2∆kf δkf ) . Here it is taken into account that the slope of the kF | | · ∆kf dispersion at the minimum is zero and therefore kF nf = kF . Assuming = 1% leads · | | kf to an 1 1 FWHM 0.01 k 0.01644Å− and σ 0.00698Å− . (4.6) ∆kf ≈ · F ≈ ∆kf ≈ The polarization at the Bragg peak is then estimated by

2 ∆kf A2 2σ2 i 2δkf ∆kf − ∆k 2k3 P = e f e F d∆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, 1 given the energy resolution of TRISP at ki = 1.7Å− . The shift is masked due to a polarized fraction of the (1 0 1) Bragg peak contaminating the signal. 80 4 NRSE line shape analysis

In order to investigate the shift of the mean energy of the excitation as a function of temperature, 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).

30 70

60 20 50 eV] eV] µ 10 µ 40 E [ E [ ∆ ∆ 0 30

20 −10 10 Energy shift −20 Energy shift 0

−30 −10 0 1 2 3 4 0 1 2 3 4 T [K] T [K]

Fig. 4.8: Temperature dependent data (black) from phase sensitive measurements 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). Energy shifts calculated from the cosine Fourier transforms of the corresponding asymmetric 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 measurements 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 scattering functions with a Lorentzian shape (left). The functions are successively shifted by 4.1 Asymmetric line shape of excitations in Cu(NO ) 2.5D O 81 3 2· 2

1 1

0.8 0.5

0.6 0 Intensity 0.4 echo amplitude −0.5 0.2

0 −1 0.2 0.3 0.4 0.5 0.6 0 5 10 15 20 25 30 35 40 45 50 E [meV] τ [ps]

Fig. 4.9: Left: Lorentzian shaped scattering function successively shifted by 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 transform.

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 functions 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.

1 T=0.5K T=0.5K 1 T=2K T=2K T=2.5K T=2.5K T=3K T=3K 0.8 0.5

0.6 0 Intensity 0.4 echo amplitude −0.5 0.2

0 −1 0.2 0.3 0.4 0.5 0.6 0.7 0 5 10 15 20 25 30 35 40 45 50 E [meV] τ [ps]

Fig. 4.10: Left: asymmetric scattering function calculated from the ﬁt parameters given in Tab. 4.1. Right: Corresponding numerical calculation of the cosine Fourier transform. 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 1 1 of Sr3Cr2O8 as a dimerized spin- 2 antiferromagnet. Sr3Cr2O8 consists of a lattice of spin- 2 Cr5+ 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 4.2 Asymmetric line shape of excitations in Sr3Cr2O8 83

J0 5.55(1) meV

J1' -0.04(1) meV

J1'' 0.25(1) meV J ' J 2 J 2''' 2'' J2'- J 3' 0.75(1) meV J J J '' 3''' J2''- J 3'' -0.54(1) meV 3' 3

J2'''- J 3''' -0.12(1) meV J J '' 1' 1 J J ' J4' 0.06(2) meV 1'' 4

J4'' -0.05(1) meV J m 0

m m

Fig. 4.11: Lowtemperature (monoclinic) crystal structure of Sr3Cr2O8 showing the magnetic Cr5+ ions only. The exchange interactions are labeled on 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 (R3¯m) to monoclinic (C2/c), which is stable for T < 120K [72, 73]. The orbital order 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◦.

M M'

7.0

6.5

6.0

5.5

5.0

4.5 Energy (meV)

4.0

Twin 1

Twin 2

3.5

Twin 3

Flex Data

3.0

(1,1,-3/2) (1,1,0) (1/2,1/2,0) (0,0,0) (-1/2,1/2,0) (-1/2,1/2,-1)

h h h h h h

Fig. 4.12: Dispersion relation of Sr3Cr2O8 (from [67]). The blue, ma- genta and black lines correspond to the ﬁtted dispersion relations of three monoclinic twins. Green points are ﬁtted centers of peaks measured at V2/FLEX at HZB. 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 intradimer 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 excitations. Additionally there are signiﬁcant interdimer interactions, which allow the dimer excitations to hop and develop dispersion. The dispersion of the one-magnon mode produces 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]. 1 TRISP was operated in a configuration with a fixed kf = 2.51Å− 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 separation 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 background 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 exponential decay of the echo amplitude. Due to the statistical error of the data at T = 0.5K a 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 scattering 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.

Spin echo signal T=0.5K Scattering function T=0.5K 0.4

0.4 0.2 0.2 Intensity

echo amplitude 0 0 0 10 20 30 40 50 −0.5 0 0.5 τ [ps] E [meV] Spin echo signal T=10K Scattering function T=10K 0.6 0.4

0.4 0.2 0.2 Intensity

echo amplitude 0 0 0 10 20 30 40 50 −0.5 0 0.5 τ [ps] E [meV] Spin echo signal T=15K Scattering function T=15K 0.6 0.4

0.4 0.2 0.2 Intensity

echo amplitude 0 0 0 10 20 30 40 50 −0.5 0 0.5 τ [ps] E [meV] Spin echo signal T=20K Scattering function T=20K 0.6 0.4

0.4 0.2 0.2 Intensity

echo amplitude 0 0 0 10 20 30 40 50 −0.5 0 0.5 τ [ps] E [meV]

Fig. 4.13: Left: Echo amplitude as a function of spin echo time τ for different temperatures (from the top: T = 0.5K, T = 10K, T = 15K and T = 20K) fitted with the phenomenological model (see equation (4.1)). Right: Line shape of the excitation in the energy domain calculated from the fit parameters obtained from the spin echo data. 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 P 0.4 (see Tab. 4.2) is significantly smaller than 1. For the chosen 0 ≈ 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 count background rate of 1 min . 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 broadening 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.

P 0.426 0.045 0 ± Γ K 0.027 0.006meV 0.5 ± α0.5K 0

γ0.5K 0

Γ K 0.028 0.011meV 10 ± α K 0.266 0.089 10 ± γ K 0.028 0.020 10 ± Γ K 0.038 0.014meV 15 ± α K 0.265 0.116 15 ± γ K 0.030 0.025 15 ± Γ K 0.058 0.023meV 20 ± α K 0.290 0.14 20 ± γ K 0.032 0.032 20 ± Tab. 4.2: Fit results of the phenomenological model (4.1) applied to the spin echo data for diﬀerent temperatures. 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.

150 0.5 0.08

0.4 0.06 100 0.3 0.04 eV] γ α µ [

Γ 0.2 50 0.02 0.1 0 0 0 0 10 20 30 0 10 20 30 0 10 20 30 T [K] T [K] T [K]

Fig. 4.14: Comparison of the ﬁt parameters of the phenomenological model obtained from the spin echo data (black) and the ﬁt parameters from [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]. 88 4 NRSE line shape analysis

1.2

0.8

0.6

0.4 E [meV] ∆ 0.2

0 NRSE TAS −0.2 0 10 20 30 40 50 60 T [K]

Fig. 4.15: Energy shift ∆E of the low energy excitation at (0.5 0.5 3) for diﬀerent temperatures. The NRSE data and the TAS data [68] are in good agreement.

4.3 Summary

Two systems, the 1-D Cu(NO ) 2.5D O and the 3-D Sr Cr O , in which the effect of tem- 3 2· 2 3 2 8 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(NO ) 2.5D O large single crystals were grown from D O-enriched solution. For the 3 2· 2 2 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 observed 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 successfully that NRSE spectroscopy can be used to reveal the effect of temperature dependent asymmetric line broadening. 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 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) are mounted between cooling plates (light green) [85]. 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 mλ Φ= γB l (5.1) 0 2π~ 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

∆Φ = √M ∆Φ . (5.2) M · 1

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

2 2 ∆Φ1 3 ∆B0 ∆l = 1 10− = + . (5.3) Φ · B l 1 s 0

∆B0 ∆l The relative deviations of the magnetic ﬁeld ( B ) and of the coil thickness ( l ) should − 0 1 10 3 4 contribute equally with · = 7.1 10 . Assuming B = 165G and l = 42mm yields for √2 · − 0 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 transmission 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. × × 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 = 120◦C, 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 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:

B [G] = 3.08 I [A]. (5.5) 0 · 0

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 (∆B = 30mG) 0 − compared to the center of the beam area (red line).

[G] 8 0 B 6

0 −20 −18 −16 −14 −12 −10 −8 −6 −4 distance [cm]

Fig. 5.3: Vertical component of the B0 ﬁeld along the coil axis in a bootstrap pair. If two coils are powered (blue line) the ﬁeld is more homogeneous and slightly enhanced as compared to a single powered coil (black line). The center of the beam area is marked red while the edges of the beam 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 42mm 165G = 2.3kG, which is far from 3mm · 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 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 W adhesive (Gap Filler 1000 [81]) with a heat conductivity of λ = 1 Km . 2 The resistance of one B0 coil is R = 0.12Ω. Using P = I R and equation (5.5) the thermal dissipation is: 2 2 P [W] = 1.26 10− B [G], (5.6) · 0 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 W flow of about q = 0.43 cm2 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

qd ∆T = 1K. (5.7) λ ≈

The cooling water passes 3 cooling plates one after another. A ﬂow rate of 2 3 l is more − min 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. 5.1 Bootstrap coils 95

Air

Air HO2 Fig. 5.4: Cooling system of the bootstrap coils. Cooling plates ﬂowed through by water are glued to the B0 coils and the whole stack is pressed together. The beam areas not covered by the cooling plates are cooled with pressurized 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

π2 ∆B 2 ∆l 2 P = 1 N RF + . (5.8) n − 4 B l " RF # 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

∆B ∆l RF = < 4%. (5.9) BRF l

According to equation (2.17) the amplitude of the HF ﬁeld is:

~ π2 1 B = 4 . (5.10) RF m γ lλ

+ The linear oscillating field BRF is divided into two counter rotating parts B1 and B1− with BRF an amplitude of 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) 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 moment of the neutron.

B 1 L2

B 1 L1

B 1 L2

Fig. 5.5: Left: Sketch of the RF coil as an assembly of three coils. The magnetic 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 . 2 The inner coil L1 is wound with four aluminum wires in parallel with a cross section of 1mm each. The straight part of the outer coils L2 is wound with three copper wires in parallel 2 with a cross section of 0.75mm 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. 5.2 Spin echo instrument arms 97

I/2 L2 I L1 L I/2 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 movable 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 Θ =Θ 35 . Combined with 1 2 ≈ ◦ 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 direct beam geometry, i.e. A4 = 0◦ and A6 = 0◦ (detector angle), in the ki-range from 98 5 Upgrade of the NRSE option at FLEXX

1 1 ki = 1.3Å− to ki = 1.7Å− 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 1 1 is between 1.37Å− and 2.37Å− 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 geometry (see section 3.3.3). The previous NRSE option allowed for a Q -range in Larmor diffrac- 1 1 | | 1 tion geometry from 2.69Å− to 3.11Å− . These values are calculated for ki = kf = 1.9Å− . Given the new geometry of the spectrometer arms, the enhanced Q -range is now from 1 1 1 | | 2.44Å− to 3.29Å− for ki = kf = 1.9Å− .

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 (B 0.3G) gives e ≈ 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 B 5mG for λ = 6Å and B 8mG for λ = 4Å. The shielding factor is defined by: r ≈ r ≈ B S = a (5.12) Br 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 × × 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 additional active shielding.

Fig. 5.7: Sketch of the magnetic shielding design of the NRSE option of FLEXX.

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 factor N and the relative flux concentration in the walls of the shielding [86]. The ballistic demagnetization factor for this geometry is given by:

4 9 1 3 3 q 5 (q 20) N = 1 20 1 − −13 , (5.13) 1+ p − q + !  − 40 13 5  3 q + 2   100 5 Upgrade of the NRSE option at FLEXX

a 1 where p = b is the ratio of the edge lengths a and b of the box and q = p +p. Approximations for the perpendicular and longitudinal shielding factor are [86]:

µd S = 0.7 (5.14) p a 2.52N µd S = . (5.15) l b b 1 + 0.56 a For the new dimensions of the shielding the approximations yield N 0.58 and S 87 for l ≈ l ≈ the longitudinal case and N 0.42 and S 184 for the perpendicular case. The calculated p ≈ p ≈ shielding factors for the old dimensions (1000 300 300mm3, thickness d = 1.5mm) are × × S 40 and S 175. The longitudinal shielding factor for the spectrometer arms is l ≈ p ≈ 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 S 184, such a field is easily shielded by the actual p ≈ 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 S 87 the remaining field within the field free l ≈ zone exceeds 5mG and will cause significant depolarization. Therefore, a parallel operation 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 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 spectrometer, 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 perpendicular 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. 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 experimentally. 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.

4 4 x 10 x 10 10 10 Coil #1 Coil #5 Coil #2 Coil #6 8 Coil #3 8 Coil #7 Coil #4 Coil #8

6 6

4 4

Intensity [Counts / 5s] 2 Intensity [Counts / 5s] 2

0 0 8 10 12 8 10 12 I [A] I [A] 0 0

Fig. 5.9: Detector counts as a function of the current of the B0-coils spectrometer arm 1 (left) and 2 (right). If the Larmor frequency of the static ﬁeld is equal to the frequency f of the RF coils the detector counts reach a minimum. The results show very little variation between the individual coils 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 oriented 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 vertically 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 minimum 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 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.

50 0.5 data 1st arm data 1st arm data 2nd arm 0.4 data 2nd arm 40 linear fit 0.3 0.2 30 0.1 0

current [A] 20 −0.1 0 B −0.2 10 −0.3 Difference Data − Fit [A] −0.4

0 −0.5 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 f [kHz] f [kHz]

Fig. 5.10: Left: Optimum currents for the first B0-coils of each spectrometer arm as function of RF frequency. A linear function is fitted to the data. 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 optimum 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. 5.3 Calibration of the new NRSE option at FLEXX 105

20 0.5 data 1st arm data 1st arm data 2nd arm 0.4 data 2nd arm linear fit 0.3 15 0.2 0.1 10 0 −0.1 −0.2

Pick−Up Voltage [V] 5 −0.3 Difference Data − Fit [V] −0.4

0 −0.5 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 f [kHz] f [kHz]

Fig. 5.11: Left: Optimum pick-up voltages Upu for the first RF coils of each spectrometer arm as function of RF frequency. A linear function is fitted 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 experimental 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 polarization 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 current 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 coupling 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. 106 5 Upgrade of the NRSE option at FLEXX

4 4 x 10 x 10 10 10

8 8

6 6

4 4

2 2 Intensity [Counts / 10s] Intensity [Counts / 10s]

0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 I [A] I [A] c1 c2

Fig. 5.12: Intensity at the detector as a function of the current of the coupling coils of spectrometer arm 1 (left) and 2 (right). While scanning the current 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

ω m A = cos Φ = f (λ)cos eff ∆Lλ dλ, (5.16) E h NSEi 2π~ Z 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 = L L . Approximating the 1 − 2 wavelength distribution with a Gaussian shape

− 2 (λ λ0) 1 σ2 f(λ)= e− 2 λ (5.17) σλ√2π with a mean wavelength λ0, the Fourier transform yields

2 1 ωeff m ∆Lσλ ωeff m A = e− 8 2π~ cos ∆Lλ (5.18) E 2π~ 0 According to equation (2.11) the beam intensity after the analyzer is

I I = 0 (1 + A ) (5.19) 2 E

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. 5.3 Calibration of the new NRSE option at FLEXX 107

4 x 10 8

Intensity [Counts / 10s] 2

0 −25 −20 −15 −10 −5 0 5 10 15 20 25 Position of fourth bootstrap coil [mm]

Fig. 5.13: Spin echo curve obtained by direct beam measurements. The ﬁt yields a wavelength of λ = 3.3498(1)Å and ∆λ 1.3%. λ ≈

The cosine term in equation (5.18) causes the rapid oscillations. By ﬁtting the period length of one oscillation 2π~ ∆L = (5.20) ωeff mλ0 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 ∆λ 1.3%. This is in good agreement with the properties of the λ ≈ focusing monochromator used during the experiment. Supplementary spin echo curves were measured in the direct beam for nominal wavevectors 1 1 1 ki = 1.37Å− and ki = 2.37Å− . Fitting the data results in ki,fit = 1.3762(1)Å− (λ = 1 4.8686(2)Å) with a corresponding ∆λ 0.8% and k = 2.3666(3)Å− (λ = 2.2666(3)Å) λ ≈ i,fit with a corresponding ∆λ 1.4%. These two wavevectors are the limits of the measur- λ ≈ able ki-range due to mechanical constraints of the monochromator shielding at the time of the experiment. The value of ∆λ decreases from ∆λ 1.4% (λ = 2.27Å) to ∆λ 0.8% λ λ ≈ λ ≈ (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 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 ( ∆λ 10%), where ∆λ is a FWHM, the spin echo λ ≈ 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.

4 x 10

8 437.5 kHz 399 kHz 7 271 kHz

Intensity [Counts / 10s] 2

0 −10 −8 −6 −4 −2 0 2 4 6 Position of fourth coil [mm]

Fig. 5.14: At the spin echo point x = 3.26 0.01mm (∆L = 0) spin echo curves for diﬀerent frequencies have− a common± minimum. The measurements were performed in 8π-mode using a wavelength of λ = 3.35Å for the frequencies 271kHz, 399kHz and 437.5kHz. 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 L2′ should increase to L L′ = = L +∆x (5.21) 2 0.97 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

0.97 L = ∆x. (5.22) 0.03

4 x 10

8 f =437.5 kHz, f =424.375 kHz 1 2 f =399 kHz, f =387.03 kHz 7 1 2 f =271 kHz, f =262.87 kHz 1 2 6

Intensity [Counts / 10s] 2

0 2 4 6 8 10 12 14 16 Position of fourth coil [mm]

Fig. 5.15: At the spin echo point x = 8.86 0.01mm (∆L = 0) all spin echo curves have a minimum. The measurements± were performed in 8π- mode with a wavelength of λ = 3.35Å. The frequencies applied to the second spectrometer arm were lowered by a factor of 0.97 compared to the ﬁrst 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 L = 391.9 0.3mm, ± ± 8π ± 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 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.

4 x 10 9 122 kHz 8 225 kHz 7 297 kHz 6 5 4 3 2 Intensity [Counts / 10s] 1 0 −20 −15 −10 −5 0 5 10 15 Position of fourth coil [mm]

4 x 10 9 f =122 kHz, f =118.34 8 1 2 f =225 kHz, f =218 1 2 7 f =297 kHz, f =288.09 1 2 6 5 4 3 2 Intensity [Counts / 10s] 1 0 −10 −5 0 5 10 15 20 25 Position of fourth coil [mm]

Fig. 5.16: Top: At the spin echo point x = 2.78 0.01mm all spin echo curves have a common minimum. The measurements− ± were performed in 4π- mode using a wavelength of λ = 3.35Å for the frequencies 122kHz, 255kHz and 297kHz. Bottom: The frequencies applied to the second spectrometer arm were lowered by a factor of 0.97 compared to the ﬁrst arm and the spin 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 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:

2L 2L L ∆Φ = 2πf coil 2πf coil = 4π coil (f f ) (5.23) 1 v − 2 v v 1 − 2 2L mλ 2L mλ = coil (f 0.97f ) = 0.03 coil f . (5.24) ~ 1 − 1 ~ 1 (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

2π~ L (f)= (5.26) 0 2Nfmλ the correction in distance can be calculated for each frequency by

∆Φ ∆L (f)= L . (5.27) corr 0 2π

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 L = ± ± 4π 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 measurement 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 rad 3h) = 98.5%. Thus, this effect is − h · negligible compared to the statistical error in most cases. 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.

5 x 10 1.8 1.82 Run 1 1.6 Run 2 1.81 Run 3 Run 4 1.4 Run 5 1.8 Run 6 1.2 1.79

1 1.78

0.8 Phase [rad] 1.77

Intensity [counts / 20s] 0.6 1.76

0.4 1.75

0.2 1.74 −0.5 0 0.5 0 2 4 6 8 Position of fourth coil [mm] Run number

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. Within 1 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,meas(f1,f2, Θ1, Θ2,ki,kf ) AE(τ)= AE(f1,f2, Θ1, Θ2,ki,kf )= . (5.29) AE0(f1,f2, Θ1, Θ2,ki,kf )

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. 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:

A (f , Θ ,k )= A (f , Θ ,k ) A (f , Θ ,k ). (5.30) E,calib 1 1 i E1 1 1 i · E2 1 1 i Assuming that the depolarizing eﬀects are the same in both arms yields

AE1 = AE2 = AE,calib(f1, Θ1,ki). (5.31) q 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:

A (f ,f , Θ , Θ ,k ,k ) = A (f , Θ ,k ) A (f , Θ ,k ) (5.32) E0 1 2 1 2 i f E1 1 1 i · E2 2 2 f = A (f , Θ ,k ) A (f , Θ ,k ). (5.33) E,calib 1 1 i · E,calib 2 2 f q q 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 1 recorded for coil tilt angles of 0◦, 10◦, 20◦, 30◦, 40◦ and 50◦ for ki = 1.8757Å− . The effective 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 114 5 Upgrade of the NRSE option at FLEXX

1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4 θ = 0° θ = 0° echo amplitude echo amplitude 0.3 θ = 10° 0.3 θ = 10° θ = 20° θ = 20° 0.2 θ = 30° 0.2 θ = 30° θ θ 0.1 =40° 0.1 =40° θ = 50° θ = 50° 0 0 0 100 200 300 400 500 600 0 500 1000 1500 f [kHz] f [kHz] eff eff

−1 Fig. 5.18: NRSE calibration data for ki = 1.8757Å . The data set was measured for coil tilt angles of 0◦, 10◦, 20◦, 30◦, 40◦ and 50◦ for both NRSE modes, 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- 1 tion sets were measured for ki = 2.3666Å− for both NRSE modes, 4π and 8π. The results are shown in Fig. 5.19.

1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4 echo amplitude echo amplitude 0.3 θ = 0° 0.3 θ = 0° θ = 20° θ = 20° 0.2 θ = 30° 0.2 θ = 30° θ θ 0.1 =40° 0.1 =40° θ = 50° θ = 50° 0 0 0 100 200 300 400 500 600 0 500 1000 1500 f [kHz] f [kHz] eff eff

−1 Fig. 5.19: NRSE calibration set for ki = 2.3666AA . The set was measured for coil tilt angles of 0◦, 20◦, 30◦, 40◦ and 50◦ for both NRSE modes, 4π (left) and 8π (right). 5.3 Calibration of the new NRSE option at FLEXX 115

1 Fig. 5.20 shows the ﬁtted intensities I0 for ki = 1.8757Å− 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.

4 4 x 10 x 10 2 2

1.8 1.8

1.6 1.6

1.4 1.4

1.2 1.2

1 1 θ = 0° θ = 0° 0.8 0.8 θ = 10° θ = 10° 0.6 θ = 20° 0.6 θ = 20° Intensity [Counts / s] Intensity [Counts / s] θ = 30° θ = 30° 0.4 0.4 θ =40° θ =40° θ θ 0.2 = 50° 0.2 = 50°

0 0 0 200 400 600 800 0 500 1000 1500 2000 f [kHz] f [kHz] eff eff

Fig. 5.20: Fitted intensities I0 according to equation (5.19) for all measured spin echo signals of the calibration set shown in Fig. 5.18 separated according 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 Θ 35 the beam cross section was reduced by a factor of 0.5. ≈ ◦ 0.5 ≈ ◦ 116 5 Upgrade of the NRSE option at FLEXX

I0 4π 1 mean I0 8π mean 0.8

0.6

0.4

0.2 normalized beam cross section

0 0 10 20 30 40 50 60 coil tilt angle [°]

Fig. 5.21: Normalized mean intensities ﬁtted to the calibration set for both modes as a function of coil tilt angle. This displays the cropping of the neutron beam. The black (green) line is a rough estimation according to the geometric dimensions of the current (previously used) bootstrap coil and 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 1 the sample position is increased by a factor of 5. This was measured for ki = 1.9Å− 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 sections 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- 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. goniometers) 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 spectrometer 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. 118 5 Upgrade of the NRSE option at FLEXX Chapter 6

Conclusion and perspectives

Two new kinds of NRSE experiments were investigated in this thesis: Resolving mode doublets 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 depolarizing 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 generalization, 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 geometry for two neighboring Bragg peaks. The results demonstrate the echo modulation in

119 120 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 overestimation 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(NO ) 2.5D O, a model system for a 1-D bond alternating 3 2· 2 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(NO ) 2.5D O were grown in the course of this thesis. 3 2· 2 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 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(NO ) 2.5D O and Sr Cr O are suggested to be 3 2· 2 3 2 8 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 operation 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 demonstrate 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(NO ) 2.5D O. 3 2· 2 122 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 spectrometer 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. 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 k k cos φ θ = I − F . (A.2) kF sin φ

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:

h   Q = k = hb1 + kb2 + lb3. (A.3)      l      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)

123 124 Appendix

This equation follows the crystallographic convention, i.e. the factor 2π is omitted. The magnitude of the wavevector is given by Q = 2π Q and B is given by M | c|

a∗ b∗ cos γ∗ B13   B = B1 B2 B3 = 0 b∗ sin γ∗ B23 (A.5)      0 0 B33      with

B13 = c∗ cos β∗ (A.6) (cos α∗ cos γ∗ cos β∗) B23 = c∗ − = c∗ sin β∗ cos α∗ (A.7) sin γ∗ − 2 2 2 B = c B B = c∗ sin β∗ sin α∗. (A.8) 33 ∗ − 3x − 3y q Note that Busing and Levy give B33 = 1/c. The usual crystallographic relations apply [47] for the reciprocal lattice

bc ca ab a∗ = sin α, b∗ = sin β, c∗ = sin γ (A.9) V V V using V = abc 1 cos2 α cos2 β cos2 γ + 2cos α cos β cos γ (A.10) − − − and p

cos β cos γ cos α cos α∗ = − (A.11) sin β sin γ cos γ cos α cos β cos β∗ = − (A.12) sin γ sin α cos α cos β cos γ cos γ∗ = − . (A.13) sin α sin β

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.

1 U = TνTc− , (A.14) 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 BQ (BQ BQ ) BQ BQ BQ T = 1 , 1 × 2 × 1 , 1 × 2 (A.15) c BQ (BQ BQ ) BQ BQ BQ | 1| | 1 × 2 × 1| | 1 × 2| where t1c is parallel to BQ1, t2c is in the plane of BQ1 and BQ2 and t3c is perpendicular to 1 this plane. Note that the matrix Tc− 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

cos ωi cos µi   uiν = sin ωi cos νi + cos ωi sin µi sin νi . (A.16)  −     sin ωi sin νi + cos ωi sin µi cos νi      The vectors u1ν and u2ν represent the two non-collinear reﬂections expressed in the laboratory 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

u (u u ) u u u T = 1ν , 1ν × 2ν × 1ν , 1ν × 2ν . (A.17) ν u (u u ) u u u | 1ν| | 1ν × 2ν × 1ν| | 1ν × 2ν|

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

QM 1 Q = (UB)− u . (A.18) n 2π nν

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 126 Appendix transformation reads

QM 1   RQ = Qθ = 0 . (A.19) ν 2π      0      The matrix R is related to the elementary rotation matrices

R = ΩMN, (A.20) where

cos ω sin ω 0 −   Ω = sin ω cos ω 0 (A.21)      0 0 1      cos µ 0 sin µ   M = 0 1 0 (A.22)      sin µ 0 cos µ   −    1 0 0   N = 0 cos ν sin ν (A.23)  −     0 sin ν cos ν      represent the rotations of the sample table and the sample goniometers, respectively. The matrix elements of R are explicitly

R R R ω µ ω µ ν ω ν ω µ ν + ω ν 11 12 13 c c c s s − s c c s c s s     R = R21 R22 R23 = ωsµc ωsµsνs + ωcνc ωsµsνc ωcνs , (A.24)    −       R31 R32 R33   µs µcνs µcνc     −      A UB matrix formalism 127 where the instrumental positioning parameters can be obtained from

µ = sin µ = R (A.25) s − 31 2 2 1/2 µc = cos µ = R11 + R21 (A.26) 2 2 1/2 νs = sin ν = R 32/ R11 +R21 (A.27) 2 2 1/2 νc = cos ν = R33/R11 + R21 (A.28) 2 2 1/2 ωs = sin ω = R21/ R11 + R21 (A.29) 2 2 1/2 ωc = cos ω = R11/R11 + R21 . (A.30) Note that the proper quadrants are obtained from

µ = arcsin( R ) (A.31) − 31 2 2 1/2 ν = arcsin R32/ R11 + R21 (A.32) and 2 2 1/2 arcsin R21/ R11 + R21 if R11 > 0 ω = (A.33)  1/2  π arcsin R / R2 +R2 if R < 0.  − 21 11 21 11 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

sin µ − plane   uν = cos µ sin ν . (A.34) ⊥ plane plane      cos µplane cos νplane      Using the unit vectors Qν and u , a right-handed orthogonal set of vectors is constructed Qν ν | | ⊥ by

Q UBQ t = u = ν = (A.35) 1ν 1ν Q UBQ | ν| | | et2ν = uν u1ν (A.36) e ⊥ × t3ν = t1ν t2ν. (A.37) e × e With e e e

T1ν = t1ν t2ν t3ν (A.38) e e e 128 Appendix the result for R is: 1 R = T1−ν . (A.39) Now the matrix elements of R are known and equations (A.31 – A.33) together with equations (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:

2D2 Ψ11 = 2 2 (B.1) −Cf NF τ2′′ ~ ~ τ N D C2N 2 Ψ = tan2 θ 2 1 I 2 2 i I (B.2) 22 −m − m 1 − τ k n − τ C2N 2 2′′ I · i 2′′ f F ~ 2 2 1 D2 ei2NI 1 Ψ33 = 2 2 2 2 2 (B.3) −m cos θ1 − τ2′′ Cf NF cos θ1 2 ~ 1 ~ tan θ2 1 D2 ef2 1 Ψ44 = 2 + 2 2 ef2 2 2 2 (B.4) m cos θ2 m cos θ2 Cf − τ2′′ Cf cos θ2 ~ 2 2 D2 ei3NI Ψ55 = 2 2 2 (B.5) −m − τ2′′ Cf NF 2 ~ D2 ef3 Ψ66 = 2 2 (B.6) m − τ2′′ Cf D2 CiNI Ψ12 = 2 2 2 (B.7) τ2′′ Cf NF D2 ei2NI 1 Ψ13 = 2 2 2 (B.8) τ2′′ Cf NF cos θ1 D2 ef2 1 ~ tan θ2 1 Ψ14 = 2 2 + (B.9) − τ2′′ Cf NF cos θ2 m cos θ2 Cf NF D2 ei3NI Ψ15 = 2 2 2 (B.10) τ2′′ Cf NF D2 ef3 Ψ16 = 2 2 (B.11) − τ2′′ NF Cf ~ 2 tan θ1 D2 CiNI 1 Ψ23 = 2 2 2 ei2 (B.12) m cos θ1 − τ2′′ Cf NF cos θ1 B Matrix elements 129

~ tan θ2 CiNI D2 CiNI 1 Ψ24 = + 2 2 ef2 (B.13) −m cos θ2 Cf NF τ2′′ Cf NF cos θ2 2 D2 CiNI Ψ25 = 2 2 2 ei3 (B.14) − τ2′′ Cf NF D2 CiNI Ψ26 = 2 2 ef3 (B.15) τ2′′ Cf NF D2 ei2ef2NI 1 ~ tan θ2 NI Ψ34 = 2 2 ei2 (B.16) τ2′′ Cf NF cos θ1 cos θ2 − m cos θ2 cos θ1 Cf NF

Ψ35 = 0 (B.17) 2 D2 ei2ei3NI 1 D2 ei2ef3NI 1 Ψ36 = 2 2 2 + 2 2 (B.18) − τ2′′ Cf NF cos θ1 τ2′′ Cf NF cos θ1 D2 ef2ei3NI 1 ~ tan θ2 1 NI Ψ45 = 2 2 ei3 (B.19) τ2′′ Cf NF cos θ2 − m cos θ2 Cf NF D2 ef2ef3 1 ~ tan θ2 1 Ψ46 = 2 2 + ef3 (B.20) − τ2′′ Cf cos θ2 m cos θ2 Cf D2 ei3ef3NI Ψ56 = 2 2 (B.21) τ2′′ Cf NF with

~ ~ NF 2 D = τ τ ′′ τ ′′ tan θ . (B.22) 2 − 2 k n − 2m 2 − 2m 2 2 F · f

I 1 B.2 The C− -matrix

1 The non-zero elements of the (6 6)-matrix I− in the resolution function given by equation × C (2.116) in section 2.3.4 are:

1 I− = N (e cos θ e sin θ ) (B.23) C,11 I i1 1 − i2 1 1 IC,− 12 = NI (ei1 sin θ1 + ei2 cos θ1) (B.24) 1 IC,− 13 = ei3NI (B.25) 1 I− = N (e cos θ + e sin θ ) (B.26) C,14 − F f1 2 f2 2 1 I− = N (e sin θ e cos θ ) (B.27) C,15 − F f1 2 − f2 2 1 I− = e N (B.28) C,16 − f3 F 1 IC,− 21 = cos θ1 (B.29) 130 Appendix

1 IC,− 22 = sin θ1 (B.30) 1 1 1 1 IC,− 32 = IC,− 45 = IC,− 53 = IC,− 66 = 1. (B.31)

B.3 The IC -matrix

The non-zero elements of the (6 6)-matrix I in the resolution function given by equation × C (2.116) in section 2.3.4 are:

1 IC,12 = (B.32) cos θ1 I = tan θ (B.33) C,13 − 1 1 IC,41 = (B.34) −NF ef2 sin θ2 + NF ef1 cos θ2 ei2NI sin θ1 ei1NI cos θ1 IC,42 = − (B.35) −ef1NF cos θ1 cos θ2 + ef2NF cos θ1 sin θ2 2 2 ei2NI cos θ1 + ei2NI sin θ1 IC,43 = (B.36) ef1NF cos θ1 cos θ2 + ef2NF cos θ1 sin θ2 ef1 sin θ2 ef2 cos θ2 IC,44 = − (B.37) −ef1 cos θ2 + ef2 sin θ2 NI ei3 IC,45 = (B.38) ef2NF sin θ2 + ef1NF cos θ2 ef3 IC,46 = (B.39) ef1 cos θ2 + ef2 sin θ2

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]:

1 T R0 X MTAS X RT AS (X)= det MT ASe− 2 (C.1) (2π)2 p where X is the four component column vector (X ,X ,X ,X )=(Q Q ,ω(q) ω (q )), 1 2 3 4 − 0 − 0 0 i.e. the resolution function is deﬁned in (Q,ω)-space. Note that X4 is the fourth com- D Dispersion relation properties of RbMnF3 131 ponent of the vector X and the quantity ∆ω = ω(q) ω (q) used in the calculations is − 0 diﬀerent and refers to energy deviations from the dispersion surface for a given wavevector q. R0 √detM is a normalization factor and R is given in equation (16) in [51]. (2π)2 T AS 0 Following Popovici the resolution matrix is given by:

1 1 1 − 1 T − T − T T MT− AS = BA D S + T FT D + G A B (C.2) h i 1 1 where G− is the covariance matrix of the distribution of the angular variables and F− 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 = IC J. (C.4)

The (6 6)-matrix I is deﬁned in Appendix B.3. Therefore, equation (C.2) can be refor- × C mulated yielding for the TAS resolution matrix in the frame of J:

1 1 1 − 1 T 1 T − − T 1 T L = I− A D S + T FT − D + G A I− (C.5) T AS C C − ( ) n o

D Dispersion relation properties of RbMnF3

The dispersion relation of RbMnF3 can be obtained from linear spin wave theory [52]:

1 E(q)= gβH + 2S (6J 12J + 8J ) + 4SJ B 2 4SJ A + 4SJ C 2 2 , (D.1) { A 1 − 2 3 2 } − { 1 3 } h i where

A = cos qha0 + cos qka0 + cos qla0 (D.2) B = cos(q + q ) a + cos(q q ) a + cos(q + q ) a h k 0 h − k 0 k l 0 +cos(q q ) a + cos(q + q ) a + cos(q q ) a (D.3) k − l 0 l h 0 l − h 0 132 Appendix

C = cos(q + q + q ) a + cos(q q q ) a h k l 0 h − k − l 0 +cos(q + q q ) a + cos(q q q ) a . (D.4) h k − l 0 h − k − l 0

J1, J2 and J3 are the exchange interactions between ﬁrst, second and third neighbors and H is the anisotropy ﬁeld. Here q = (2π/a Q ) and Q are in r.l.u.. Magnetic A h,k,l 0 · h,k,l h,k,l zone centers are located at ( 0.5, 0.5, 0.5). According to [52] it is assumed that H = 0, ± ± ± A J2 = 0, J3 = 0, S = 5/2, a0 = 4.204Å and J1 = 0.293meV. Hence, the dispersion relation reduces to 1 E(q)= ~ω(q) = 4SJ 9 A(q) 2 2 . (D.5) 1 − { } h i 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 A = 4SJ1a0 sin a0qh,k,l . (D.6) ∂qh,k,l √9 A2 − The elements of the curvature matrix can be calculated from the analytical expressions

2 2 ∂ E 4Sa0J1 2 3 = 3 9sin qha0 9A cos qha0 + A cos qha0 (D.7) ∂qh∂qh −(9 A2) 2 − − and 2 2 ∂ E 36SJ1a0 sin qha0 sin qka0 = 3 . (D.8) ∂qh∂qk − (9 A2) 2 − Other second derivatives are obtained from cyclic permutation of h, k and l.

E Growth of copper nitrate single crystals

Required deuteration ratio

x Using the incoherent cross sections listed in Tab. 6.1, the incoherent cross section σinc of Cu(NO ) 2.5H O, Cu(NO ) 2.5D O and a mixture of both can be calculated. Here x 3 2· 2 3 2· 2 is deﬁned as the percentage fraction of Cu(NO ) 2.5D O in the mixture. Therefore, the 3 2· 2 incoherent cross section σx of pure Cu(NO ) 2.5D O and pure Cu(NO ) 2.5H O yields: inc 3 2· 2 3 2· 2 1 0.55 + 2 0.5 + 8.5 0 + 5 2.05 σ100 = · · · · b = 0.72 b (E.1) inc 16.5 1 0.55 + 2 0.5 + 8.5 0 + 5 80.26 σ0 = · · · · b = 24.42 b. (E.2) inc 16.5 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 compounds 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 σx is comparable to the incoherent cross section σ100 of pure Cu(NO ) 2.5D O. Thus, a inc inc 3 2· 2 deuteration ratio of 98.5% resulting in

1 0.55 + 2 0.5 + 8.5 0 + 5 (80.26 0.015 + 2.05 0.985) σ98.5 = · · · · · · b inc 16.5 100 = 1.0707 b = 1.487 σinc , (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, g g g · D2O and H2O are 232.59 mol , 20 mol and 18 mol , respectively. For the distillation process 140.99g Cu(NO ) 2.5H O powder material, containing 27.04g 3 2· 2 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. 134 Appendix

added distilled thereof thereof H2O in D2O in D2O D2O D2O liquid H2O D2O solution solution fraction fraction [g] [g] [g] [g] [g] [g] [g-%] [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 deuteration level 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 spectrometer 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 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.

Z L

C2 R

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 1 = iωC2 + . (F.2) Z 1 + iωL + R iωC1 136 Appendix

The real part of equation (F.2) should be 50Ω while the imaginary part should be zero:

1 R Re = = 50Ω (F.3) Z 2 R2 + ωL 1 − ωC1 1 ωL 1 Im = ωC − ωC1 = 0 (F.4) Z 2 − 2 R2 + ωL 1 − ωC1 Therefore, the resulting capacitances are:

1 C1(ω) = (F.5) ω2L ω√ZR R2 − 1 − L 2 − ω C1 C2(ω) = 2 . (F.6) R2 + ωL 1 − ωC1 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.

f = 100kHz f = 200kHz f = 300kHz 360 2.4 175 3 140 350 1.3 2.2 170 2.5 135 340 2 1.2 165 130 330 1.8 2 [nF] [nF] [nF]

2 1.1 2 160 2

C C C 125 320 1.6 155 1.5 1 120 310 1.4 150 115 1 300 0.9 1.2 145 700 750 800 80 85 90 95 28 30 32 34 C [nF] C [nF] C [nF] 1 1 1

Fig. F.2: Resonance of the induced voltage as a function of the capacities C1 and C2 for frequencies of 100kHz, 200kHz and 300kHz. The measurements were done in 8π-mode (4 parallel RF coils). The black line shows the 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 connected, resulting in R 0.25Ω and L 11µH. The maximum value of the resonance tot ≈ tot ≈ shows little sensitivity to C2. For f = 100kHz the resonance is quite broad, while it gets G Adiabatic and non-adiabatic transitions of the magnetic moment of theneutron 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 starting 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

ω ω , (G.1) L ≫ B d(B/~ B~ ) where ωL = γ B is the Larmor frequency and ωB = | | is the frequency of the change | |· dt 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

L ω 1. (G.2) L v ≫ 138 Appendix

For the non-adiabatic transition the analogous condition reads:

ω ω (G.3) L ≪ B and can be rewritten as L ω 1. (G.4) L v ≪ Adiabatic transition: The guide ﬁeld from the monochromator to the coupling coil is oriented 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 m thus, a neutron velocity of v = 1675 s equation (G.2) yields:

L Hz 0.18m ωL = 2916 30G m 9.4 1, (G.5) v G · 1675 s ≈ ≫ 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 m and therefore a neutron velocity of v = 659 s equation (G.4) yields:

L Hz 0.0005m ωL = 2916 30G m 0.066 1 (G.6) v G · 659 s ≈ ≪

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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 reinforced 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 compensate 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

145 146 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 components as well as setting up the NRSE instrument. Thanks to Norbert Beul for designing and manufacturing the coupling coils and the compensation coils.

Thanks to Dr. Diana Quintero-Castro for the Sr3Cr2O8 crystal and a lot of fruitful discussions. 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.