Development of Triplet Dynamic Nuclear Polarization for Polarization Analysis in Small-Angle Neutron Scattering

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Polarization for Polarization Analysis in Small-Angle Neutron Scattering

Inauguraldissertation zur

Erlangung der W¨urde eines Doktors der Philosophie vorgelegt der

Philosophisch-Naturwissenschaftlichen Fakult¨at der Universit¨at Basel

von

Yifan Quan

2021

Originaldokument gespeichert auf dem Dokumentenserver der Universit¨at Basel edoc.unibas.ch

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Genehmigt von der Philosophisch-Naturwissenschaftlichen Fakult¨at auf Antrag von

Prof. Dr. Michel Kenzelmann Dr. Patrick Hautle

Prof. Dr. Martino Poggio Prof. Dr. Robert Griffin

Basel, 17.11.2020

Prof. Dr. Martin Spiess

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This thesis work was performed in the Laboratory for Neutron and Muon Instrumentation (LIN) of the Paul Scherrer Institute in Switzerland.

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Acknowledgements

I would like to thank many people who offered help and support in completing this thesis.

First, I would like to give credits to all my colleagues at PSI. My PhD supervisor Patrick Hautle, who brought me to the field of DNP and strongly supported me in performing all the experiments. His great scientific assistance and guidance on how to conduct them independently had a tremendous impact on my development as a physicist. To Ben van den Brandt, who unfortunately passed away while I was working on this thesis and is deeply missed. Ben was not only very encouraging and supportive during our experiments, but also gave me great advice for my daily life in Switzerland. To Tom Wenckebach, who knows everything about DNP and shared his wisdom with me. I very much enjoyed the highly stimulating discussions that often resulted in fantastic projects. To my PhD fellows, especially to my successor Jakob Steiner, who will continue the work and hopefully achieve great results, and to my predecessor Nemanja Niketic, who transferred to me all his knowledge about the triplet DNP project. I enjoyed the time of working with them very much. To Joachim Kohlbrecher, who gave me great support during all SANS experiments at PSI. To Paul Schurter, who provided me with considerable technical support. To Michel Kenzelmann, who hosted me at PSI and gave me scientific advice.

My thanks also go to our collaborator in Luxumbourg, Andreas Michels, who has shared his knowledge in magnetism and SANS with me.

Finally, I would like to thank my family, friends and especially my beloved Anqi Sun.

Their constant support makes me not afraid of failure and keeps me pursuing my goal.

PSI Villigen, December 2020

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Abstract

We present the status of the development of a novel neutron spin filter based on the dynamic nuclear polarization (DNP) of protons in a naphthalene single crystal that uses highly polarized optically excited triplet states of pentacene as the polarizing agent (PA).

The filter is applied as spin analyzer in small-angle neutron scattering (SANS) experiments to study magnetism.

In order to improve the spin filter performance, a better understanding of the electron polarization creation was essential. For this purpose, careful light absorption measurements have been performed an a theory developed to describe the light propagation and absorption in the biaxial anisotropic absorptive pentacene:naphthalene single crystal and the subsequent triplet production of pentacene molecules. The DNP build-up in a crystal of given size can now be simulated and optimized by the proper choice of the experimental parameters, e.g. the type of excitation light source or the pentacene dopant concentration.

As a result 80% proton polarization can now be routinely achieved, close to the theoretical maximum, with extremely long relaxation times. This significantly improved the figure of merit of the spin filter and furthermore allowed to implement a new scheme of filter operation that greatly facilitates its operation in the environment of a large-scale neutron scattering facility. We have made the device transportable, i.e. the filter is conveniently polarized under optimum conditions in the laboratory and then transferred to the neutron beam line where it can be operated during several days with practically frozen polarization while requiring only a minimum of equipment. This saves cost on instrumentation, beam time and work.

These improvement allowed to apply the spin filter as neutron polarization analyzer in a series of polarized SANS experiments to study magnetism on the nano-scale. These studies focused on an exotic and elusive physical phenomena – the defect-induced Dzyaloshinskii- Moriya interaction (DMI) in a nanocrystalline two-phase alloy Fe₇₃Si₁₆B₇Nb₃Cu₁. An asymmetric signal is observed in the difference between the two spin-flip cross sections, which is a key signature directly related to the DMI. The result supports the generic relevance of the DMI for the magnetic structure of defect-rich ferromagnets.

Two additional studies are presented that do not directly relate to the spin filter subject but further exploit the unique properties of highly polarized proton spin systems with extremely long relaxation times that now can be prepared as a result of all the optimized processes. The first addresses a fundamental issue of DNP that storage and transport of hyperpolarized samples is severely restricted. A procedure and equipment is presented to transport polarized samples over long distance and provide hyperpolarized nuclear spins to users that are not in the possession of DNP equipment. The second studies the long-range nuclear magnetic ordering (ferromagnetic or antiferromagnetic) that is created by adia- batic demagnetization in the rotating frame. We focus on the antiferromagnetic nuclear spin configuration of hydrogen nuclei in a naphthalene single crystal, similar to the antiferromagnetic structure of electron spins ordered by the Heisenberg exchange interaction.

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List of Figures

2.1 Wavelength dependence of the neutron-proton scattering cross sections. . . 7

2.2 Performance of a naphthalene neutron spin filter based on polarized protons. 8 2.3 Figure of meritQ of the triplet spin filter. . . 8

2.4 Pentacene molecule and the crystal structure of pure naphthalene. . . 11

2.5 A schematic drawing of the electronic levels and transitions in a pentacene molecule after photo-excitation. . . 12

2.6 The effective frequency vector in the rotating frame. . . 16

2.7 The transitions which are induced by the spin operators in the various terms in the super-hyperfine interaction between the electron spin and the nuclear spin. . . 17

2.8 Scheme of the solid effect in the rotating frame. . . 18

2.9 Scheme of the ISE in the rotating frame. . . 19

2.10 Scheme of the ISE sequence. . . 22

2.11 Proton polarization build-up in a pentacene:naphthalene crystal. . . 23

2.12 The cryostat and the ESR cavity. . . 24

2.13 Set up of the 556 nm laser. . . 25

3.1 Index ellipsoid of the naphthalene crystal. . . 29

3.2 Incident light beam perpendicular to the cleavageab-plane. . . 33

3.3 Refractive ellipsoid with light propagating perpendicular to the ab-plane. . 34

3.4 Apparatus scheme used for the transmission measurements. . . 36

3.5 Transmission spectra at room temperature with the incident light beam~k perpendicular to the ab-plane. . . 38

3.6 Transmission (incident light beam~k ⊥abplane) as a function of the polarization angle at λ= 597 nm. . . 38

3.7 Transmission spectra (incident light beam~k ⊥ab plane) at different temperatures. . . 40

3.8 Light absorption coefficients as function of temperature for the 0-0 transition at the peak wavelength near 600 nm. . . 41

3.9 Light absorption coefficients as function of temperature at 556 nm. . . 41

3.10 Light absorption coefficients as function of temperature at 515 nm. . . 42

3.11 Level scheme for optical excitation. . . 43

3.12 Pulse shapes of the Yb:YAG 515 nm disk laser and the solid state 556 nm laser operating at 1 kHz. . . 49

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3.13 The proton polarization homogeneity of a sample after 30 min ISE, compared to the simulation. The 515 nm laser system was deployed with light propagating along the b-axis and operated at 1 kHz. The solid line is the simulation of the polarization homogeneity (3.75) with the corresponding experimental parameters. Additionally we estimate the beam area to be A = 36 mm² and a very efficient ISE with E_ISE = 0.68 close to theoretical maximum. . . 51 3.14 Simulation of the triplet probability of a N₁ = 1.88×10²³ m⁻³ sample at

25 K shortly after each laser pulse of the two laser systems. . . 52 3.15 ISE polarization build-ups of aN₁ = 1.88×10²³m⁻³ concentrated crystal

at 25 K with the two laser systems running at 1 kHz and the corresponding simulations. . . 52 3.16 Simulation of the triplet probability of a N1 = 4.45×10²³ m⁻³ sample at

25 K shortly after each laser pulse of the two laser systems. . . 53 3.17 ISE polarization build-ups of aN₁ = 4.45×10²³m⁻³ concentrated crystal

at 25 K with the two laser systems running at 1 kHz and the corresponding simulations. . . 53 3.18 Simulation of the polarization build-up for samples with different pentacene

concentrations using the X-band system and the 556 nm laser. . . 54 3.19 Simulation of the polarization homogeneity for samples with different pen-

tacene concentrations after a 1000 min DNP build-up using the X-band system and the 556 nm laser. . . 55 3.20 Simulation of a 1000 min polarization build-up and the achieved homogene-

ity of a sample with a 1.88×10²³ m⁻³ pentacene concentration using the X-band system and the 556 nm laser with and without considering relaxation. 56 4.1 Simulation of the electromagnetic B component. The colors give the inten-

sity of the z component of the B field. . . 60 4.2 ESR signal of a standardγ-irradiated E’ sample at room temperature mea-

sured with the 5 GHz spectrometer using a stimulated echo sequence with a 50 ns π/2 pulse at 5.1474 GHz and B= 0.182 T. . . 61 4.3 ESR signal and spectrum of pentacene in a naphthalene single crystal mea-

sured with the 5 GHz system . . . 61 4.4 The polarization build-up by ISE of a 5.5 × 4.5 × 4 mm³ pen-

tacene:naphthalene sample using the 5 GHz system and the NMR signal. . . 62 4.5 Polarization homogeneity of a naphthalene sample polarized by the 5 GHz

system. . . 62 4.6 Simulation of the polarization build-up using the 5 GHz system and the

556 nm laser for different sample concentrations with and without taking into account a 40 h homogeneous relaxation. . . 63 4.7 Simulation of the polarization homogeneity after 1000 and 2000 min build-

up using the 5 GHz system and the 556 nm laser with different sample concentrations. . . 64 4.8 Simulation of a 2000 min polarization build-up and the homogeneity after-

wards of a 1.88×10²³ m⁻³ sample using the 5 GHz system and the 556 nm laser with and without considering relaxation. . . 64 4.9 Simulation of the polarization build-up and the homogeneity afterwards of

a 1.88×10²³m⁻³ sample with the 556 nm laser exciting from one and both sides considering a 40 h homogeneous relaxation. . . 65

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4.10 Nuclear spin lattice relaxation of several pentacene:naphthalene crystals at

a temperature of T = 25 K. . . 67

4.11 Nuclear spin lattice relaxation of the pentacene:naphthalene spin filter crystal at a temperature ofT = 80 K. . . 67

4.12 Nuclear spin lattice relaxation of the pentacene:naphthalene spin filter crystal at a magnetic field ofB = 24 mT. . . 68

4.13 The sketch illustrates the procedure of removing the cryostat from the DNP setup. . . 69

4.14 Proton polarization build-up in a pentacene:naphthalene crystal at 25 K and 0.36 T monitored by pulse NMR. . . 70

4.15 Temperature evolution recorded during the transport. . . 70

5.1 Scheme of a typical setup for a SANS experiment with polarization analysis. 76 5.2 Concept of the SANS experiment with the polarization analysis. . . 77

5.3 Result of the spin leakage correction. . . 80

5.4 Corrected intensity plots of four spin channels by SANS polarization analysis on the nanocrystalline alloy FINEMET. . . 81

(a) S++ . . . 81

(b) S₋₋ . . . 81

(c) S₊₋ . . . 81

(d) S₋₊ . . . 81

5.5 Normalized room temperature magnetization curve of Vitroperm (Fe73Si16B7Nb3Cu1). . . 84

5.6 Nuclear and magnetic SANS cross sections at magnetic saturation. . . 87

5.7 Magnetic field dependence of the DMI asymmetry in Vitroperm (Fe₇₃Si₁₆B₇Nb₃Cu₁) at selected applied magnetic fields. . . 88

5.8 DMI asymmetry at 20 mT and 14.5 mT measured with better statistics. . . 89

5.9 Azimuthally-averaged ∆Σ(q) at 14.5 mT and 20 mT. . . 89

5.10 Estimation of the DMI strength for nanocrystalline Vitroperm. . . 90

5.11 Polarization homogeneity of the spin analyzer after transportation and in- stallation on the neutron instrument measured by neutron transmission. . . 92

5.12 Polarization evolution of the spin analyzer over the 2 days of the neutron experiment. . . 92

5.13 Spin leakage corrected intensity plots of four spin channels of polarized SANS on the Vitroperm sample at 20 mT. . . 93

(a) Σ⁺⁺ . . . 93

(b) Σ⁻⁻ . . . 93

(c) Σ⁺⁻ . . . 93

(d) Σ⁻⁺ . . . 93

5.14 Difference between the two spin-flip cross sections. . . 94

5.15 Polarized neutron beam traversing a sample. . . 97

5.16 Depolarization of the transmitted neutron beam as a function of the Vit- roperm sample thickness at different external magnetic fields. . . 100

5.17 Depolarization of the transmitted neutron beam as a function of the external magnetic field for different Vitroperm sample thicknesses. . . 101

5.18 Simulation of the Vitroperm SANS contrast signal as function of the sample thickness with and without depolarization. . . 102

5.19 Normalized contrastI⁺−I⁻ and the unpolarizedI⁺+I⁻ scattering inten- sities as a function of the sample thickness. . . 103

6.1 The procedure of sample extraction from the DNP setup. . . 107

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6.2 Devices for transport polarized samples: the permanent holding magnet, the Halbach transport magnet and the cryogenic dry shipper. . . 108 6.3 Proton thermal equilibrium signal of pentacene:naphthalene crystal at a

temperature of T = 80 K at a field of B = 0.55 T with a polarization of P = 7.083×10⁻⁴%. . . 109 6.4 oss of polarization during the process of extraction from the cryostat, trans-

fer into the dry shipper and loading back into the cryostat. . . 110 6.5 Decay of the polarization during storage in the dry-shipper at T ∼ 80 K

and B ∼0.75 T. . . 110 6.6 Proton NMR signals of sample #3 before extraction from the DNP appa-

ratus (left) and 8 h later after transport over 500 km. . . 112 6.7 Proton NMR signals of the polarized sample before extraction from the

DNP apparatus and after crushing to a powder and re-insertion. . . 113 7.1 The five anti-ferromagnetic configurations projected on the crystalline ac-

plane. . . 131 7.2 The two anti-ferromagnetic configurationsm= 495 andm= 426 projected

on the crystalline ab-plane. . . 133 7.3 Precise determination of the Larmor frequency by AFP. . . 134 7.4 Proton NMR signal before and after ADRF from a negative spin temperature.135 7.5 Proton NMR signals in the dipolar state after ADRF from negative spin

temperatures with different initial proton polarizations. . . 136 7.6 Proton NMR signals in the dipolar state after ADRF from negative spin

temperatures, where the ADRF center frequency is set slightly off the Lar- mor frequency. . . 136 7.7 Protons NMR signals after ADRF from positive spin temperatures. . . 137 7.8 Protons NMR signal after a (quasi) ADRF sequence, where the B1 field is

much stronger than the dipolar interaction. . . 138 7.9 Comparison of¹³C pulse NMR signals with the protons in the demagnetized

state that is achieved by ADRF from a negative spin temperature with different initial proton polarizations. . . 139 7.10 Comparison of¹³C pulse NMR signals with the protons in the demagnetized

state that is achieved by ADRF from a negative (ordered) or a positive (not ordered) spin temperature. . . 140 A.1 Temperature sensitivity of the NMR circuit. . . 146 A.2 NMR signal as function of the position of the 5 mm long sample within the

NMR coil. . . 147 A.3 Saturation of NMR measurement used for the polarized sample with 16

weak pulses (42 dB pules attenuation) averaged. . . 147 A.4 Ratio of NMR signal integrals measured with non-saturating and 45^◦ pulse. 148 A.5 Proton thermal equilibrium signal of pentacene:naphthalene crystal at a

temperature of T = 150 K at a field ofB = 0.55 T (f = 23.615MHz). . . 149 B.1 Reference sample before (left) and after (middle) baking and mixing. Before

baking, the pentacene stays at the bottom of the ampoule and fully covered by the naphthalene powder. . . 152 B.2 Pentacene transmission spectrum taken 2 min after the sample transferred

into the heated holder and completely melted. . . 153 B.3 Degrading of the pentacene concentration as a function of time. . . 153

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List of Tables

3.1 Notation. . . 28 3.2 Results of light absorption measurements of the sample in liquid and solid

phases. . . 39 3.3 The wavelengths of the three absorption peaks at different temperatures. . 40 6.1 Details of the long range transport experiment. . . 111 6.2 List of the samples stored and transported over 500 km (PSI - Geneva - PSI).111 7.1 Results for the maximum and minimum values of the dipole sum. . . 130 7.2 The anti-ferromagnetic configurations. . . 131 7.3 Normalized polarizations for the structure 495 near the transition temper-

ature. . . 133 A.1 Error table for the polarization calibration with the NMR system. . . 149 C.1 Light absorption coefficients of pentacene in the naphthalene single crystal

for the absorption peak at around 600 nm. . . 155 C.2 Light absorption coefficients of pentacene in the naphthalene single crystal

at 556 nm. . . 156 C.3 Light absorption coefficients of pentacene in the naphthalene single crystal

for the absorption peak at around 556 nm. . . 156 C.4 Light absorption coefficients of pentacene in the naphthalene single crystal

at 515 nm. . . 157 C.5 Light absorption coefficients of pentacene in the naphthalene single crystal

for the absorption peak at around 515 nm. . . 157 D.1 The crystal parameters of naphthalene and the positions of the hydrogen

atoms. . . 159

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Chapter 1

Introduction

The use of spin-polarized neutrons is a powerful tool to study a wide range of scientific phenomena in fundamental and condensed matter physics. Polarized magnetic neutron scattering e.g. has contributed more than any other technique to the understanding of magnetism at the atomic level.

Already in 1939 Halpern and Johnson predicted in their seminal paper [1] that the polarization of magnetically scattered neutrons depends on the orientation of the scattering vector with respect to the polarization of the incident neutrons. Therefore polarization analysis in neutron scattering provides an excellent method to distinguish between magnetic and nuclear scattering which is isotropic. Only much later Moon, Riste and Koehler [2] introduced in their classical experimental study the one dimensional polarization analysis and demonstrated the polarization dependence of nuclear and magnetic scattering, which allowed the unambiguous separation of magnetic scattering from nuclear scattering, and nuclear coherent scattering from spin-incoherent scattering. The technique was later expanded to the xyz polarization analysis [3] and paved the way for today’s zero field spherical polarization analysis devices [4, 5]. However, neutron polarization analysis for the scattering along the forward direction became only recently possible with the development of³He spin filters [6] that allow the neutron spins from a divergently scattered beam to be assayed. This rather new option triggered the interest in polarized small-angle neutron scattering (SANS), and using the polarization analysis possibility. It now becomes a most powerful tool for studying the magnetic microstructure of bulk materials [7–9], as it can measure the change of nuclear and magnetization contrast across interfaces on length scales between a few nanometers and microns.

However integrating a³He spin filter in an experimental environment typical for magnetic scattering is demanding due to its sensitivity to stray magnetic fields [10]. To limit relaxation of the ³He polarization, i.e. the decrease of filter analyzing efficiency, the filter cell has to be placed in a very homogeneous magnetic field environment at some distance (∼1 m) to the sample under investigation [11, 12].

At PSI the development of an alternative neutron spin filter based on the strong spin dependence of the neutron scattering on protons has been pioneered. It is small and works in inhomogeneous fields. The necessary large proton polarization in a single crystal of naphthalene doped with pentacene is created with a recent method of dynamic nuclear polarization (DNP) [13] that uses photo-excited triplet states of pentacene [14] as polarizing agent and requires only moderate experimental means. Prior to this thesis work several development steps have already brought the spin filter performance to a considerable level. In an initial step it has been demonstrated that the method can be used to build a reliably working spin filter for neutrons operating at 0.3 T at a temperature of 100 K [15, 16] and next showed that by using deuterated pentacene the DNP efficiency can

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be much improved [17]. An extensive study of the triplet-DNP mechanism then enabled to optimize the polarization process close to the theoretical maximum [18, 19], which resulted in proton polarization levels of 70% at a field of 0.36 T using a simple helium flow cryostat for cooling [20]. In a proof of principle experiment it was finally demonstrated that the

“triplet spin filter” is well suited to perform polarization analysis in SANS in a magnetic field environment where a³He filter can only be operated with a proper magnetic shielding [21].

These achievement build a sound basis from which the present thesis starts out with the goal to address the open fundamental and technical questions that need to be solved for the transformation of the experimental setup into a device that can be implemented on a neutron beamline or instrument. Here the development of a proper understanding of the optical absorption and subsequent triplet creation processes is key to fully exploit the potential of triplet-DNP, to create a filter with extremely high polarization and long decay time. For its operation in a neutron experiment ways had to be found to simplify the handling of the spin filter and to decrease its footprint. The new device then enables longitudinal polarization analysis in magnetic SANS that allows extracting information on the spatial distribution of magnetization inside the material under study. Specifically the novel analyzer is key to study the chiral magnetism displayed in defect rich nanocrystalline materials.

Structure of the thesis

Inchapter 2an overview of the triplet spin filter is given. The filtering principle based on the spin dependence of the neutron proton scattering is outlined and the nuclear scattering cross sections are given that allow to estimate the figure of merit of the device. Further- more the pentacene:naphthalene system is introduced and the efficient DNP mechanisms using triplet states are described together with the main parts of the apparatus. Chap- ter 3 starts with a formal description of the light propagation in the biaxial anisotropic absorptive media of a pentacene doped naphthalene single crystals and derives expressions that allow to extract the light absorption coefficients as a function of wavelength, light polarization and temperature from a transmission measurement performed with a simple experimental setup. In the second part of the chapter a model is developed to describe the photo-excitation profile and the subsequent triplet state distribution. Using the numerical values of the light absorption coefficients as input, the proton polarization profile along the laser beam direction can be calculated. The model can guide the choice of experimental parameters to achieve the optimum DNP performance. Two important improvements of our spin filter are addressed in chapter 4. In order to increase the angular acceptance as well as the figure of merit of the filter, larger crystals need to be used. This required the set up of a new pulsed ESR/DNP system with a larger microwave resonance structure operating at 5 GHz. Furthermore, the extremely long relaxation times achieved allowed to implement an ex-situ operation procedure, i.e. the filter is polarized in the laboratory under optimum conditions and then transfer it to the neutron beamline where it can be operated during several days with almost constant polarization while requiring only a minimum of equipment. The spin filter then serves as neutron polarization analyzer in a series of polarized SANS experiments to study magnetism on the nano-scale. Chapter 5 introduces the concept of magnetic SANS and the specifics of performing polarization analysis with the triplet spin filter and then presents a study focusing on the defect-induced Dzyaloshinskii-Moriya interaction (DMI) in a nanocrystalline two-phase alloy.

Two additional studies are then presented that do not directly relate to the spin filter subject but further exploit the unique properties of highly polarized proton spin systems

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with extremely long relaxation times that now can be prepared as a result of all the optimized processes. Chapter 6addresses a fundamental issue of DNP: as a result of fast spin-lattice relaxation induced by the polarizing agent, long time storage and transport of hyperpolarized samples is severely restricted and the apparatus for DNP is necessarily located near or integrated with the apparatus using the hyperpolarized spins. In triplet- DNP the PA induced relaxation can be easily eliminated by switching off the light after the sample being hyperpolarized. The idea of the transportable spin filter presented in chapter 4 is taken a step further and a procedure is presented to transport polarized samples over long distance and provide hyperpolarized nuclear spins to users that are not in the possession of DNP equipment. Finally inchapter 7 nuclear dipolar magnetic ordering is discussed, an elusive phenomena that has been observed only in a very few cases.

At very low temperature a nuclear spin system subjected to dipole-dipole interaction can reach a long range ordering. We focus our study on the antiferromagnetic nuclear spin configuration, similar to the antiferromagnetic structure of electron spins ordered by the Heisenberg exchange interaction.

At the end in chapter 8 an overall conclusion is drawn and perspectives related to this work are highlighted

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Chapter 2

Principle of the triplet-DNP neutron spin filter

In the following chapter we give an overview of the basic principles governing the real- ization and application of a neutron spin filter based on polarized protons. The filtering mechanism is outlined and the nuclear scattering cross sections are given that allow to estimate the efficiency of the device. Furthermore the methodology of triplet-DNP applied to the system of pentacene doped naphthalene crystals is introduced and the main parts of the experimental apparatus are described.

2.1 Neutron polarization techniques

For the measurement of the polarization dependence of neutron scattering cross sections several techniques for producing and analyzing polarized neutron beams have been developed.

Supermirror polarizers which rely on spin-dependent neutron optical reflection are the standard device [22]. They are easy to setup and use, have a high polarizing efficiency for cold and thermal neutrons, are stable in time and need no maintenance. However, these devices are restricted in their angular acceptance and not suited for polarization analysis of divergent beams. Furthermore the neutron energy is limited to energies below 20 meV (corresponding to wavelengths longer than 2 ˚A) because their transmission efficiency drops dramatically for shorter wavelength.

These limitations do not exist for neutron spin filters. They rely on spin-dependent nuclear scattering, e.g., on polarized protons [23], or absorption on polarized ³He [24]. The analyzing power and transmission of a homogeneous spin filter with parallel entrance and exit windows perpendicular to the beam are practically independent on the beam divergence, making them predestined for polarization analysis and also leads to a principle advan- tage for high precision neutron polarimetry [25, 26]. Furthermore, spin filters can polarize the whole range of cold, thermal and hot neutrons. With the advent of strong pulsed neutron spallation sources this property becomes even more important, since they allow time-of-flight methods to be employed to analyze all wavelengths in a single experiment [27].

Spin filters of optically polarized ³He are available at several neutron research centers and present efforts (see for example [28]) aim to further optimize this technique. ³He is hyperpolarized either via Spin-exchange Optical Pumping (SEOP) [29] or Metastability- Exchange Optical Pumping (MEOP) [30] and employs the spin-dependent neutron absorption [31] to polarize neutrons. Filter cells suited to polarize or analyze wide beams

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are available, but the fact that the ³He filter cell has to be placed in a very homogeneous magnetic field (∆B/B < 5×10⁻⁴cm⁻¹) makes it difficult to use as analyzer in a sample environment with a magnetic field, typical for the investigation of magnetic scattering. Furthermore, however not so much relevant for conventional SANS setups that use a monochromatic beam, the strong energy dependence of the absorption cross section does not allow optimizing the filter thickness for a large neutron energy range.

Shapiro and co-workers demonstrated that dynamically polarized proton spins offer an attractive possibility to realize a broad band neutron spin filter, as the spin-dependent neutron-proton cross section is large in a broad wavelength range [23]. An actual imple- mentation of a proton spin filter has been so far restricted to a few special cases, e.g., for a precise beam polarization determination [32] or to polarize neutrons of thermal [33] and epithermal energy where it is the only method currently available [34]. This is most prob- ably due to the necessary cryogenics and magnets needed for the classical DNP systems employed so far.

The problems and constraints of both types of the spin filters can be largely solved using photo-excited triplet states for DNP of proton spins (see section 2.3).

2.2 Using polarized protons as neutron spin filter

¹

2.2.1 Spin-dependence of the neutron-proton scattering

The working principle of a polarized proton spin filter is based on the fact that the singlet cross section for neutron-proton scattering is much higher than the triplet cross section [36].

Neutrons with spin anti-parallel to the proton polarization will be much stronger scattered, while neutrons with spin parallel to the proton polarization are preferably transmitted.

The Bragg cutoff wavelength is about 14 ˚A, above which only incoherent scattering on bound protons takes place and the spin dependent incoherent cross section per nucleus is given by [36]

σ_inc=πb²_N

I(I+ 1)−I²P²−IpP

(2.1) wherebN =−(5.824±0.002)×10⁻¹²cm is the spin-dependent scattering length difference [37],pthe neutron polarization,I = ¹₂ andP the proton polarization. Putting in numerical values and assuming a fully polarized neutron beam (p= 1) we obtain

σ_±= 79.9

1−1

3P²∓ 2 3P

×10⁻²⁴cm² (2.2)

where the ± stands for the two eigenstates of the neutron spin. In the intermediate energy range between 3 and 0.3 ˚A, the cross sections become very complex and cannot be theoretically predicted due to interference and inelastic scattering. We generalize Eq.

(2.2) to the form [23]

σ_±=σ₀⁰−σ₀^PP²∓σ_PP (2.3) where σ⁰₀, σ^P₀ and σP are empirical parameters. The neutron spin-independent cross section includes two termsσ₀⁰andσ₀^P, where the latter depends on the proton polarization, while σ_P is the so called ”polarization cross section”.

For neutron energies larger than inter-atomic bonds (E > 1 eV, λ < 0.3 ˚A) the cross sections approach those of the isolated free nuclei. Then for protons σ_P = 16.7 barn, σ₀= 20.36 barn [23] and for carbon nucleiσ₀= 4.7 barn [38].

1adapted from: Y. Quan, N. Nekitic, B. van den Brandt, and P. Hautle, A novel broad-band neutron spin filter based on dynamically polarized protons using photo-excited triplet states, EPJ Web of Conferences 219(2019) 10006 [35].

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When passing through a spin filter with a proton density N and a thickness d, the two spin components of the neutron beam will be attenuated according to

T_±=e^−σ^±^{N d} (2.4)

and an initially unpolarized beam will acquire a polarization A= T+−T₋

T₊+T₋ = tanh(−σ_PP N d) (2.5) which is called the filter analyzing power. The total transmission T through the filter increases with its polarizationP:

T = T₊+T₋

2 = exp

−(σ₀⁰−σ₀^PP²)N d

cosh(−σ_PP N d) (2.6) The filter quality factor is often judged by its figure of merit

Q=A²T (2.7)

2.2.2 Cross-sections and spin filter efficiency

In order to better quantify the performance of the triplet spin filter, a sound data set for the relevant neutron-proton cross sections as a function of the wavelength is needed for the filter material, which is in our case naphthalene. First measurements of the polarization dependent cross section have been presented in [15], while an extension of the data set became necessary to properly quantify the efficiency of the spin filtering process.

Figure 2.1: Wavelength dependence of the neutron-proton scattering cross sectionsσ₀⁰,σ^P₀ andσ_p for naphthalene. The values plotted without error bars are adapted from literature (see text). The lines are drawn to guide the eye.

Experiments were performed at the BOA beam line [39], at the continuous spallation neutron source SINQ at the Paul Scherrer Institute (PSI) in Switzerland. Figure 2.1 com- piles our results for the three cross sections [35] together with values published previously [23, 33]. Forσ₀⁰ we have smoothed the data, i.e. we have neglected the effect of any Bragg scattering as it depends on the exact orientation of the crystal which we are free to choose.

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For σ₀^P we give an estimate which is based on the theoretical values for the long neutron wavelength range and on literature values for the short wavelength range [23]. Moreover, below 1 ˚A, the scattering cross section for carbon has also been included [38].

With the effective cross sections at hand, we can now calculate the efficiency of the spin filter, i.e. the transmissionT, analyzing powerAand the figure of meritQ=A²T at a given proton polarization P, crystal thickness d and neutron wavelength λ. Figure 2.2 shows the results of simulations for two specific wavelengths, 6 ˚A and 0.33 ˚A, respectively. The former is a typical wavelength used in small angle neutron scattering (SANS) experiments and the latter corresponds to the 0.734 eVp-wave resonance in ¹³⁹La used in time reversal invariance violation searches [40]. In the long wavelength range the optimum thickness is around 7 mm, whereas for epithermal neutronsσ_P is smaller and thus the optimum filter thickness is larger, around 15 mm.

λ λ

Figure 2.2: Performance of a naphthalene neutron spin filter based on polarized protons.

Plotted are the analyzing (polarizing) efficiencyA, the total transmissionT and the figure of merit Q as a function of the thickness d for two wavelengths, (top): 6 ˚A [21] and (bottom): 0.33 ˚A.

λ W

F

Figure 2.3: Figure of meritQof the triplet spin filter for different thicknesses compared to high opacity polarized ³He cells. The plotted values forQ are determined from the data presented in figure 2.1 and the error is estimated to be about 5%. The black dots give the Q of the filter presently used for spin analysis in SANS experiments [41].

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Furthermore we may plot the figure of merit Q of the triplet filter as a function of the wavelenght and compare it with polarized gaseous ³He (Figure 2.3). The triplet filter gives aQcomparable to that of ³He filter with P = 0.7, optimized for 3 ˚A. By choosing a specific opacity factor of the³He gas cell, given by the pressure-length-wavelenght product in bar-cm-˚A, the Q of the ³He spin filter is optimized for a specific wavelength. In the case of naphthalene, as illustrated in figure 2.3, one either optimizes the filter thickness for cold neutrons (d ∼ 7 - 8 mm) or for the thermal and epithermal range (d > 10 mm) and obtains a high Q in a broad energy range. The triplet filter might be the preferred way to polarize high energy neutrons, since the opacity factor of the ³He spin filter gets limited by practical restrictions. Notice, an increase in polarization results not only in an improvement of the filter analyzing power A but even more of the transmission T. Thus the improvement in polarization from 0.7 to 0.8 would lead to a 40 % gain in the figure of merit compared to [20].

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2.3 Dynamic nuclear polarization (DNP) with photo- excited triplet states

The necessary high proton polarization required for an efficient filter, ideally above 70 %, can be achieved by dynamic nuclear polarization (DNP) [13]. The gist of this method is to exploit a high polarization of an electron spin system and transfer it to the nuclear spin system. In the first three decades after the inception of DNP in the 1950s [42, 43], its development was mainly driven by studies of the role of spin in nuclear and particle interactions—for a review see [44]. Subsequently its potential to enhance nuclear magnetic resonance (NMR) signals drove its development for applications in various fields of NMR [45]. In 2003 a breakthrough was achieved with the introduction of dissolution DNP, which enabled a huge increase in contrast in magnetic resonance imaging (MRI) [46, 47] and which opened the way to clinical applications [48]. In the usual classical scheme of DNP employed in these applications nuclear spins of interest are embedded in an insulating solid doped with an agent typically radicals—denoted as a polarizing agent (PA)—providing unpaired electron spins. The polarization of the latter is increased by cooling to about 1 K and applying a magnetic field of several tesla. Then a microwave field is applied to transfer this electron spin polarization to the nuclear spins. The necessary elaborate cryogenic equipment and superconducting magnet strongly limit applications.

furthermore the paramagnetic dopants are also the main source of nuclear spin lattice relaxation limiting the lifetime of the created nuclear polarization [49, 50].

DNP using photo-excited triplet states, also known as triplet-DNP, can solve these problems. The electron spin polarization of the triplets is basically independent of temperature and magnetic field. Thus, the protons can be hyperpolarized at moderate conditions and a compact apparatus can be developed (see section 2.4). Furthermore, the relaxation path- way via paramagnetic centers can be fully eliminated by switching off the photo-excitation after reaching high polarization. This fact we will later exploit to implement a more conve- nient way of filter operation in neutron scattering experiments (see section 4.3). A proven system for triplet-DNP, used for the triplet filter development, is a naphthalene molecular host crystal doped with a small concentration of pentacene guest molecules, typically a few 10⁻⁵mol/mol.

2.3.1 The pentacene:naphthalene system

Generally the valence electrons in aromatic molecules occupying the πbond have antipar- allel spins and form a singlet in the ground state. The spin part of the wave function is antisymmetric while the orbital part is symmetric. Upon photo-excitation, one of the electrons can be excited to another orbit with a higher energy. In principle the total spin is conserved so the excited state is still a singlet state, which will decay back to the ground state via fluorescence. However, spin-orbit coupling can induce a transition to a spin triplet state²[51]. This process is the so called intersystem crossing (ISC). Due to the fact that the ISC is spin selective, the triplet sublevels are not equally populated. Thus, the paramagnetic triplet states can have a high electron spin polarization and be used as polarizing agent (PA) for DNP. Figure 2.5 shows an example of the triplet creation process in an aromatic molecule (pentacene) and a more detailed discussion on the energy levels of the triplet state is given in the following.

Among the aromatic molecules pentacene has the advantages: it can be photo-excited in the visible wavelength range, has a high spin polarization in the triplet state and has a reasonable triplet lifetime of several tens of µs. These properties make pentacene

2the spin part of the wavefunction becomes symmetric while the orbital part becomes antisymmetric.

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an excellent candidate for triplet-DNP. Actually fully deuterated pentacene has a better DNP performance [17], due the weaker hyperfine interactions that lead to a narrower ESR line. Up to date only a very limited number of aromatic molecules have been used for triplet-DNP [52] and present efforts are directed towards developing water-soluble variants of aromatic molecules for biological applications [53].

The energy splitting (ESR frequency) of the triplet highly depends on the relative orientation of the aromatic molecule with respect to the external magnetic field (see section 2.3.2).

Thus, in order to achieve a high nuclear polarozation, a single crystal is indispensable. For this reason, most of the triplet-DNP using pentacene is performed in naphthalene or ter- phenyl [54] single crystals, which are well compatible with pentacene. We use naphthalene for the spin filter application due to its higher proton density and longer spin-lattice relaxation time that leads to superior DNP and spin filter performances.

The naphthalene crystal structure, as shown in Figure 2.4, was determined in [55, 56], refined in [57] and more recently in [58]. The crystal structure of pure naphthalene is monoclinic and belongs to the space group C_2λ⁵P21/a [59]. The ab-plane of the naphthalene crystal is the cleavage plane and the c-axis of naphthalene has approximately the same length as the X-axis of pentacene. It is possible to build a pentacene molecule into the crystal structure of naphthalene by replacing two naphthalene molecules by a single pentacene molecule.

Y

X

Figure 2.4: Top: Pentacene molecule with its principle axes X and Y (in plane) and Z (normal axis). Bottom: The crystal structure of pure naphthalene is monoclinic with a unit cell with axes a ∼ 8.2 ˚A, b ∼ 6.0 ˚A, c ∼ 8.7 ˚A and β ∼ 123^◦ [58] and belongs to the space group C2λ5P21/a [59]. The ab-plane is the cleavage plane. Two naphthalene molecules can be replaced at two positions by one pentacene molecule. Both sites share the pentaceneX-axis while theZ and Y-axes, respectively are at 47±3^◦. TheX-axis lies in theac-plane at an angle of 10^◦ to thec-axis [60].

To achieve high nuclear polarizations and long relaxation times naphthalene is extensively purified in a home-built zone-refinement apparatus [16] in typically more than 300 zone passes to eliminate possible paramagnetic impurities. The deuterated pentacene has been custom synthesized and purified by ISOTEC (Sigma-Aldrich group). All handling of the materials during the preparation process is conducted under a nitrogen atmosphere inside a glovebox. Large single crystals of several cm³ size are then grown in specially designed double walled ampoules employing a self-seeding vertical Bridgman technique [61, 62].

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They typically contain pentacene with a relative dopant concentration between 2 and 10×10⁻⁵mol/mol, determined by transmission spectroscopy (see section 3.3). Keeping the concentration levels in this range still assures a high crystal quality, which has been confirmed by neutron diffraction and furthermore proves to be well-adapted to the available laser pulse energies. For the spin filters cubic samples of typically 4 to 5 mm side length are cut out of the large single crystals, while the top/bottom planes correspond to the crystal ac-plane.

2.3.2 The electronic levels of the pentacene molecule

Figure 2.5 shows a detailed scheme of the electronic energy levels and transitions in a pentacene molecule with a magnetic field applied along the its molecular X-axis. The magnetic field is so strong that the Zeeman energy is much higher than the zero-field or fine splitting of the triplet (see below). These are our typical DNP conditions. With a short laser pulse the pentacene molecule is excited from a singlet ground state S₀ to the lowest excited singlet state S1. From S1 the system may decay back to the ground state under the emission of fluorescence in about 20 ns, or decay via intersystem crossing (ISC) to the lowest triplet state T₁. The selection rules for ISC strongly favor one of the three triplet levels [60].

Figure 2.5: A detailed schematic drawing of the electronic levels and transitions in a pentacene molecule after light excitation with the a high magnetic field applied along the pentacene molecular X-axis.

Fine structure of the triplet state

We continue with a more detailed discussion of the lowest triplet stateT1. A fine splitting of the triplet sublevels is induced by the spin-orbit coupling and the dipolar interaction between the two electron spins involved in the triplet state. Experimentally, [60] shows that the g-tensor is almost isotropic and the value is very close to that of a free electron. Hence, the spin-orbit coupling makes an extremely small contribution to the triplet structure and can be neglected.

Then the Hamiltonian describing the fine structure caused by the dipolar interaction is given by

HF =~(D(S_Z² −¹₃S(S+ 1)) +E(S_X² −S_Y²)), (2.8)

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where X, Y, Z corresponds to the pentacene molecular axes and S_X,Y,Z are the spin operators. The values for the fine splitting constants D/(2π) = 1381 MHz and E/(2π) =

−42 MHz are experimentally determined in [60]. We define the eigenstates of the S_Z as

|SZ,+i,|SZ,0i,|SZ,−i. The eigenvalue of the Hamiltonian are ^D₃ −E, ^D₃ +E and −^2D₃ , corresponding respectively to the eigenstates

|T_Xi= √¹

2(|S_Z,+i − |S_Z,−i),

|TYi= ^√¹

2(|SZ,+i+|SZ,−i),

|T_Zi=|S_Z,0i,

(2.9)

This energy splitting is the so called zero-field splitting and we immediately realize that the three zero-field eigenstates are the eigenstates ofS_X, S_Y, S_Z with the eigenvalue 0, i.e.

|T_Xi=|S_X,0i,|T_Yi=|S_Y,0i,|T_Zi=|S_Z,0i.

Next we want to calculate the energy splitting when applying an external magnetic field.

We restrict ourselves to the DNP condition where the field is parallel to the pentacene molecular X-axis. We define the z-axis of the laboratory frame along the external field direction and x-axis makes an angle φ with the molecular Y-axis. The Hamiltonian becomes

H=Z+HF =~[ω0SSz+D(S_Z² −¹₃S(S+ 1)) +E(S_X² −S_Y²)], (2.10) where the first term is the Zeeman interaction (ω_0S = −γ_S|B₀|, γ_S being the electron gyromagnetic ratio). The relation of spin operator between the laboratory frame and the molecular frame is

S_X =S_z

S_Y =S_xcosφ+S_ysinφ S_Z =−S_xsinφ+S_ycosφ ,

(2.11) and the Hamiltonian in the laboratory frame becomes

H=~{ω0SSz−¹₂(D−3E)[S_z²−¹₃S(S+ 1)]

−¹₂(D+E)[(S_x²−S_y²) cos 2φ+ (S_xS_y+S_yS_x) sin 2φ]}. (2.12) The matrix representation on the basis of the eigenstates ofSz is

H= ^~₂





2ω_0S−¹₃(D−3E) 0 −(D+E)e⁻^2iφ

0 ²₃(D−3E) 0

−(D+E)e^2iφ 0 −2ω_0S−¹₃(D−3E)



 . (2.13)

We assume that the external field is so high that the Zeeman interaction is much stronger than the dipolar interaction. Hence the eigenstates of the Hamiltonian approximates to be the eigenstates of the Zeeman interaction orS_z, denoted as|T₊i,|T₀i,|T₋i. We know thatSz=SX and we can immediately write down eigenstates in the basis of the zero-field eigenstates from (2.9) by cyclic permutation,

|T₊i= √¹

2(|T_Yi+|T_Zi)

|T₀i=|T_Xi

|T₋i= √¹

2(−|T_Yi+|T_Zi).

(2.14)

Two transitions should then be observed and the frequencies can be easily calculated from 2.13 to be

ω_± =ω_0S±¹₂(D−3E). (2.15)

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This is confirmed by the observation of two ESR lines separated by 1.51 GHz at ∼0.3 T, corresponding to a low field (|T₋i ↔ |T₀i) and a high field (|T₀i ↔ |T₊i) transition.

The three triplet energy levels are not equally populated as the selection rule of ISC strongly favors in populating the zero-field state |TXi. Van Strien gives the relative population of the state |T_X,Y,Zi at an applied field of 0.3 T [60]. With an external magnetic field applied, the zero-field states mix and govern the population of each triplet energy level. For our DNP condition, whenBkXandB is strong, the populations are calculated from (2.14) to be

ρ_T₀ =ρ_T_X = 0.91 for |T₀i,

ρ_T₊ =ρ_T₋= ¹₂(ρ_T_Y +ρ_T_Z) = 0.045 for|T₊i and|T₋i. (2.16) Triplet-DNP exploits the large population difference between (|T₋i,|T₀i) or (|T₀i,|T₊i), which is know as a high spin alignment. Notice that this is not the actual electron polarization −_S¹Tr(ρSz) that is calculated to be 0. This triplet electron spin alignment is practically independent of temperature and strength of the applied field. As discussed, it depends only on the direction of the external field, thus single crystals are indispensable.

Moreover, the lifetime of each triplet level is different and determined to be [63], τ_T₀ = 61±1µs,

τT+ =τT− = 290±70µs. (2.17)

At temperatures of 100 K and below the triplet spin lattice relaxation of the triplet (ther- malization of the triplet sublevels) [64] is much longer than its lifetime, thus the electron spin alignment can remain high for reasonably long time (several tens of µs) after the creation of the triplet state

2.3.3 DNP mechanisms and the solid effect

There are two families of mechanisms for DNP, known as the solid effect and the thermal mixing. Due to the low pentacene concentration in the samples mutual interaction between triplet states can be neglected and only the solid effect mechanisms – well resolved solid effect, differential solid effect (DSE), nuclear orientation via electron spin locking (NOVEL), integrated solid effect (ISE) – are present in triplet-DNP. Here we give a brief illustration of these, which is based on [65], where more details can be found.

As pointed out by Jeffries [43], the microwave field combined with the hyperfine interaction can induce a transition that flips an electron spin and a nuclear spin simultaneously. This is the basic transition of the so called solid effect. In order to explain the mechanism, we consider a simple system of a single triplet S = 1 interacting with a single nuclear spin I = ¹₂. A more realistic system is a singlet triplet interacting with many nuclear spins but the principle remains the same [65]. We align the pentacene molecular X-axis along the external fieldB₀, which we set as thez-axis of the laboratory frame. The microwave field B₁cosω_mt for the polarization transfer is applied perpendicular to B₀ along the x-axis.

The Hamiltonian is

H =~{ω_0SS_z+ 2ω_1SS_xcosω_mt+D⁰(S_z²−¹₃S(S+ 1)]

+E⁰[(S_x²−S_y²) cos 2φ+ (SxSy+SySx) sin 2φ]−ω0IIz+S·A·I}. (2.18) Here, the first term is the electron Zeeman interaction (ω_0S = −γ_S|B₀|, γ_S being the electron gyromagnetic ratio) and the second term is the interaction with the microwave field (Rabi frequency ω0S = −γS|B1|, B1 being the microwave field amplitude). The third and fourth terms are the fine splitting terms. Compared to (2.12), D⁰/(2π) =

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−¹₂(D−3E)/(2π) =−755 MHz andE⁰/(2π) =−¹₂(D+E)/(2π) =−670 MHz are defined.

The fifth term is the nuclear Zeeman interaction (ω_0I = γ_I|B₀|, γ_I being the nuclear gyromagnetic ratio), and the last term is the hyperfine interaction between the electron spin and the nuclear spin (A being the hyperfine tensor). The gyromagnetic ratio of the electron is set to be negative while that of the nuclei to be positive.

We transform the Hamiltonian to the rotating frame, H = ~

(ω_0S−ω_m)S_z + ω_1SS_x⁰ + D⁰S_z² − ω_0II_z + 1

2S_z(A_zxI_x+A_zyI_y)

. (2.19) Since the aim is to extract the slow evolution of the spin system, we average the Liouville von Neumann equation over a time 2π/ω. Then all terms oscillating with a frequency ω or 2ω vanishes. We assume the system is under a strong external magnetic field so that the nuclear Larmor frequencyω_0I is much larger than the hyperfine interaction. Thus we neglect theA_zzS_zI_z term, which only only slightly shifts energy levels, but does not cause transitions. Moreover the constant term−¹₃D⁰S(S+ 1) is eliminated by a shift of the zero point of the energy.

The spin 1 triplet and the fine splitting add complications. However, the two transitions are separated by−2D⁰/(2π) = 1508 MHz (see (2.13)), which is much larger than the ESR line width, hence the microwave can selectively excite one of the transitions, e.g. the high field transition (T₀ ↔ T₊). Then the microwave induced mixing of the T₋ state with the other states is negligible. The pentacene molecules excited to the T₋ remain in that state until they decay back to the ground state S0, without participating in the DNP process. Therefore we can treat the triplet as fictitious spin 1/2 system, withS_z =s_z±¹₂ and S_x⁰ =√

2s_x⁰ [19]. The “+” and “−” signs correspond to the high field and low field transitions respectively. The Hamiltonian becomes

H = ~

ω_0S−ω_m±D⁰

s_z + ω_1S√

2s_x⁰ − ω_0II_z + 1

2(s_z±1

2) (A_z−I₊+A_z+I₋)

. (2.20) Here we eliminate the constant term ¹₂(ω0S−ωm±D⁰) by a shift of the zero point of the energy. The step operators areI_±=I_x±iI_y and we defineA_z±=A_zx±iA_zy.

On a basis of the eigenstates of|m_s, m_Ii ofs_z andI_z:

|m_s, m_Ii

|+¹₂,−¹₂i

|+¹₂,+¹₂i

| −¹₂,−¹₂i

| − ¹₂,+¹₂i,

(2.21)

the matrix representation of the Hamiltonian is

1 2~







ω0S−ωm±D⁰+ω0I (¹₂±¹₂)Az+

√2ω1S 0

(¹₂±¹₂)Az− ω0S−ωm±D⁰−ω0I 0 √

2ω1S

√2ω1S 0 −(ω0S−ωm±D⁰) +ω0I (−¹₂±¹₂)Az+

0 √

2ω1S (−¹₂ ±¹₂)Az− −(ω0S−ωm±D⁰)−ω0I





. (2.22) There are two types of non-diagonal elements including√

2ω_1S that only induce a transition of the electron spin and ¹₂A_z± that only induce a transition of the nuclear spin. We now seek for a transformation that creates non-diagonal matrix elements which couple two states with different values of bothmS and mI. This is done with a transformation to the tilted rotating frame.