List of Tables

to return the magnetization to its equilibrium direction. In some DNP experiments, when the spins are negatively polarized, this can lead to sustained M/RASER pulses. We have recently observed such pulses appearing for short times (100 ms) followed after their disappearance by a persistent signal for several tens of seconds in deuterated DNP-hyperpolarized samples; but also as a series of maser pulses separated by several seconds and of quasi-constant intensities. This behavior can be attributed to the combination of two competing mechanisms, namely the loss of the nuclear magnetization of the proton by "radiation damping" and the repolarization of the spins of the proton from either the deuterium spins in the sample through their interactions with the electron spins, or from the electron spins alone, the absence of deuterium in the sample.

These observations can be qualitatively interpreted using the Bloch-Maxwell equations for radiation damping coupled to the Provotrov equations (BMP equations) for the thermal mixing model (TM) of dynamic nuclear polarization (DNP) which account for the spin temperature fluxes between the nuclear and non-electronic Zeeman reservoirs. However, this model fails to explain some details of the observations. In particular, in the experiments performed previously, the maser pulses show an asymmetric profile. Moreover, we observe some kind of "echoes" of the most intense maser pulses, which cannot be reproduced by the BMP equations and require a more complete description of the spin system. In this context, the magnetization intensity of these highly polarized samples is accompanied by large dipole fields that contribute to the

Résumé

Les expériences de polarisation nucléaire dynamique (DNP) à des températures cryogéniques peuvent conduire à des polarisations extrêmement élevées qui donnent lieu à des effets non linéaires qui ne se produisent pas dans la RMN conventionnelle à l'état solide. Pour des polarisations aussi élevées, le couplage de la grande aimantation avec le circuit de détection devient extrêmement intense et donne lieu à un amortissement par rayonnement cohérent [1,2] (ou "radiation damping" (RD) en Anglais), qui tend à ramener l'aimantation vers sa direction d'équilibre.

Dans certaines expériences DNP, lorsque les spins sont polarisés négativement, cela peut conduire à des impulsions M/RASER entretenues [3,4]. Nous avons récemment observé de telles impulsions apparaissant pour des temps courts (100 ms) suivies après leur disparition par un signal persistant pendant plusieurs dizaines de secondes dans des échantillons DNP-hyperpolarisés deutérés [4] ; mais aussi à type de séries d'impulsions maser séparées de plusieurs secondes et d'intensités quasi-constantes. [5]. Ce comportement peut être attribué à la combinaison de deux mécanismes concurrents, à savoir la perte de l'aimantation nucléaire du proton par "radiation damping" et la repolarisation des spins du proton à partir, soit des spins du deutérium dans l'échantillon par l'intermédiaire de leurs interactions avec les spins électroniques, soit des seuls spins électroniques, l'absence de deutérieum dans l'échantillon. Ces observations peuvent être qualitativement interprétées à l'aide des équations de Bloch-Maxwell pour le "radiation damping" couplées aux équations de Provotrov [6,7] (équations BMP) pour le modèle de mélange thermique (TM) de la polarisation nucléaire dynamique (PND) qui rendent compte des flux de températures de spin entre les réservoirs Zeeman nucléaires et non-Zeeman électroniques.

Cependant, ce modèle ne parvient pas à expliquer certains détails des observations. En par-

Expériences

Lors d'expériences de DNP à 1.2 K sur un échantillon uniquement protoné, nous avons pu observer des impulsions maser entretenues, en présence d'irradiation microonde, pendant des durées de l'ordre de l'heure(des extraits sont présentés en figure 1). L'absence de deutération des composés, bien qu'inhabituelle pour les expériences DNP, s'est avérée nécessaire pour obtenir une aimantation suffisante de l'échantillon dans notre cas pour produir un "radiation damping" est un phénomène bien documenté et observé en RMN liquide [10,11,8]. Cette dépendance a été confirmée par le calcul du premier moment (m 1 ) du spectre, qui fournit une mesure du décalage de résonance.

Simulation numérique

B ⃗ dip (r ⃗ i ) = µ 0 4π ∑ j̸ =i 1 -3 cos 2 θ ij 2 |r ij | 3 × [ 3M z (r ⃗ j )z ˆ-M ⃗ (r ⃗ j ) ] (1)
Un autre ingrédient essentiel est la présence du champ de "radiation damping" B ⃗ rd [2] pour un échantillon en forme de disque et

ω dip = γ H µ 0 2M 0
6 pour un cylindre [8].

Instrumentation

En 1995, Louis-Joseph [20]

Introduction

A large magnetization produced by a nuclear spin ensemble, when efficiently coupled to the LCR resonant circuit of the the nuclear magnetic resonance (NMR) detection probe, can create a coherent radio frequency (rf) field with significant magnitude. If strong enough, such a field will efficiently act back on the precessing magnetization to rotate the latter back to its equilibrium direction, giving rise to a so-called maser pulse. This well-known phenomenon, improperly termed radiation damping (RD), has been studied decades ago [1,2] and has motivated a wealth of methodological work aiming at its suppression or control in solution NMR [3,4,5]. Such a magnetization behaviour can also be generated in the solid state, when the spins are brought to a high level of polarization, as has been demonstrated using dynamic nuclear polarization (DNP) at liquid helium temperatures.

Even less common is the observation of multiple, self-sustained, maser (an acronym for microwave amplification by stimulated emission of radiation) type pulses. This unusual magnetization dynamics requires an additional mechanism that restores the Zeeman energy of the spins that is lost to the coil through radiation damping as a consequence of the precession of the magnetization about the radiation feedback field towards its equilibrium direction. The antagonist process allows to restore a large magnetization pointing in a direction opposite to the magnetic field, resulting, in certain conditions, in the renewing of the process. A combination of such effects has been obtained in various contexts of NMR spectroscopy. In solution-state NMR in ambient conditions, sustained maser pulses have been achieved by use of an electronic control of radiation damping that inverts the feedback process and drives the magnetization towards the south pole of the Bloch sphere. [6] In the context of hyperpolarization of nuclear spins, sustained masers have been obtained recently, where a hyperpolarization process drives xxxiv xxxv the polarization towards a large negative value, generating a large negative magnetization. These were described in the context of parahydrogen-based hyperpolarization techniques using SABRE ("signal amplification by reversible exchange") or PHIP. [7,8] In this work, we study the nonlinear dynamics of the magnetization of an ensemble of spins in the context of dynamic nuclear polarization (DNP) in high field and at cryogenic temperatures.

When negatively polarized, 1 H spins can nevertheless be repolarized by electron spins as well as, possibly, by other DNP-hyperpolarized nuclear spin species that are uncoupled to the NMR probe. [9] In this case, the additional nuclear spins represent a reservoir of polarization that is indirectly transferred to the 1 H spins through their interactions with the electron spins. [10] Such sustained masers followed by long lasting tails extending over several tens of seconds have been recently observed in DNP-hyperpolarized samples. [11] Following previous analyses of analogous experiments, [12,9] the experiments were analyzed in a classical framework using the nonlinear Bloch-Maxwell equations to take into account the coupling of the magnetization with the detection circuit [1,2].

An additional level of complexity is the description of the DNP mechanism itself. In the experimental conditions of these works, DNP was assumed to be in the thermal mixing regime, [13,14] and described by the Provotorov equations [15,16]. The combination of both phenomena results in the Bloch-Maxwell-Provotorov (BMP) equations that allow to shed some light onto the unusual nonlinear magnetization dynamics. However, the Bloch-Maxwell-Provotorov equations are deceptively simple, and are not amenable to an analytical analysis, except in particular cases where analytical solutions of the linearized Bloch-Maxwell-Provotorov equations can be derived. [11] Such a study nevertheless provides insight into the non intuitive behaviour of the magnetization dynamics. If one is interested in the magnetization dynamics rather than in the details of the DNP mechanisms, a number of idealizations and simplifications can be helpful to get a better understanding.

Many aspects of this nonlinear dynamics still remain unclear, and a model based on the simple framework of the Bloch equations fails to capture details of the observations, which call for a more complex and complete description of the spin system. In this context, the large magnetization of such highly polarized samples entails the presence of large distant dipolar xxxvi fields, [17,18] which contributes to the dynamics of the magnetization. This is unusual in solid state NMR, where dipolar interactions between nuclear spins are of paramount importance, but have no other effect on the magnetization dynamics other than a fast decoherence, therefore a large line broadening [19,20] with line widths of tens of kHz. In contrast, possible effects of distant, much weaker, dipolar interactions, requires the persistence of coherent magnetization on a much longer time scale. The effects of the associated distant dipolar field (DDF) has been studied in the particular context of solid 3 He in a seminal paper by Deville et al. [21]. In that work, the unusual long transverse relaxation time allowed to observe such effects. These were subsequently investigated mainly in solution NMR, where the combination of radiation feedback and the distant dipolar field have been the subject of a number of studies. Thus, the combination of both these effects have been shown to produce even more complex magnetization dynamics, such as spin turbulence [22] and spectral clustering [23,24,25]. In typical solid state NMR samples, short NMR decoherence times prevent the development of dynamical effects of the distant dipolar field (DDF), and to our knowledge, no study of the combined collective effects of radiation feedback and the distant dipolar field have been investigated in this context.

In this thesis, our main objective is to explore the dynamics of a magnetization subject to these combined effects in the context of DNP-hyperpolarized spins at liquid helium temperatures, both numerically and experimentally. We provide evidence for the manifestation of the DDF at long evolution times of the magnetization and show that the fine features of the observed maser pulses can be related to the interplay between the latter and radiation damping [26].

In addition, with the aim of providing a tool to control these maser effects and to experimentally investigate the non-linear dynamics of DNP-hyperpolarized spins at 1.2 K in more details, we present the implementation of an electronic feedback control unit of radiation damping to be used on a DNP polarizer.

Chapter 1

A classical description of NMR beyond the

Bloch equations

This thesis deals with the collective dynamics of a spin ensemble subject to the combination of several phenomena of different natures, namely, radiation feedback from the probe, the dipolar field effects and DNP. The collective interaction of the spins with the detection circuit (radiation damping), as well as the dipolar nuclear spin, and electron-nuclear spin interactions of dynamical nuclear polarization (DNP) make the description of dynamics particularly complex. However, the macroscopic magnetization dynamic effects dealt with in this work are essentially classical and can be described in terms of interacting classical moments. Nevertheless, even with such a drastic simplification, the complexity of this system requires several simplifications and hypothese which will be introduced below.

Nuclear Spin

The question as to the nature of the spin of a particle is beyond the scope of this thesis. Spin is a pure quantum mechanical concept without a classical counterpart, that has the properties of an intrinsic angular momentum. In 1922 Otto Stern and Walther Gerlach [1] showed experimentally the existence of a quantized spin angular momentum of a single electron using a beam of silver atoms. Details of this experiment is available in all introductory quantum mechanics books [2].

Consider a particle with intrinsic spin angular momentum S ⃗ = S x x ˆ+ S y y ˆ+ S z z ˆplaced in a external magnetic field B o z ˆ. The angular momentum is quantized, and can take integer or half-integer values S. Moreover the projection of the spin along the direction of external magnetic field is also quantized, and the spin angular momentum operators are: S 2 ˆand S z ˆwith eigenvectors |S, m⟩ defined as:

S 2 ˆ|S, m⟩ = h2 S(S + 1) |S, m⟩ (1.1)
and:

S z ˆ|S, m⟩ = hm |S, m⟩ (1.2)
. S and m are the spin and magnetic quantum numbers respectively. The spin quantum number can take the values S = 0 ( 12 C, 16 O), 1 2 ( 1 H, 13 C, 15 N , electron), 1 ( 2 H, 6 Li), 3 2 ( 7 Li), and the magnetic quantum number can take the 2S + 1 values m = -S, -(S -1), ..., (S -1), S.

For a spin-

Boltzmann Distribution

In statistical mechanics, Boltzmann distribution [3] is a probability distribution that gives probability p i of a state |i⟩ as a function of the energy of the state E i and temperature T of the system.

p i = N i N = 1 Q e -E i /K B T (1.3) where Q = ∑ i e -E i /K B T is called the canonical partition function [3]
, N i is the population of state |i⟩, N is the total population and K B is the Boltzmann constant.

Considering an ensemble of N non interacting spin half nuclear spins at thermal equilibrium, the population of each state is given by the Boltzmann distribution, according to which the

N m = N e -Em/k B T ∑ m e -E i /k B T (1.4) where m = {-1 2 , 1 2 }n.

Magnetic Dipole Moment

To the spin 1 2 particle is associated a magnetic moment µ ⃗ S such that:

µ ⃗ S = γS ⃗ = gq 2m o c S ⃗ (1.5)
where γ is the gyromagnetic ratio, g is the g-factor, q is the charge of particle and m o its mass, and c is the speed of light in vacuum. The z ˆcomponent of magnetic dipole moment is given by:

µ S z = γhm (1.6)
When γ > 0 the magnetic dipole moment and spin are parallel and γ < 0 the magnetic dipole moment and spin are anti-parallel. The gyromagnetic ratio and g-factor of electron and proton

is shown in table 1.1.
The energy of magnetic dipole with dipole moment placed in a field B 0 z ˆis given by:

E m = -(µ x , µ y , µ z ) • (0, 0, B z ) = -mhγB o (1.7) Therefore energy of state |α⟩ = | 1 2 , 1 2 ⟩ and |β⟩ = | 1 2 , -1 2 ⟩ are -
N m = N e mhγBo/k B T ∑ m e mhγBo/k B T (1.8) ≈ N 2 ( 1 + mhγB o k BT )
where the approximation in equation 1.8 holds in the high temperature approximation, i.e., for k B T >> mhγB o , ie. thermal energy of the spin is much larger than magnetic dipole energy of spin half particle. For a spin 1/2, the polarization, i.e., the relative difference of populations of the ±1/2 states is given by:

P z = tan ( γhB o 2k B T ) (1.9)
where γ is the gyromagnetic ratio, k B the Boltzmann constant, and T the temperature.

Magnetic Resonance

Stern [4] and Rabi [5] had independently done experiments to measure the magnetic moment of nuclei by deflecting atomic beams, but the new method introduced by adding additional oscillating magnetic field orthogonal to the strong field with radio frequency resonant to Larmor frequency of nuclei, Rabi [6,7,8] gave birth to the new approach, the magnetic resonance.

In 1946 Edward Mills Purcell [9] in Harvard and Felix Bloch [10]

τ = d dt S ⃗ = µ ⃗ × B ⃗ (1.10)
which, using equation 1.5 can be rewriten as:

d dt µ ⃗ = γµ ⃗ × B ⃗ (1.11)
Bloch in his 1946 paper Nuclear Induction [12] has mentioned possibility of applying strong radio frequency for a short period, a pulse to change the orientation of nuclear spin θ from constant B o and when the pulse disappear the nuclearr spin precess about B o z ˆwith Larmor frequency and create free induction signal in the resonant coil circuit. In 1946, H. C. Torrey [13] and in 1950 E.L. Hahn [14] started working on the pulsed magnetic resonance inspired from the Bloch and recorded the first free induction signal from free precession of proton nuclear spins in water and glycerine. In 1966 a milestone in nuclear magnetic resonance spectroscopy was laid Richard

Ernst by introducing Fourier transform nuclear magnetic resonance FT-NMR [15]. The main advantage of this technique is to speed up the acquisition of NMR spectra and increase the signal to noise ratio with respect to continuous wave nuclear magnetic resonance spectroscopy.

Magnetization

For an ensemble of N magnetic moments in a volume V the magnetization, M ⃗ = N µ ⃗ V , i.e., the magnetic dipole moment per unit volume, is defined as:

M ⃗ = lim ∆v→0 1 ∆v N ∑ i=1 µ i ⃗ (1.12)
In thermal equilibrium, the sum of the transverse component of dipole moment µ ⃗ T is zero due to the random phases of the µ ⃗ vectors in space. Only the z ˆcomponents of the nuclear magnetic moment add up to yield the equlibrium magnetization M o :

M o = 1 V N ∑ i=1 µ z,i = γh V ( - 1 2 N -1/2 + 1 2 N 1/2 ) (1.13) ≈ nγ 2 h2 B o 4k B T
where the second line was obtained using equation 1.8. Here, n = N V is the number density. Now, rewriting equation 1.11 in terms of the magnetization, we get the Bloch equation:

d dt M ⃗ (t) = M ⃗ (t) × γB ⃗ (t) (1.14)
Rotating Frame of Reference Equation 1.14 describes the dynamics of the magnetization in the laboratory frame. Let us assume a frame rotating with angular frequency ω ⃗ . The equation of motion of magnetization can be transformed to the rotating frame [16,17] and can be written as follows:

( d dt M ⃗ (t) ) rot = ( d dt M ⃗ (t) ) lab -ω ⃗ × M ⃗ (t) = M ⃗ (t) × (γB ⃗ + ω ⃗ ) (1.15)
where the quantities are expressed in the rotating frame of coordinates. Thus, in the rotating frame, the magnetization precesses about the effective magnetic field B ⃗

′ (t) = B ⃗ (t) + ω ⃗ /γ. When
the total magnetic field is the sum of a constant induction field B o z ˆalong z ˆand of a weak radio frequency field oscillating in ths xy plane , B 1 (cos(ω rf t + ϕ)x ˆ+ sin(ω rf t + ϕ)y ˆ), this law of motion composition yields the precession equation of the magnetization in a frame rotating around z ˆwith the angular velocity ω ⃗ = ω rf z ˆ:

( d dt M ⃗ (t) ) rot = γM ⃗ (t) × B ef f (1.16)
where

B ef f = -ω 1 /γ(cos ϕx ˆ+ sin ϕy ˆ) -Ω/γz ˆ, Ω = ω o -ω and ω o = -γB o . Thus, resonance
occurs for the condition ω o = ω rf , where the effective field is just

B ef f = -ω 1 /γ(cos ϕx ˆ+ sin ϕy ˆ)
in the rotating frame. Bloch, in the 1946 magnetic resonance experiment, [11] used a strong constant field along z ˆ, B o z ˆand a weak linearly polarized oscillating radio frequency magnetic field along x ˆ, 2B 1 cos(ω rf t + ϕ). Any linearly polarized oscillating field can be decomposed into two fields rotating in opposite directions, and in a frame rotating with angular frequency ω, the oscillating radio-frequency field can be written as:

B ⃗ rot rf = B 1 (cos(A + B) cos(A -B), sin(A + B) sin(A -B), 0) (1.17)
Where A = ω rf t + ϕ and B = ω. Higher frequency terms can be neglected due to the sum frequency beyond the absorption range of the nuclei (rotating wave approximation).

Thus the motion takes place in a time-independent field in the rotating frame. If we apply the radio frequency field for a duration τ , the magnetization vector M rot (t) precesses around the effective magnetic field B ⃗ rot ef f by an angle α = ω nut τ in the rotating frame, with the nutation frequency ω nut = -γB ef f . The time-independent Bloch equation in rotating frame can be written as:

d dt ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ M x M y M z ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ rot = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 0 -Ω -γB rot y,ef f Ω 0 γB rot x,ef f γB rot y,ef f -γB rot x,ef f 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ M x M y M z ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ rot (1.18)
The Bloch Equation

Spin relaxation was introduced in this simple classical model by Bloch [12] through empirical time constants, describing the return of the population to equilibrium (longitudinal T 1 ) and of the transverse components of the magnetization to zero (T 2 relaxation, where T 1 ≥ T 2 . Longitudinal relaxation drives exponentially the M z (t) component of the magnetization back to equilibrium value M o with the time T 1 :

d dt M z = - (M z -M o ) T 1 (1.19)
Alternatively, the decay of the transverse component M x,y (t) to 0 is given by rate equation:

d dt M x,y = - M x,y T 2 (1.20)
Inserting these into equation 1.18 yields the Bloch equations:

d dt ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ M x M y M z ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ rot = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ -1 T 2 -Ω -γB rot y,ef f Ω -1 T 2 γB rot x,ef f γB rot y,ef f -γB rot x,ef f -1 T 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ M x M y M z ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ rot + ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 Mo T 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ (1.21)
1.2 The Origin of Non-Linearity in the dynamics of a spin ensemble

The Bloch equations are a satisfactory representation of the dynamics of an ensemble of isolated spins with magnetic dipole moment µ ⃗ i , i.e., for uncoupled spins. However, the uncoupled spin picture is an idealized case, as in many situations of interest spins mutually interact, for instance through dipole dipole interactions. [2] Such interactions may be of little relevance in liquids, where in most cases dipolar interactions are averaged out through molecular motions and contribute to relaxation [18,19,20]. However, in some cases, for large samples and large magnetizations, the contributions of long distance dipolar interactions, small but in large numbers, dramatically influence the magnetization dynamics. [21,22,23,24,25] Alternatively, in the solid, dipolar interactions are of paramount importance, but these take place mainly between neighbouring spins, and are therefore strong interactions that lead to fast decoherence and broad dipolar lines. [26,27] Long range interactions require much longer decoherence time to be observed.

Apart from possible spin couplings through dipolar interactions, if the magnetization is sufficiently large, a coupling of a different nature between the precessing magnetization and the high-Q resonant circuit may also affect the dynamics of the magnetization, and results in the well-known phenomenon of radiation damping [28]. In this section we review both these phenomena.

Radiation Damping

In 1949, Suryan [29] first pointed to the role of the quality factor (Q factor) of a resonant detection circuit on the decay time of a free induction decay signal, considered so far as caused only by transverse relaxation. Later, Bloembergen and Pound [28] described the effect of the coupling between a precessing magnetization with the resonant circuit and showed that this led to a damping of the induction signal, hence the name "radiation damping". Although this term may lead to some confusion with the phenomenon of radiation damping in electrodynamics, [30] it has remained in use so far.

Bruce et. al. [31] discussed the effect of radiation damping on the line width and the maximum signal intensity in continuous wave high resolution nuclear magnetic resonance spectra of water. Bloom, [32] unlike Bloembergen et. al., focused on the dynamics of the magnetization and proposed modified Bloch equation for radiation damping, and derived their solutions in various situations (steady state slow passage, adiabatic fast passage). Bloom also derived analytical solutions of the modified Bloch equations in the presence of transverse relaxation. However, it is only later that Szöke and Meiboom [33] first demonstrated experimentally a nuclear magnetic maser based on radiation damping, with the typical signal envelope predicted by Bloom that when the equilibrium magnetization was flipped between 90 o and 180 o , the magnetic resonance signal increased to a maximum before decaying.

Radiation damping is a now well-known phenomenon, [34] widely encountered in high resolution NMR in liquids, and has been the subject of many studies, such as the design of radiation damping-insensitive composite and shaped pulses [START_REF] Warren | Dynamics of radiation damping in nuclear magnetic resonance[END_REF]; the effect of radiation damping on multiplets [START_REF] Hervé | The behavior of multiplet signals under "radiation damping" conditions. i. classical effects[END_REF]; solvent suppression [START_REF] Guéron | Solvent signal suppression in nmr[END_REF]; the study of the interplay between radiation damping and transverse and longitudinal relaxation rates [START_REF] Xi An Mao | Competition between radiation damping and transverse relaxation effects on nmr signal intensities[END_REF]; the interpretation of artifacts due to radiation damping in two 2D NMR experiments [START_REF] Ball | Radiation damping artifacts in 2d cosy nmr experiments[END_REF] and radiation damping in inhomogeneous fields, [40,41,[START_REF] Mao | Nuclear-magnetic-resonance line-shape theory in the presence of radiation damping[END_REF][START_REF] Vlassenbroek | Radiation damping in high resolution liquid nmr: A simulation study[END_REF] among others.

Theoretical Description of Radiation Damping

We here follow the theoretical derivation of Jeener [START_REF] Vlassenbroek | Radiation damping in high resolution liquid nmr: A simulation study[END_REF] to describe the magnetization dynamics upon its coupling with a resonant circuit. When the magnetization is tilted away from equilib- two voltage sources, V s (t) corresponding respectively to the electromotive force induced in the coil, and to V RF (t), the voltage applies by radio frequency generator (see figure 1.1, adapted from Ref. [START_REF] Vlassenbroek | Radiation damping in high resolution liquid nmr: A simulation study[END_REF]). Using the principle of reciprocity [START_REF] Hoult | The principle of reciprocity[END_REF], the voltage generated by the precessing magnetization can be written as:

V s (t) = - d dt ∫ V b ⃗ 1 (r ′ ⃗ ).M ⃗ (r ′ ⃗ , t)dv ′ (1.22)
where b ⃗ 1 (r ⃗) is the magnetic field induced by a unit current in the coil and M ⃗ (r ⃗, t) is the magnetization at point r ⃗ in volume V . For simplicity assume b ⃗ 1 (r ⃗) is assumed constant over the sample volume and points along x ˆ. The (energy) filling factor, defined as the ratio of the magnetic energy stored in the sample volume V to the magnetic energy stored in all space, η [START_REF] Hill | Limits of measurement in magnetic resonance[END_REF] is:

η = ∫ V b ⃗ 2 1 (r ⃗)dv ∫ space b ⃗ 2 1 (r ⃗)dv = b 2 1x V ∫ space b ⃗ 2 1 (r ⃗)dv (1.23)
The magnetic energy stored in the inductance L when a current i flows is given by 1 2 Li 2 .[46] Therefore, from equations 1.23 and 1.22, the magnetic field produced by a unit current is b 1x = (µ o ηL/V ) 1/2 where µ o is the permeability of free space, and equation 1.22 can be written as:

V s (t) = - ( µ o ηL V ) 1/2 ∫ V M x (r ′ ⃗ , t)dv ′ (1.24) Applying Kirchhoff's law in this series (L,C,R) circuit, [46] V R + V C + V L = V s + V RF , where V R = iR, V C = 1/C ∫ idt and V L = Ldi/dt, so that: d 2 dt 2 i + R L d dt i + 1 LC i = 1 L d dt (V s + V RF ) (1.25)
The precessing magnetization induces an alternating current i in the coil that produces a magnetic field (the "radiation damping" field) B ⃗ 1 = b ⃗ 1 i which acts back on the spins. Therefore, Equation 1.25 also provides the field equation. Equation 1.25 represents a damped harmonic driven oscillator with natural resonance frequency ω LC = 1/ √ LC and damping factor 1/2Q, where Q = ω LC L/R is the quality factor of the circuit. Its solution has a transient part that can be neglected as it decays much faster than the characteristic evolution time of the magnetization, and only the steady state part, determined by the driving force V s needs to be considered.

Therefore, the steady state solution [START_REF] Vlassenbroek | Radiation damping in high resolution liquid nmr: A simulation study[END_REF]

i(t) = V s (t)/Z(ω o ) is written as: V s (t) = |Z(ω o )|e jΨ(ωo) i(t) (1.26)
where: and ω o is the spin Larmor frequency. Note that when the circuit is exactly tuned to the Larmor frequency of the spins, the phase Ψ between the current and the induced voltage V s is zero and

|Z(ω o )| = ω LC L Q (1 + ∆ 2 ) 1/2 , Ψ(ω o ) = tan -1 (∆), ∆ = Q ω LC ω o ( ω 2 o -ω 2 LC ω 2 LC ) (1.27)
|Z(ω o )| = ω o L/Q.
The radiation damping field can be written as:

B ⃗ 1 (t) = b 1x x ˆi(t) = -x ˆ( µ o ηL Z(ω o )V ) d dt ∫ V M x (r ⃗, t)dv ′ (1.28)
It can the be shown that in the rotating frame, equation 1.28 becomes: [START_REF] Vlassenbroek | Radiation damping in high resolution liquid nmr: A simulation study[END_REF] B ⃗ rot

+RD (t) = - ( µ o ηL Z(ω o )V ) d dt ∫ V M rot + (r ⃗, t)e -ω RF t dv ′ ≈ j ( ω RF µ o ηLe -Ψ(ωo) |Z(ω o )|V ) ∫ V M rot + (r ⃗, t)dv ′ (1.29)
Expressed in terms of the angular frequency, we can write equation 1.29 as (the superscript rot has been dropped):

ω +,RD = -j ζ V e -jΨ(ωo) ∫ V M + (r ⃗, t)dv ′ (1.30)
where ω +,RD = ω x,RD + jω y,RD , M + = M x + jM y and ζ = γω RF µoηLe -Ψ(ωo)

|Z(ωo)|

Combining equation 1.30 with the Bloch equation one can write a set of non-linear differential (Bloch-Maxwell) equations in the rotating frame as:

d dt M x (r ⃗, t) = -δωM y (r ⃗, t) -ω y,RD M z (r ⃗, t) -γ 2 M x (r ⃗, t) d dt M y (r ⃗, t) = δωM x (r ⃗, t) + ω x,RD M z (r ⃗, t) -γ 2 M y (r ⃗, t) d dt M z (r ⃗, t) = ω y,RD M x (r ⃗, t) -ω x,RD M y (r ⃗, t) -γ 1 (M z (r ⃗, t) -M o (r ⃗, t))
where δ = ω RF -ω o , γ 1 = 1/T 1 and γ 2 = 1/T 2 are the longitudinal and transverse relaxation rate. In the absence of relaxation, these have the analytical solution:[47]

M rot + (t) = M o sech ( t -t o τ RD -log [ tan(ϕ f lip /2) ] ) e jϕ(t) M rot z (t) = M o tanh ( t -t o τ RD -log [ tan(ϕ f lip /2) ] ) (1.31)
Equation 1.31 shows the law of return to equilibrium direction of the magnetization upon rotation by the radiation damping field, illustrated in figure 1.3. The time dependent phase, ϕ(t) is given by :

ϕ(t) = ϕ(t = 0) + δωt + tan(Ψ) log [ sin(ϕ f lip (t = 0)) sin(ϕ f lip (t)) ] (1.32)
and the radiation damping rate γ RD = 1/τ is given by: From equation 1.32 it is seen that the magnetization precesses about the B o with a timedependent frequency: Here the volume average magnetization in the rotating frame is given by The effect of spin-spin relaxation on the Maxwell-Bloch equations Bloom [START_REF] Bloom | Effects of radiation damping on spin dynamics[END_REF] has shown that the effect of spin-spin relaxation (T 2 ) on the radiation damping field yields to the following solution of the modified Bloch equations for T 1 = ∞:

γ RD = γω o µ o ηL o M o |Z(ω o )| cos(Ψ(ω o )) (1.33)
ω(t) = d dt ϕ(t) = δω + [ tan(Ψ(ω o )) τ M o ] M z (t) (1.34)
M avg x,y (t) = 1 V ∫ V M x,y (r ⃗, t)dv ′ (1.36) (a) (b) (c) (d)
M T ∝ (γτ RD ) -1 sech t -t m τ RD M z ∝ (γτ RD ) -1 tanh t -t m τ RD -(γT 2 ) -1 (1.37)
and t m is determined by:

exp(2t m /τ RM ) = 1 -((τ RD /T 2 ) + (τ RD /τ ∞ RD ) cos(θ(t = 0))) 1 + ((τ RD /T 2 ) + (τ RD /τ ∞ RD ) cos(θ(t = 0))) (1.38)
where τ RD is the radiation damping time, τ RD (T 2 = ∞) = τ ∞ RD = 2/γηQµ o M o and the relation between τ RD and τ ∞ RD is given by:

τ ∞ RD τ RD = √ (τ ∞ RD /T 2 ) 2 + 2(τ ∞ RD /T 2 ) cos(θ(t = 0)) + 1 (1.39)
When the initial flip angle cos(θ(t = 0)) is close to π and τ ∞ RD τ RD is close to unity, the sech profile of the transverse magnetization becomes broader, indicating a longer lasing coherence, whereas the longitudinal magnetization does not vary much from its initial value (see Figure 1.4), in contrast with the situation where transverse relaxation is absent and M z always ends up positive, with

M z = M o , as in Figure 1.3.

Dipolar Interactions and the distant dipolar field

Dipolar interactions in liquid and solid-state magnetic samples usually impact the dynamics of spins in different ways. In liquids, short distance dipolar interactions are usually averaged out by molecular motions, and so do not contribute coherently to the spin dynamics. However, their fluctuations provides an efficient relaxation mechanism. Alternatively, long-range dipolar interactions can be effective for large samples at high field, producing various effects, such as magnetization dependent chemical shifts, [22] multiple spin echoes, [START_REF] Deville | Multiple echoes observed in solid 3 he[END_REF] cross-precession effects, [START_REF] Goldman | Cross-precession induced by the average dipolar field in high-resolution nmr[END_REF] spectral clustering and instabilities, [START_REF] Jeener | Dynamical effects of the dipolar field inhomogeneities in high-resolution nmr: Spectral clustering and instabilities[END_REF][START_REF] Sauer | Nmr instabilities and spectral clustering in laser-polarized liquid xenon[END_REF][START_REF] Nacher | Nmr instabilities in highly magnetised liquid helium solutions[END_REF][START_REF] Henner | Collective effects due to dipolar fields as the origin of the extremely random behavior in hyperpolarized nmr maser: a theoretical and numerical study[END_REF] and have found a number of application in multidimensional NMR [21,[START_REF] Mccoy | Three-quantum nuclear magnetic resonance spectroscopy of liquid water: Intermolecular multiple-quantum coherence generated by spin-cavity coupling[END_REF][START_REF] Warren | Mr imaging contrast enhancement based on intermolecular zero quantum coherences[END_REF]. In solids, dipole-dipole interactions usually provide an efficient decoherence mechanism, leading to broad lines and to fast decaying induction signals, on the order of a few hundreds of microseconds. This is particularly effective for near neighbor interactions, where dipolar coupling is strong. Alternatively, because the dipolar interactions fall off relatively fast as ∼ r 3 , these effects may be assumed much weaker at long distances, where the interaction energies are smaller and the associated time scales accordingly longer.

However, strikingly, the first multiple spin echoes caused by the distant dipolar field, by Deville et al. [START_REF] Deville | Multiple echoes observed in solid 3 he[END_REF] were observed in solid 3 He. This was allowed due to the unusually long T 2 relaxation time in this case due to exchange interaction between neighbouring spins, which is 10 3 stronger than dipolar interaction [START_REF] Landesman | L´échange dans l´hélium trois solide[END_REF]. The exchange interaction is short range and becomes weaker when the distance between spins increases, and for some critical distance r c the exchange interaction becomes weaker than the dipolar interaction, and can be neglected. The distant dipolar field acting on a moment m i at point r ⃗ i can be written as:

B ⃗ dip (r ⃗ i ) = µ 0 4π ∫ d 3 r ′ 3[m ⃗ (r ⃗ ′ ).r ˆij ]r ˆij -m ⃗ (r ⃗ ′ ) ⃓ ⃓ ⃓r ⃗ i -r ⃗ ′ ⃓ ⃓ ⃓ 3 (1.40)
where

r ˆij = (r ⃗ i -r ⃗ ′ )/|r ⃗ i -r ⃗ ′
| is the unit vector joining the positions r ⃗ i and r ⃗ ′ In a frame rotating at the angular frequency ω o , the non secular part averages out to zero [START_REF] Deville | Multiple echoes observed in solid 3 he[END_REF] and therefore the Equation 1.40 can be written as (see appendix B):

B ⃗ dip (r ⃗ i ) = µ 0 4π ∫ d 3 r ′ 1 -3 cos 2 θ ij 2 |r ij | 3 × [ 3m z (r ⃗ ′ )z ˆ-m ⃗ (r ⃗ ′ ) ] (1.41)
where θ ij is the angle between z ˆand the position vector r ˆij . Since the distant dipolar field depends on the shape of sample, it is difficult to solve equation 1.41 analytically for arbitrary sample shapes. However, analytical solutions do exist for ellipsoidal samples. [23,24] Thus the distant dipolar field contribution from a uniform magnetization in a perfect ellipsoidal sample can be written as:

B ⃗ dip (r ⃗ i ) = nµ o [ 3 cos 2 (θ) -1 2 ] [3m z z ˆ-m ⃗ ] (1.42)
where θ is the angle between z ˆand the axis of the ellipsoid, n = 0 for sphere, n = 1/6 for a long cylinder and n = -1/3 for thin disks.

B o (T ) Since the first observation in ethanol of the effect of the electronic environment of nuclear spins and the discovery of the chemical shift, [START_REF] Arnold | Variations in absolute chemical shift of nuclear induction signals of hydroxyl groups of methyl and ethyl alcohol[END_REF] nuclear magnetic resonance has proven a unique method in the panoply of analytical methods in chemistry. However, the ever growing range of NMR applications has also prompted the demand for higher sensitivity and resolution. The signal to noise ratio (SNR), i.e., the power ratio between the signal to the background noise and a measure of sensitivity, depends, among others, on the static magnetic field B o . For a solenoid detection coil, one has:

[60] SN R ∝ ω 1/2 o M o ∝ γ 5/2 B 3/2 o (1.43)
So, the larger B o , the larger M o (∝ P z , the z equilibrium polarization), [START_REF] Wenckebach | Essentials of Dynamic Nuclear Polarization[END_REF] and the higher the SNR. The proton polarization for different B o and T values is shown in table 1

.2. From table 1.2,
for a magnetic field of 23.5 T and at a temperature of 1.2 K, the thermal proton polarization is 2%. Therefore, creating a nuclear spin polarization beyond the thermal equilibrium polarization represents an important step forward. In the early 1950s the possibility to create non Boltzmann spin hyperpolarized states, i.e., more populated than Boltzmann, was theoretically predicted.

Some efficient techniques to achieve hyperpolarization were later introduced: dynamic nuclear polarization [START_REF] Overhauser | Polarization of nuclei in metals[END_REF], spin exchange optical pumping (SEOP) [START_REF] Bouchiat | Nuclear polarization in he 3 gas induced by optical pumping and dipolar exchange[END_REF][START_REF] Grover | Noble-gas nmr detection through noble-gas-rubidium hyperfine contact interaction[END_REF][START_REF] Appelt | Theory of spin-exchange optical pumping of 3 He and 129 Xe[END_REF], metastability exchange optical pumping (MEOP), [START_REF] Gentile | Optically polarized 3 He[END_REF] chemically induced dynamic nuclear polarization (CIDNP) [START_REF] Bargon | Kernresonanz-emissionslinien während rascher radikalreaktionen[END_REF][START_REF] Roy | Nuclear magnetic resonance emission and enhanced absorption in rapid organometallic reactions[END_REF], parahydrogen induced polarization (PHIP) [START_REF] Bowers | Transformation of symmetrization order to nuclear-spin magnetization by chemical reaction and nuclear magnetic resonance[END_REF][START_REF] Bowers | Parahydrogen and synthesis allow dramatically enhanced nuclear alignment[END_REF][START_REF] Michael | Net nmr alignment by adiabatic transport of parahydrogen addition products to high magnetic field[END_REF], signal amplification by reversible exchange (SABRE) [START_REF] Adams | Reversible interactions with para-hydrogen enhance nmr sensitivity by polarization transfer[END_REF][START_REF] Adams | A theoretical basis for spontaneous polarization transfer in non-hydrogenative parahydrogen-induced polarization[END_REF].

Dynamic nuclear polarization was first suggested (in the early 1950s by Overhauser [START_REF] Overhauser | Polarization of nuclei in metals[END_REF]) as a hyperpolarization technique, able to enhance the nuclear spin polarization beyond the one predicted by a Boltzmann statistics and to increase the signal to noise ratio of nuclear magnetic resonance spectra. Since then several mechanisms of DNP have been described that depend on the type of paramgnetic impurity (the "polarizing agent") and external parameters such as the temperature, the microwave frequency. These mechanisms are the Overhauser effect (OE) [START_REF] Abragam | Overhauser effect in nonmetals[END_REF],

the solid effect (SE) [START_REF] Abraham | γ-ray anisotropy of co 60 nuclei polarized by paramagnetic resonance saturation[END_REF][START_REF] Jeffries | Polarization of nuclei by resonance saturation in paramagnetic crystals[END_REF]77,[START_REF] Sofia | Dynamic nuclear polarization for sensitivity enhancement in modern solid-state nmr[END_REF][START_REF] Hovav | Theoretical aspects of dynamic nuclear polarization in the solid state -the solid effect[END_REF], the cross effect (CE) [START_REF] Hwang | New effect in dynamic polarization[END_REF][START_REF] Hwang | Phenomenological model for the new effect in dynamic polarization[END_REF]77,[START_REF] Th | Electron spin-spin interactions in dnp: Thermal mixing vs. the cross effect[END_REF][START_REF] Sofia | Dynamic nuclear polarization for sensitivity enhancement in modern solid-state nmr[END_REF] and thermal mixing (TM) [START_REF] Th | Dynamic nuclear polarization via thermal mixing: Beyond the high temperature approximation[END_REF][START_REF] Th | Electron spin-spin interactions in dnp: Thermal mixing vs. the cross effect[END_REF]. They will be briefly introduced below.

The Overhauser Effect and the Solid Effect:

The Overhauser effect, proposed theoretically in 1953 [START_REF] Overhauser | Polarization of nuclei in metals[END_REF] was demonstrated experimentally a couple of years later by Carver and Slichter [START_REF] Carver | Polarization of nuclear spins in metals[END_REF][START_REF] Carver | Experimental verification of the overhauser nuclear polarization effect[END_REF]. Hyperpolarization relies on the cross relaxation between mixed electron-nucleus states [17] by hyperfine coupling between them, and is achieved by saturating the electron Zeeman transition of the paramagnetic impurities with microwaves. The mechanism mechanism can be explained as follows. Consider electron and nuclear spin at static magnetic field (B o )) and constant irradiation of microwave (Larmor frequency of electron, ω o,e = ω M W and pumping rate, Γ M W ), Hamiltonian [START_REF] Michaelis | Handbook of High Field Dynamic Nuclear Polarization[END_REF][START_REF] Wind | Applications of dynamic nuclear polarization in 13c nmr in solids[END_REF] for electron S nuclear I spin system in rotating frame of microwave field can be written as:

H = H Z,e + H Z,n + H e-n + H e-e + H n-n (1.44)
where H Z,e = (ω o,e -ω M W )S z ˆand H Z,n = -ω o,n I z ˆare the electon and nuclear Zeeman hamiltonians. Here, ω M W is the microwave irradiation frequency. The dipolar part of the electronnuclear hyperfine interaction Hamiltonian is given by:

H e,n = A e,n S ˆzI ˆz + 1 2 B + S ˆzI ˆ+ + 1 2 B -S ˆzI ˆ- (1.45)
Here only the terms of the electro-nuclear spin dipolar Hamiltonian that commute with S ˆz are considered, and the hyperfine tensor coefficients are A e,n = µoγeγnh 2 2πr 3 (3 cos 2 (θ) -1) and B ± =

3µoγeγnh 2 2πr 3
sin(θ) cos(θ), where r ⃗ is the interspin vector between electron and nuclear spin and θ is where Γ 0 , Γ 1 , Γ 2 and γ 1 are respectively the zero-, single-, double-quantum transition, and longitudinal relaxation rates of the nuclear spin. In this expression ,

ζ = (Γ 2 -Γ 0 )/Γ 0 + 2Γ 1 + Γ 2
is the coupling factor, f = (Γ 0 + 2Γ 1 + Γ 2 )/γ 1 is the leakage factor, and s = (⟨S⟩ z,T hermal -⟨S⟩ z,M W )/ ⟨S⟩ z,T hermal is the saturation parameter. The Overhauser effect is the only DNP mechanism that is effective in the liquid state, and has been observed in insulating solids at 90K. [START_REF] Can | Overhauser effects in insulating solids[END_REF][START_REF] Delage-Laurin | Overhauser dynamic nuclear polarization with selectively deuterated bdpa radicals[END_REF] In a solid at cryogenic temperature, if the fluctuations of the hyperfine interaction can be neglected, due for instance to the absence of molecular motions, the Overhauser effect can be discarded. The solid effect was first described [START_REF] Abragam | Spin temperature[END_REF]93,94]

′ 1 ⟩ = |Ψ 1 ⟩ + q |Ψ 2 ⟩, |Ψ ′ 2 ⟩ = |Ψ 2 ⟩ -q |Ψ 1 ⟩,|Ψ ′ 3 ⟩ = |Ψ 3 ⟩ -q |Ψ 4 ⟩ and |Ψ ′ 4 ⟩ = |Ψ 4 ⟩ + q |Ψ 3 ⟩
, where q = (-3/4)γ e γ n ω -1 o,n r -3 cos(θ) sin(θ)e -jϕ are the mixing coefficient. [START_REF] Wind | Applications of dynamic nuclear polarization in 13c nmr in solids[END_REF][START_REF] Maly | Dynamic nuclear polarization at high magnetic fields[END_REF] The double and zero quantum transition probabilities are proportional to 4q 2 , [START_REF] Abragam | Principles of dynamic nuclear polarisation[END_REF][START_REF] Duijvestijn | A quantitative investigation of the dynamic nuclear polarization effect by fixed paramagnetic centra of abundant and rare spins in solids at room temperature[END_REF] hence the transition probability and dynamic nuclear polarization enhancement is proportional to ω -2 o,n , ie., the solid effect less likely the mechanism in high field experiments.

The solid effect requires paramagnetic impurities with a homogeneous electron paramagnetic resonance linewidth (δ epr ) and inhomogeneous spectral breath (∆ epr ) smaller than nuclear spin Larmor frequency (δ epr , ∆ epr < ω o,n ) like Triphenylmethyl radical (trityl), in order make sure only one of the forbidden transition is excited. If both transition are exited at the same time it leads to more ore less complete cancellation of the enhancement.

The Cross Effect and Thermal Mixing

Both the Overhauser and solid effects describe processes where the polarization enhancement is based on the coupling of a pair of electron and nuclear spins, and electron-electron dipolar coupling is neglected. In solids, better DNP enhancement was observed using biradicals, with broad inhomogeneous spectral breath ∆ epr and strong intra-molecular dipolar coupling between electrons [START_REF] Kan-Nian | Dynamic nuclear polarization with biradicals[END_REF]. The cross effect mechanism, first introduced in 1963, [99,[START_REF] Farrar | Mechanism of dynamic nuclear polarization in high magnetic fields[END_REF] involves allowed transitions and depends on dipolar coupling between two electrons in a inhomogeneously broadened electron paramagnetic resonance spectrum. It is an efficient dynamic nuclear polarization mechanism that is the basis of most high-field MAS-DNP experiments. [START_REF] Hall | Polarization-enhanced nmr spectroscopy of biomolecules in frozen solution[END_REF][START_REF] Farrar | Mechanism of dynamic nuclear polarization in high magnetic fields[END_REF][START_REF] Kan-Nian | Dynamic nuclear polarization with biradicals[END_REF]. At high magnetic field, the cross effect can be described as a triple spin process, involving dipolar coupling between two electron and a nuclear spin with Larmor frequencies ω o,e1 , ω o,e2 , and ω o,n . This process favored when the mutual spin transitions represent an energy-conserving process, i.e., when the condition: polarizations. [START_REF] Maly | Dynamic nuclear polarization at high magnetic fields[END_REF] The cross effect requires1.47 broad EPR line widths ∆ epr , with the g-anisotropy larger than the nuclear spin Larmor frequency: ∆ epr > ω o,n . This is also the case for EPR line widths smaller than the nuclear spin Larmor frequency δ epr < ω o,n in the case of bi-radicals such as AMUPol or TOTAPOL as paramagnetic agents. There, the dipolar coupling between them can make the cross effect an effective. The cross effect is the dominant DNP mechanism in magic angle spining (MAS) DNP at high field.

ω o,e2 -ω o,e1 = ω o,n (1.47
Another type of DNP enhancement of nuclei with large gyromagnetic ratio was observed with high concentration of nitroxide monoradicals at cryogenic temperature [START_REF] Vuichoud | Measuring absolute spin polarization in dissolution-dnp by spin polarimetry magnetic resonance (spy-mr)[END_REF][START_REF] Bornet | Cross polarization for dissolution dynamic nuclear polarization experiments at readily accessible temperatures 1.2< t< 4.2 k[END_REF], which can be described using thermodynamical concepts. Thus, the concept of spin temperature was used by Provotorov in his theory of saturation [START_REF] Provotorov | Magnetic saturation in crystals[END_REF][START_REF] Provotorov | A quantum-statistical theory of cross relaxation[END_REF] to describe the polarization enhancement in DNP. With this approach any interaction inside the spin system affecting the spins can be represented as thermodynamic heat reservoir and characterized by its spin temperature (T ) or its inverse (β = 1/k b T ). In the late sixties, Borghini used the concept of spin temperature [START_REF] Abragam | Spin temperature[END_REF] to ), in solution NMR using hyperpolarized substrates using PHIP ( [3,4,5]), dissolved hyperpolarized Xenon [6,7], or in Dissolution-DNP experiments [8], and even on water in ambient conditions. [9] The work of Bösiger et. al., [10,11] demonstrated the presence of sustained NMR masers in DNP experiments in Ruby at liquid Helium temperature, and recently,such effects have been observed in frozen solutions [12] at low temperatures, and in MAS-DNP experiments [13].

propose

A DNP NMR MASER at ENS

In a previous work, [14] the first observations of a DNP-NMR maser on a sample of hyperpolarized proton spins at temperature of 1.2 K and in a field of 6.7 T. In order to achieve this magnetization behaviour, the protons were first polarized negatively, that is, the initial magnetization pointing to the -z direction, before a short and low intensity pulse was applied. In a preparatory experiment, a buildup experiment with the monitoring of the signal allows to estimate the time needed to reach the stationary polarization. For the maser experiment itself, no monitoring of the buildup was used in order to avoid a premature triggering of the maser.

The maser was triggered by a short, low power, pulse. A typical induction signal is shown in NMR signal that typically decays in a few hundreds of micro seconds. Interestingly, these maser pulses could also be observed even when the microwave source was switched off once the maximum polarization has been reached and before the trigger radio-frequency pulse is applied. [14] The induction signal, shown in Figure 2.2, is still persistent at the end of the 30 s acquisition(see Figure 2.2(b)). These observations [12] were explained in the framework of Provotorov's theory of thermal mixing. [15,16].

However, it was noticed that in these experiments, the initial maser bursts exhibited features

Experimental setup

The experiments were performed in the polarizer of a dissolution-DNP system. The principle of the technique is to bring a sample of interest to low temperature (below 4.2 K) in a cryostat mounted inside a high field (6.7 T) NMR magnet, to irradiate the electrons of the sample with a microwave source so as to hyperpolarize the nuclei present. The sample is subsequently brought to liquid by flushing hot water at ∼ 10 bars of pressure, then pushed by pressurized helium to a liquid state NMR spectrometer through a magnetic tunnel made of permanent magnets so designed as to preserve the polarization of the nuclei during transfer by avoiding passages through zero field. In this work, only the polarizer part of the setup is used. The polarizer consists of a superconducting NMR magnet (6.7 T) with a bore that accommodates variable MHz. Besides, the microwave power was measured at the output of the µw source (ELVA-1 VCOM-10/94/400-DD), and found to be ≈ 32 mW. The microwave power loss during the propagation through the cylindrical wave guide was not determined.

All the experiments were performed on fully protonated compounds and with large volume (800µL) samples containing a mixture of 250µL H 2 O, 400µL unlabelled Glycerol, and 50 mM of the TEMPOL radical obtained from a 150µL stock solution. TEMPOL, because of its broad EPR resonance line typical of nitroxides, efficiently polarizes nuclear spins with large gyromagnetic ratios, such as 1 H, and was therefore used as the polarizing agent.

The absence of deuteration of the compounds is unusual for DNP experiments, but turned out to be necessary to increase the 1 H spin density and to obtain a sufficiently large sample magnetization in our setup to yield intense enough radiation damping. The need for such a sample composition can be explained by the low power of the microwave source that was used in this work, and possibly by the lower Q factor of the NMR resonator than in previous work. [14] 2.3 Polarization buildup The wobble curve of the reflected power from the probe measured after hyperpolarization of the proton spins, shown in figure 2.6(b) exhibits an increased reflected power at the Larmor frequency of the protons, resulting in a W-shaped wobble curve. This is an effect of the modification of the probe impedance caused by the coupling between the spins and the coil [18] ((see figure 2.6(d)).

Resonance shift and line narrowing

The observation of the NMR line upon polarization buildup revealed modifications of both the line width and of the frequency of resonance line. Figure 2.7 illustrates the line narrowing obtained with increasing polarization.

A resonance shift proportional to the longitudinal component of the magnetization caused by the presence of a strong dipolar field is a well documented phenomenon and is observed in NMR of liquids [20,21,22]. In the case of hyperpolarized solids the resonance shift effect due to the dipolar field is usually explained by the theory of moments [23,24]. Moreover, a direct measurement of the 1 H line shift is often difficult and inaccurate in these dynamic nuclear polarization experiments because the determination of the frequency of the maximum intensity is prone to large errors on such broad and possibly non symmetrical lines [25]. In these conditions, the first moment (m 1 ) of the spectrum provides a measure of the resonance shift in the case of a symmetrical line shape, and is defined as: line was computed at several stages of the polarization buildup. m 1 was not estimated by direct numerical computation of Equation 2.1, but because of its large width, especially at lower polarization levels, the resonance lines may not have completely returned to the baseline within the spectral width. Therefore, in order to avoid large numerical errors, the first moments were and higher moments for the case of dipolar interactions do not apply. Thus, the decrease of the line width with increasing magnetization (Figure 2.8) is a first manifestation of the qualitative change of dynamics induced by radiation feedback from the probe upon hyperpolarization, and illustrates its interference with dipole dipole interactions between nuclear spins.

m 1 = ∫ ∞ -∞ ωL(ω)dω , ( 2
Finally, we may add a comment on the first moment analysis presented above to relate the resonance line shift to polarization. A model of the magnetic interactions governing the dynamics is required to interpret the changes of m 1 computed from the resonance spectrum. In the problem at hand, these include the spin dipole-dipole interactions and the coupling of the spins with the resonance circuit. However, because the spins are on resonance with the probe circuit, radiation damping induces no frequency shift, [19] the latter being thus caused by the presence of the dipolar field only. Consequently, the first moment can indeed be related to the spin polarization, therefore the magnetization, in the sample in these experiments.

Observation of pulsed masers

Once the polarization reached a constant value after a buildup time of ≈ 40 min, without monitoring in order to avoid triggering an unwanted radiation damping pulse, the resonant circuit was again tuned and matched, and the maser was initiated by a short and low power pulse (8 µs at 1.23 µW) with a small initial flip angle (0.01 o ). The pulse sequence is shown in one. This first maser burst immediately follows the radio frequency trigger pulse, and its peak intensity is roughly ∼ 10 times larger and its duration accordingly shorter, than those of the next maser pulses. These features was reproducible in all the experiments giving rise to sustained masers. However, although such sustained maser pulses were easily reproduced, their duration and the delay between them could vary between different experiments in an uncontrolled way.

A striking feature of the observed sustained masers is the absence of decay of the intensities of the maser pulses with time (apart from the initial one), suggesting that the magnetization reaches a pseudo-periodic asymptotic dynamics kind of regime. However, some irregularities appeared in these experiments, with the occurrence of maser pulses of various widths and at random delays between them as shown in figure 2.11(c). However, in all our experiments, these were transient behaviours that never led to a erratic dynamics. The occurrence of maser pulses separated by delays on the second time sale is surprising, as one would expect the signal to have totally vanished during such time intervals. However, a closer look at the induction signal shows that, although weak, it actually remains above the noise level in these experiments. This is illustrated in figure 2.12 and shows that the magnetization is persistently coherent, suggests that in this sequence, maser bursts are not triggered by noise, but by persistent coherent in-plane magnetization throughout the experiment.

Another surprising feature of these maser pulses is their unusually long duration, on the order of ∼ 200 ms, as shown in Fig. 2.13(b). As mentioned above, the first maser was shorter, with a total duration of ∼ 50 ms (see Figure 2.13(a)). Therefore, applying a Fourier transformation to the decaying part of the pulse, one obtains an narrow line. This is exemplified by the spectrum obtained after processing the maser pulse occurring around 2.3 s in Figure 2.11(a).

The obtained line width is ∼ 10 Hz, a rather unusual value for a non rotating solid (Fig. 2.13(c), and significantly smaller than the dipolar line width.

Some manifestations of the dipolar field

A closer look at the maser pulses (Figure 2.13(a) and 2.13(b)) allows to identify additional features. Indeed, one notices that the envelopes of these maser pulses show an asymmetric profile, with a faster rise than decay, a characteristics that is not explained by a simple Bloch-Maxwell equations of the radiation damping theory. [28,19] Moreover, the frequency offset of the signal with respect to the carrier frequency is not constant in time, which can be seen by superimposing time traces of different bursts and comparing the periodicity of the signals. Mere visual inspection of such a superposition, depicted in Fig. 2.14 for the first five maser pulses of This phenomenon can be ascribed to the presence of the distant dipolar field, which will be further investigated by numerical simulations in the following chapter. Further dramatic manifestations of the frequency shift caused by the dipolar field were also observed. One such example is shown in Fig. 2.15(a), where the induction signal has a quasi rectangular profile -actually slightly decaying -that lasts several tens of ms and abruptly returns to zero. Such a signal shape is clearly quite remote from the expected hyperbolic secant (or approximately so) maser pulse envelope. In addition, these signals exhibited a time-varying frequency, giving rise to a squared chirped maser pulse. The time-frequency analysis of this induction signal performed by the short-time Fourier transform [29] depicted in Fig. 2.15(b) shows the frequency change with time during the pulse. The frequency shift between the beginning and the end of the pulse was about 7500 Hz. The combined resonance and intensity changes are characteristic of the presence of the distant dipolar field [30], and such induction signals are yet a surprising manifestation of nonlinear dynamics of the magnetization in the simultaneous presence of a strong dipolar field and an efficient radiation field.

Conclusion: Comparison with previous experimental results

Our observations are in line with previous work [31,14] in many respects, but also show strikingly distinctive features. In reference [14] the induction signals consistently exhibited a short series These discrepancies are quite puzzling, but some significant differences between experimental conditions in which experiments were performed here and in reference [14] should be pointed out that may help their interpretation. Here, experiments were performed with a probe of moderate Q, and radiation damping accordingly less efficient. Therefore, in order to achieve a large enough magnetization, larger volume samples were used (800 µL instead of 150 µL typically used in our dissolution-DNP setup) and only unlabelled (i.e., non deuterated) compounds were used for the sample preparation in order to increase the 1 H density. Besides, the complete absence of 2 H nuclei in the sample precluded repolarization of 1 H through a cross-talk mechanism, [32,33,34] in contrast with our previous study (reference [14]). Finally, MASER pulses were not observed in the absence of continuous µw irradiation, which was deemed necessary in all cases in the present study. This is explained by the absence of deuterium in the sample, which is not coupled to the probe but serves as a repolarization reservoir for 1 H (reference. [14])

Chapter 3

Modeling the DNP NMR MASER: RD,

DDF and DNP

In a previous work [1], a number of aspects of the observed signals of the hyperpolarized spin sample remained unclear, and remained unexplained in the simple framework of the Bloch-Maxwell-Provotorov equations. In particular, this model seemed to be unable to account for the asymmetric envelopes of the maser pulses. Furthermore, "bursts within burst" or "echo-like" patterns were observed for the most intense maser pulses. None of these can be reproduced by the simple (modified) Bloch equations, which call for a more complex and complete description of the spin system. As was shown in the previous chapter, one ingredient was missing in the simplified BMP model, namely, the presence of the dipolar field that couples the nuclear spins.

It is usually assumed that, in solids, this dipolar interaction among the close neighbouring spins is responsible for the decoherence and the fast decay of the induction signal (typically a few hundreds of µs). For distant spins the dipolar interaction is weaker and it affects the spin dynamics at time scales which can be much longer than the inverse line width. As was discussed in chapter 1, when the ratio of the radiation damping rate to the transverse relaxation rate approaches unity, the transverse coherence may persist for much longer times so that the effects of this weak distant dipolar interaction becomes effective. For these reasons, we decided to introduce a model with short-and long-distance dipolar interactions, where the former is responsible for the fast decoherence, and the distant dipolar field is treated explicitly, similarly 57 to the model proposed by Deville et. al. [2].

Distant Dipolar Field (DDF) in Solids

In order to take into account both these short and long range dipolar effects on the magnetization dynamics, the environment of each spin i may be decomposed as follows. A small region centered on i, the dimensions of which are large with respect to the first neighbor distances but much smaller than the sample dimensions, is assumed to be responsible for decoherence and the fast decay of the induction signal. Alternatively, the long distance, and accordingly weaker, interactions with remote spins affect the magnetization dynamics only on time scales that are much longer than the typical inverse line width. This may nevertheless be the case if phase coherence between spins is maintained long enough, as in the presence of a strong radiation damping. These considerations lead us to envisage a simplistic model where all the spins in the system have a common T 2 relaxation time (and a γ 2 = T -1 2 rate) that represents the effect of the local dipolar interactions. Alternatively, the dipolar contribution from remote spins, an active ingredients of the magnetization dynamics in this context, is treated explicitly. In this model sample, the magnetic moments are sitting on a regularly spaced lattice and the local dipolar field at site i, B ⃗ dip (r i ), due to the long range dipole dipole interactions can be explicitly computed for all moments in the model sample as:

B ⃗ dip (r ⃗ i ) = µ 0 4π ∑ j 3[m ⃗ (r ⃗ j ).r ˆij ]r ˆij -m ⃗ (r ⃗ j ) |r ⃗ i -r ⃗ j | 3 (3.1)
In these equations, m ⃗ (r ⃗ i ) and m ⃗ (r ⃗ j ) are the moments at sites i and j, and

r ˆij = (r ⃗ i -r ⃗ j )/|r ⃗ i -r ⃗ j |
is the unit vector joining the positions i and j. For a regularly spaced lattice, we can re-scale the position vector as r ⃗ = ar ⃗, where a is the spacing between the lattice points. Thus equation 3.1 can be written as:

B ⃗ dip (r ⃗ i ) = µ 0 4π ∑ j 3[M ⃗ (r ⃗ j ).r ˆij ]r ˆij -M ⃗ (r ⃗ j ) |r ⃗ i -r ⃗ j | 3 (3.2)
here M ⃗ (r ⃗ j ) is the magnetization (dipole per unit volume) at the position vector r ⃗ j . In a rotating frame with angular frequency ω o , the position vector can be written as r ⃗ = (r sin(θ) cos(ω o t), r sin(θ) sin(ω o t), r cos(θ))

where the θ is the polar angle. Therefore the time average of equation 3.2 retaining only the secular part becomes:[3]

B ⃗ dip (r ⃗ i ) = µ 0 4π ∑ j̸ =i 1 -3 cos 2 θ ij 2 |r ij | 3 × [ 3M z (r ⃗ j )z ˆ-M ⃗ (r ⃗ j ) ] (3.3)
The distant dipolar field is a nonlocal quantity, making its computation difficult. However, as has been recognized long ago by Deville et al. [2] that Equation 3.3 has the form of a convolution in the real space, so that its Fourier transform with respect to the spatial coordinates is local quantity, which is expressed as a product in the reciprocal space [2]. In terms of the threedimensional Fourier transforms of the field and the magnetization:

B ⃗ dip (k ⃗ i ) = 1/(2π) 3 ∫ d 3 r exp ( -jk ⃗ i • r ⃗ ) B ⃗ dip (r ⃗ i ) (3.4)
and:

M ⃗ (k ⃗ i ) = 1/(2π) 3 ∫ d 3 r exp ( -jk ⃗ • r ⃗ ) M ⃗ (r ⃗ i ) (3.5)
of the distant dipolar field and magnetization. The dipolar field in the Fourier space, B ⃗ dip (k ⃗ i ), can thus be written as (for more details, see appendix C):

B ⃗ dip (k ⃗ i ) = µ 0 6 [ 3 ( k ˆi • z ˆ)2 -1 ] × [ 3M z (k ⃗ i )z ˆ-M ⃗ (k ⃗ i ) ] (3.6)
This expression can be numerically computed using the strategy introduced by Enss et al. [4].

The equation 3.6 has a singularity at k ⃗ = 0, so care should be exerted when performing the inverse Fourier transform ( F -1 ):

B ⃗ dip (r ⃗ i ) = F -1 {B ⃗ dip (k ⃗ i )} k ⃗ =0 + F -1 {B ⃗ dip (k ⃗ i )} k ⃗ ̸ =0 (3.7)
Sphere (all direction) 0 long cylinder ( along parallel axis) µ o /3 long cylinder (perpenducular to parallel axis) -µ o /6 flat disk ( perpendicular to axis) -2µ o /3 flat disk ( in plane)

µ o /3

F -1 {B ⃗ dip (k ⃗ i )} k ⃗ ̸ =0
is computed directly using Equation 3.6, whereas the term

F -1 {B ⃗ dip (k ⃗ i )} k ⃗ =0
is obtained from the average nonzero magnetization in the sample [4,3,5]. Since the distant dipolar field depends on the shape of sample, it is difficult to solve equation 3.6 analytically for arbitrary sample shapes. However, analytical solutions do exist for ellipsoidal sample. [3,5] Thus the distant dipolar field contribution from uniform magnetization can be written as:

F -1 {B ⃗ dip (k ⃗ i )} k ⃗ ̸ =0 = 1 2 [ ∑ α=U,V,W k αα e ˆα • z ˆ][ 3M avg z z ˆ-M ⃗ avg ] (3.8)
where U, V, W are the axis of the ellipsoidal sample (see Figure 3.1) and the values for k αα are shown in table 3.1 for different cases. In particular, for a long cylinder and a disk the distant dipolar field contribution from uniform magnetization can be calculated as follows:

F -1 {B ⃗ dip (k ⃗ i )} k ⃗ ̸ =0 = µ o 6 [ 3 cos 2 (θ) -1 2 ][ 3M avg z z ˆ-M ⃗ avg ] cylinder (3.9) F -1 {B ⃗ dip (k ⃗ i )} k ⃗ ̸ =0 = - µ o 3 [ 3 cos 2 (θ) -1 2 ][ 3M avg z z ˆ-M ⃗ avg ] disk (3.10)

Hyperpolarization of the Nuclear Spins-A simplification of the Provotorov equations

The last ingredient of our dynamical model is the dynamic nuclear polarization hyperpolarization process of the nuclear spins, which provides the magnetization build-up mechanism. In our experiments, performed at or below 4 K with TEMPOL radical concentrations above 50 mM, DNP is assumed to be in the thermal mixing regime [6] a process described by the Provotorov equations [7,8,9,10]. In this thermal mixing description, electron-electron and electron-nuclear spin interactions are associated to several coupled energy reservoirs, each of which characterized by a spin temperature T i (or its inverse β i = 1/k b T i ). Thus, Nuclear Zeeman (NZ) and electron dipole-dipole, non Zeeman, (eeD) reservoirs are in thermal contact with each other [11,12,13,14] and when several nuclear species are present, each one constitutes a distinct reservoir coupled to the eeD reservoir, hence indirectly to one another [15,6] adding further complexity to the associated Provotorov rate equations. The time evolutions of these inverse spin temperatures are governed by Provotorov's rate equations describing the flows of inverse spin temperatures between the various reservoirs [7,8,9]:

dβ n dt = -γ n,ee (β n -β ee ) dβ ee dt = -γ ee,n (β ee -β n ) -γ ee,d (β ee -β d ) -γ ee,L (β ee -β L )
+πω

Model A

An in-depth description of thermal mixing is beyond the scope of this thesis, which focuses on the collective and macroscopic aspects of the nuclear ùspin dynamics, rather on the details of the spin polarization process itself. Besides, because the latter is difficult to access, it is desirable to rely on a simplified model that captures the effect of the phenomenon on the magnetization dynamics. Recently Weber et. al. proposed a simplified model [1] to empirically capture the effect of dynamical nuclear polarization on the collective and macroscopic aspects of the nuclear spin dynamics, and to reproduce experimental observations with a drastic simplification of the Provotrov's rate equation by a simple mono-exponential function for β n . Such a simplification is likely justified when the 1 H are repolarized towards their initial hyperpolarized state under microwave irradiation or in the absence of latter, when γ ee,n , γ ee,d >> γ ee,L and γ ee is in fast equilibrium with abundant 1 H and other nuclear Zeeman reservoirs.

In this model, the nuclear inverse spin temperature β n of the 1 H Zeeman reservoir relaxes exponentially with time constant γ n,ee , to the electron non-Zeeman spin temperature β ee , which also exponentially relaxes (with rate γ st ) to a stationary value that represents either its polarization upon µw irradiation, or the lattice temperature. This toy model of DNP is relevant to the case where 2 H nuclei are present in large quantities in the sample, as these provide a source of polarization to the 1 H nuclei by an indirect "cross-talk" mechanism through their interaction with the electron non Zeeman reservoir [15,6,16]. Thus, the deuterium spins contribute to the repolarization of the protons and buildup of the nuclear magnetization, and compensate at least partially the 1 H Zeeman energy lost to the probe through radiation damping.

In this model setup, the z component of the nuclear spin magnetization density at point r ⃗, M z (r ⃗, t), therefore tends to the time-varying value M th oz (r ⃗, t), that "relaxes" to a stationary value M st 0 (r ⃗). Again, the quantity M st 0 (r ⃗) pertains either to the stationary value of the 1 H magnetization upon µw irradiation, or to its thermal equilibrium, when µw irradiation is turned off. This model therefore writes:

dM z (r, t) dt (r ⃗, t) = -γ n,ee [ M z (r ⃗, t) -M th oz (r ⃗, t)
] (3.12)

and:

dM th oz dt (r ⃗, t) = -γ st [ M th oz (r ⃗, t) -M st 0 (r) ] (3.13)
In Eq. 3.12, M ⃗ z (r ⃗, t) is related to the spin temperature β n of the 1 H:

M z (r ⃗, t) = γ 2 n h2 N h B 0 4 β n (t) (3.14)
where N h is the proton spin density; and β n itself relaxes to the temperature of the eDD reservoir β ee (t), so that one has:

M th oz (r ⃗, t) = γ 2 n h2 N h B 0 4 β ee (t) (3.15)
This model, introduced in ref. [1] will be referred to as model (A).

Model B

In the experiments performed for this thesis, however, only 1 H are present (no deuterated compounds were used, so there are no 2 H, and thus model A is not relevant to this situation), and once polarized, 1 H can only be repolarized by DNP from direct interactions with the electrons, under µw irradiation. Therefore, it is assumed that the temperature of the eDD reservoir, β ee is a constant due to the fast equilibration between electron Zeeman and dipolar heat reservoirs.

Thus, according to Equation 3.11, the nuclear spin magnetization density at point r ⃗, M z (r ⃗, t), builds up exponentially towards its stationary value

M st 0 (r ⃗) = γ 2 n h2 NhB 0 4
β ee , taken in this case as the stationary polarization level obtained by DNP, with the time constant γ n,ee :

dM z (r ⃗, t) dt (r ⃗, t) = -γ n,ee [ M z (r, t) -M st o (r ⃗) ] (3.16)
This will be referred to as model (B). Finally, in the classical and empirical model proposed here, the modified Bloch-Maxwell equations obeyed by the magnetization are:

d dt M ⃗ (r ⃗, t) = γM ⃗ (r ⃗, t) × (B ⃗ rd (t) + B ⃗ dip (t)) -γ 2 (M x (r ⃗, t)x ˆ+ M y (r ⃗, t)y ˆ) -γ n,ee (M z (r ⃗, t)) -M th oz (r ⃗, t))z ˆ(3.17)
where, for model (B), M th oz (r, t) is replaced by its constant value M st o (r). Numerical simulations of Equation 3.17 allows one to study the time evolution of the total magnetization, and provide better insight into the largely non intuitive dynamics of this non local and nonlinear system.

Numerical simulations

Numerical simulations aimed at better understanding the possible role of the dipolar field in the experiments presented here, and also in previous work [1]. In all simulations, classical moments with magnetization density M ⃗ (r ⃗ i ), were located at fixed positions on a grid. In this coarse grained representation of the sample, each moment represents a "cluster" of spins undergoing decoherence resulting from the strong, local dipolar interactions within the cluster, and characterized by a unique relaxation rate γ 2 for all clusters, and given in Equation 3.17. In addition, the different clusters (i ̸ = j) mutually interact through long-distance dipolar interactions.

Numerical simulations were performed on an ensemble of spins located on a regularly spaced lattice. In the experiments presented in this work, the sample is held inside a cylindrical cup. However, it is not possible to determine the region of the sample that is actually hit by the microwaves, so that the shape of the "effective" sample, that is, the region of the sample that . (adapted from [4]) is highly polarized and therefore contributes most effectively to radiation damping and to the distant dipolar field, cannot be easily determined. For this reason, in order to qualitatively investigate the effect of the shape of the sample on the magnetization dynamics, we decided to simulate two extreme cases for sample shape: a thin disk (21 × 21 × 7 grid points, figure 3.4) and an elongated cylinder (11 × 11 × 21 grid points, figure 3.5), shown in thick line inside the unit cell of figure 3.3. In order to avoid numerical artifacts due to the periodicity induced by the FFT, zero padding was applied. [4,3] Satisfactory conditions were obtained by increasing the padding dimensions in test simulations until no significant changes on computed magnetization trajectories were observed. The retained padding was 100 × 100 × 150 padding in the x, y, and z dimensions for all simulations, effect of padding on transverse magnetization is shown in figure 3.6. Finally, because of the singularity at k = 0 this component was computed from the average magnetization using the equation 3.9 [4,3]. Equations 3.17 were numerically solved using explicit Runge-Kutta method of order 8 [17] implemented in Scipy [18] by the scipy.integrate.solve_ivp adaptive stepsize integrator, figure 3.7 shows integration flow chart for solving equations 3.17.

The simulations shown in this article were run on a 10-core Dell Precision 5820 desktop computer. We used an idealized initial state in which all moments⃗M i have the same initial value (0, -M o sin(ϕ), M o cos(ϕ)), where M o is the magnitude of M ⃗ i and ϕ is the initial flip angle, which was set to -1 o . The control parameter of the dipolar field effects in our simulations was the magnetization density parameter m o , which can be related to the average dipolar field ω dip at equilibrium:

ω dip = γ H µ 0 2M 0
3 for a disk-shaped sample and ω dip = γ H µ 0 2M 0 6 for a cylinder [3]. Using these numerical tools, we performed simulations of the magnetization dynamics that follows an initial period of spin DNP-hyperpolarization ending by a small trigger pulse, for models (A) and (B) above that pertain to different experimental conditions, namely, DNP samples composed of a mixture of 1 H/ 2 H, or of protons only.

Numerical Results

Sustained masers under µw irradiation -1 H only (model B)

In a first round of simulations we investigated situations described by model (B), an idealization of the experiments presented above. In order to ensure the presence of maser pulses in the simulations, we set the condition γ 2 /γ rd ≤ 1 at the initial time of the trajectory, when the magnitude of the total magnetization is largest. The latter was equal to the stationary hyperpolarized magnetization M st 0 , which was set to a negative value, as only a negative polarization is expected to lead multiple masers.

In Figure 3.8, the traces of the magnitude of the transverse magnetization, M x (t)+M y (t) (red)

and of the M z component of the magnetization are shown. The typical multiple maser pattern is recovered, and is understood to be due to the presence of two antagonist processes, namely, the radiation damping that rotates the magnetization vector towards its equilibrium direction, and the (negative) polarization process that builds up the z component of the magnetization towards its stationary state M st 0 . [19,20] As shown by our experiments, one of the ingredients involved in the dynamics is the presence of a strong enough dipolar field in the sample to affect the evolution of the magnetization.

Therefore, we next investigated these through a ensemble of simulations. In our numerical model, the dynamics can be studied by varying its strength through the (uniform) spin density M o assigned to each moment on the lattice, the control parameter for this purpose.

In this series of simulations we assumed continuous a µw irradiation, which is part of model (B), with γ n,ee = 20.0Hz. Besides, the radiation damping time constant and the transverse relaxation rate γ 2 were set to the fixed value γ rd = 2.0 kHz and γ 2 = 650 Hz, so that the ratio γ rd /γ 2 was above unity, the theoretical maser threshold. Note that changing the value of M o also affects the value of the radiation damping constant, so keeping γ rd fixed in these simulations accordingly. The value of γ n,ee = 20.0Hz was chosen so as to mimic a buildup process that occurs on a much longer time scale than individual maser pulses, and therefore likely not interfering with the latter.

The effect of the DDF on the dynamics is presented in Figure 3.8, where the simulations show typical series of sustained maser bursts for different values of the spin density M o .

Interestingly, the envelope of the transverse signal shown in Figure 3.8(a) qualitatively reproduces the main features of the experiments, in which an apparent "steady-state" of regularly spaced masers was observed. Alternatively, in the simulations performed with decreasing values of M o (from top to bottom), the intensities of the bursts monotonously decay and the transverse magnetization eventually reaches a constant intensity, indicating a steady-state precession about the z axis. Besides, the details of each maser pulse envelope are also affected. Indeed, one observes that for large values of M o , each pulse exhibits an asymmetric profile with several pulses within it, whereas in the absence of any distant dipolar field, this "bursts within burst" pattern disappears, and the maser shape recovers the conventional symmetric, hyperbolic secant profile, as expected from the simpler BMP model used in previous work [19,1].

The effect of the decoherence rate γ 2 on the maser pulse dynamics was also investigated, and results are illustrated in Figure 3.9. Two such simulations, performed at constant γ rd and M 0 , therefore identical RD and DDF effects, and for different values γ 2 leading to γ rd /γ 2 = 3.1 and γ rd /γ 2 = 2, are shown. In both cases a continuous series of sustained maser are obtained.

For larger γ rd /γ 2 ratios, maser pulses are sharper and have shorter duration, which attests for a more efficient RD for longer transverse relaxation, a known and expected feature. This is associated with larger excursions of M z during the pulses and larger the maximum excursion from its initial value (see Figure 3.9). Nevertheless, it is noteworthy that in all the simulations, the m z component remains close to its initial value, within ∼ 79-86% of its minimum, indicating that the trajectory of the magnetization vector lies in the proximity of the south pole of the Bloch sphere. Thus, with these parameters, the maser effect is actually rather inefficient, due to too fast decoherence. Again, as shown in the previous example, the envelope of the maser pulse exhibits a clear asymmetry, with the presence of multiple, smaller, maser pulses within the main pulse. But for the lower γ rd /γ 2 ratio, this asymmetry vanishes, suggesting that the presence of the faster decoherence prevents the distant dipolar field to significantly impact on the magnetization dynamics.

Sustained masers -the "cross-talk" situation (model A)

We also performed numerical studies of model (A), which was introduced to mimics the magnetization dynamics of the 1 H nuclear spins in the presence of a 2 H reservoir. The presence of this additional nuclear spin reservoir allows for the repolarization of the former, both with and without µw irradiation, by an indirect "cross-talk" mechanism through the electron non

Zeemann reservoir [15,6]. In this case, only the 1 H spins are coupled to the detection circuit of the spectrometer, to which they lose energy, whereas the likewise hyperpolarized 2 H only exchange energy with other spins. As detailed in ref. [1] and explained above, this process leads to partial repolarization, i.e., re-cooling, from deuterium spins, together with a loss of polarization, therefore heating, of the 1 H nuclei. Thus, model (A) introduces a simplified description of this process, where the z magnetization M z (r ⃗, t) tends to the value M th 0z (r ⃗, t) that corresponds to the deuterium spin temperature, whilst deuterium spins itself relax to thermal equilibrium, in the absence of µw irradiation (see Eqs. 3.12 and 3.13).

In our simulations, the equilibrium magnetization M st o was set to M st o = -0.05 × M mw oz , where A/m, γ 2 = 650 Hz, γ rd = 2 kHz, so that γ rd /γ 2 = 3.1 and ω dip /γ rd = 22.22. In this model, the rates of return to the lattice temperature of the 2 H reservoir is set to γ st = 0.030 Hz and the 1 H/ 2 H equilibration rate of spin temperatures is γ n,ee = 20 Hz (see Eq; 3.13. During the early stage of the dynamics, the in-plane magnetization envelope exhibits a succession of a few (∼ 7 -8) maser bursts of decaying intensities with time, until it reaches a long-term evolution of monotonous decay. Due to the presence of the dipolar field the asymmetry and the "bursts within burst" profiles are also reproduced. At longer evolution times the amplitude of the in-plane magnetization decays monotonously, whereas the z component of the magnetization returns to its thermal equilibrium value, in the absence of µw irradiation, with the γ st time constant. These simulations thus reproduce satisfactorily the basic features of the experimental observations described in Reference [1], i.e., asymmetric masers at short times and long lasting monotonous decay at long times.

How to interpret the effect of the dipolar field on the dynamics

The work presented here aims at better understanding the complex non linear dynamics of a large magnetization subject to efficient radiation damping. To do so, the dynamical effects of the dipolar field were investigated through numerical simulations of a simplified model, which provided results that are consistent with several observed effects, both in this and previous work. In particular, it was found that for strong dipolar fields, the predicted evolution of the magnetization In our view, these elements strongly suggest that the kind of induction signal observed in this work attests for the presence of an effective DDF. The noted asymmetry and the "bursts within burst" profile are clearly also due to the presence of the DDF. However, the mechanism through which the dipolar field produces these effects is somewhat unclear. The dipolar field essentially contributes an average z component that is much larger than its transverse

(Figures 3.8
x and y components. The average ⟨ω d,z ⟩ = -⟨B d,z (r)⟩/γ component is depicted in Figure 3.11 for a trajectory of model (B). It appears that its time evolution mirrors the M z component of the magnetization, and therefore contributes a time-varying precession frequency to the magnetization. Considering its effect on an isolated burst, one would expect its effect to be that of a chirp pulse, with a time-varying offset frequency. However, the local dipolar field B ⃗ dip (r ⃗ i ) at each site of the sample is not uniform, achieving a distribution of its z ˆcomponents, therefore of precession frequencies across the sample, as shown in Figure 3.11(b). From the dynamical viewpoint, this amounts to the existence of an inhomogeneity of B 0 , albeit a time-dependent one. This interesting, as it points to an effect that was already invoked in the context of solution NMR, where B 0 inhomogeneity was shown to lead to such non-decaying sustained maser pulses, an effect not explained by the Bloch-Maxwell equations [21].

A comment on time scales

The simulations presented in this work qualitatively explain a number of features of the magnetization dynamics under DNP and radiation feedback conditions. The models introduced in this work involve a large number of parameters and time scales (γ n,ee , γ rd , γ st , γ 2 , shape of the sample and spin density) that control the kind of dynamics that the magnetization undergoes, and exploring the complete parameter space is a formidable task. We have restricted the study to ranges of parameters that were able to reproduce experimental observations and, as it turns out, some of them may seem remote from the experimental ones. Although these values were arbitrary to some extent, the important point made in our study are the roles played by the relative values of the ratios γ rd /γ 2 and ω dip /γ rd : the former determines the maser threshold, whereas the latter controls the manifestation of DDF effects. Moreover, we have assumed a time scale separation between γ rd , γ 2 and ω dip , on the one hand, and on the much slower rates γ n,ee and γ st , on the other hand. This corresponds to the much longer polarization buildup times.

Experimentally, one of the striking facts is that the observed duration of a typical maser pulse is on the order of ms to tens of ms despite the broad lines (∼ 25 kHz). A crude analysis of the classical Bloch-Maxwell equations may lead to the erroneous conclusion that such a large relaxation rate γ 2 necessarily implies very narrow maser bursts. However, this is not necessarily true, as can be understood from the analysis made by Bloom [22] of radiation damping in the absence of T 1 relaxation. Indeed, analytical solutions derived there show that the duration of a single maser pulse can actually be long. This can be seen from the relation between the radiation damping rate in the presence and in the absence of transverse relaxation, γ rd and γ ∞ rd : [22]

γ rd γ ∞ rd = √ ( γ 2 γ ∞ rd ) 2 + 2 cos θ 0 γ 2 γ ∞ rd + 1 (3.18)
where θ 0 is the initial flip angle. Then, for ratios γ 2 γ ∞ rd = 1 + ϵ larger but close to unity, and for flip angles close to π, θ 0 = π -h so that cos θ 0 = -cos h = 1 -h 2 2 to first order, one sees that the ratio of Equation 3.18 can be made arbitrarily small, meaning that γ rd ≪ γ ∞ rd . This is illustrated in Figure3.12 This allows to explain why, even for very broad lines such as the ones in our experiments (∼ 25 kHz), for extremely small flip angles (and low rf power on the order of the mW that excite only a small fraction of the NMR line) and a radiation damping rate that is almost equal to the line width, maser bursts lasting tens of ms are observed. In fact, this situation was met experimentally when monitoring the polarization buildup with small pulses that trigger the DNP-NMR maser when the growing magnetization approaches the condition

γ 2 ≈ γ rd .
In our simulations, we have used values of γ 2 that are much lower than what the experimental line widths suggest (∼ 25 kHz). This was imposed by numerical issues and we were not able to perform simulations with such large γ 2 and the required small flip angles values. However, based on the above discussion, we argue now that this does not affect the relevance of our calculations that provide insight into these complex phenomena.

Sample shape

Because it is determined by a large nonlinear differential system that, moreover, depends on many parameters, a complete exploration of the magnetization dynamics defined by the models Figure 3.12: Dependence of a single maser time duration on the γ ∞ rd /γ 2 ratio, for a fixed flip angle θ = 179.999 o . The curves were obtained from Ref. [22] with γ 2 = 25 kHz. For ratios approaching unity, the duration increases roughly a tenfold.

of the DNP NMR maser proposed in this work is a formidable task. Nevertheless, we here illustrate by dynamics simulations the effect of the shape of the sample on the DDF, and on the magnetization dynamics. We considered the dynamics of model (A) for a specimen with the shape of a flat disk or an elongated cylinder. Results are depicted in Figure 3.13). Simulations were performed with identical parameter values to show the pure effect of the sample shape on the dynamics. They show similar behaviors, with at short times, the series of maser bursts, followed by a transient stage of sustained precession in the intermediate region, and an evolution towards thermal equilibrium in the long term, when no significant radiation damping is present.

However, the details of the maser pulses , in particular, are different, and can be ascribes to the different values of the DDF, therefore of the ω dip /γ rd ratios due to the different shapes of the samples. Nevertheless, as expected, the overall characteristics of the dynamics are conserved. Chapter 4

An Electronic Control of Radiation Damping

Motivation for Radiation damping control

Radiation damping is known to be a nuisance in high-field solution NMR and to create numerous spectral artifact in magnetic resonance spectra. In conventional high resolution liquid state NMR, radiation damping essentially originates from the solvent (the water). It has been shown

to not only affect the resonances close to the solvent but can also influence resonance lines away from it, up to the kHz. [1,2] It can also lead to artifacts in 2D experiments, such as pseudomultiple quantum peaks. [3,4] Recently, it was also observed that during spin-locking NMR experiments, radiation damping can affect the spin-locked sample nuclei spins. [5] Many approaches have been proposed to eliminate radiation damping or to prevent its effects in solution state NMR pulse sequences. Detuning the probe away from solvent resonance may be used to suppress the effect of radiation damping by sacrificing the signal to noise ratio, but this can create dynamic frequency shift as predicted in [6] and observed by Hwang et. al. [7] It is also possible to suppress radiation damping by acting on the detection system itself. This can be achieved either by use of a radio-frequency feedback circuit, or Q-factor switching device.

In 1959 Chidambaram [8] suggested a method to reduce radiation damping electronically by applying a negative feedback to the resonant probe circuit without degrading the signal to noise 82 ratio of the spectra. Jeener [9] demonstrated the suppression of radiation damping by using a radio frequency feedback. The idea behind this techniques is to minimize the current in the coil, except during radio frequency pulses, by applying a feedback signal obtained from radiofrequency pre-amplifier of the spectrometer thus opposing the electromotive force produced in the coil by the rotating bulk magnetization and suppress the radiation damping field. In this method the feedback signal is injected into the circuit by way of inductive coupling to the detection coil, which required a modification of the probe. The same year Louis-Joseph et. al.

[10, 11] demonstrated a slightly different feedback system. Here, a small radio frequency signal is picked up from the pre-amplifier of the spectrometer and re-injected into the probe, after phase modulation and amplification, using a directional coupler. This method allows one not only to suppress radiating damping but also to enhance it. This radiation damping control scheme will be discussed in detail in the next chapter. An alternative method to suppress radiation damping was based on the switching of the Q-factor of the resonant circuit from a high value during radio-frequency pulses and acquisition periods to a low value during evolution delays by grounding the coil, [1] or even during acquisition [12].

In the context of this thesis, we have implemented and tested an electronic feedback radiation control unit based on a previous approach. [13] This device was designed and built by Alain Louis-Joseph (Laboratoire de Physique de la matière condensée, Ecole Polytechnique). The objective is to implement this device on the 6.7 T polarizer in order to achieve control of the onset of the sustained DNP-NMR maser. This may eventually represent an exploratory route to access the mechanisms of DNP and thermal mixing in particular, and to explore nonlinear dynamics in this context.

Principle of the radiation damping control unit

From the expression of the radiation damping field (see chapter1), when the spins are exactly tuned to the coil, the radiation damping field lags behind the transverse magnetization by Ψ = π/2. So if we can induce a feedback current into the coil, i F B that generates a magnetic feedback field B +,F B e Ψ F B , we can suppress radiation damping when the feedback phase with respect to the transverse magnetization is Ψ F B = -π/2 (correction field pointing in the opposite direction) and enhance the radiation damping field when Ψ F B = π/2 (correction field pointing in the same direction as the radiation damping field). The second stage consists of a low noise amplifier, a quadratic demodulator, a low pass filter, a second quadratic modulator, RF phase modulator (0 to 360 o ), a variable attenuator and a low power radio frequency amplifier. The input of this second stage is also amplified by way of a low noise amplifier. The amplified radio frequency signal is demodulated with respect to the reference generated by a local oscillator and using a pair of mixers into two, in-phase (0 0 ) and in quadrature (90 0 ) signals. The in-phase and quadratic signals are then passed through a low-pass filter in order to remove the higher frequency components. The cut-off frequency of the low pass filter is set to 100 kHz and can be adjusted to change the bandwidth of the device. The low frequency components are then re-modulated with the reference frequency to generate radio-frequency signals (quadratic modulation) using an additional pair of mixers, and then combined to radio frequency signal. The output of the quadratic modulator is then fed into a rf phase modulator acting from 0 -360 o then amplified using a low power amplifier. The The third stage (purple) consist of a power amplifier that allows to adjust the power of the signal fed back to the coil, by way of a directional coupler (green, CPL 1) inserted between the probe and the pre-amplifier input.

Radiation damping control on a 400MHz spectrometer

The implementation of this radiation damping control unit on a commercial spectrometer is a crucial, and far from straightforward, step in this work as many technicalities were encountered.

The tests were performed on a 9.4 T liquid state commercial Bruker spectrometer, using a triple resonance probe (TXI 5 mm), controlled through the Topspin software. This observation, similar to previous work, [14] also resembles the maser evolution shown in In the DNP experiments, this was ascribed to the presence of the dipolar field. Here, the exact origin was not identified with certainty. However, earlier studies [14] suggest that the presence of a B 0 inhomogeneity can lead to both these features. In the case of the DNP maser, it may be argued that these features may be due to the distribution of dipolar field across the sample, which would act like an inhomogeneity of the (time-varying in this case) z component of B o .

A Two-Mode Liquid State Maser

A single pulse two-mode maser in liquid state was obtained by enhancing the radiation damping field acting on proton magnetization corresponding to water and methanol using the radiation damping control unit. observes a single maser from methanol, whilst the water signal is not intense enough to trigger a maser itself (see orange trace in Figure 4.16). Alternatively, when radiation damping is amplified, a two-mode (blue) maser pulse is observed (blue trace in Figure 4.16). In the latter case, the magnetization was flipped to 180 o using a short pulse 22 µs, the radiation damping control unit was switched on with the phase (129 0 ) and power (0dB) in order to enhance the radiation damping field acting on both water and methanol. These experiments therefore illustrate that for resolved resonance lines and for spins sufficiently abundant, distinct feedback fields are produced (spontaneously or after enhancement), therefore distinct maser modes.

Conclusion

One of the goals of this thesis was to use an electronic feedback control unit of radiation damping on the DNP polarizer in order to generate masers in a controlled way. As this thesis has suffered from various unfavorable circumstances (COVID crisis and various instrumentation problems such as the failure of the DNP probe and of the microwave source) that have prevented the full developments of the project, the radiation damping control unit could not be implemented on the polarizer. The device was successfully tested at 300 MHz and 400 MHz spectrometers at the Ecole Polytechnique and at ENS, which shows its versatility. Radiation damping suppression and enhancement were both achieved on these spectrometers. Interestingly we were able to generate a two-mode maser in liquid state using the device by enhancing the radiation damping field.

Chapter 5

Conclusion and Perspectives

We have pursued the work initiated in the thesis of Emmanuelle Weber, who first observed multiple maser on this instrument. Although these observations were rationalized using a Bloch-Maxwell-Provotorov set of equations, a number of fine features could not be explained by this approach. This was one of the motivations of this thesis. The other motivation was to try an evaluate the possibility to control the onset of these masers, in order to indirectly access the parameters of the Provotorov equations.

However, we have adapted our approach, due to various instrumentation issues (NMR probe, microwave source failures, among others). Due to the degradation of the quality factor of the probe, initially designed and developed by the manufacturer of the polarizer prototype (Bruker), much weaker radiation damping was obtained that initially prevented us from reproducing previous maser observations. This led us to modify the sample composition dramatically in order to be able to observe maser effects again. The important change, sample-wise, was the total suppression of deuterated compounds in the sample, whereas in the optimization process of dissolution-DNP experiments, the use of 1 H is usually limited to maximize the polarization.

Moreover, the total volume of the sample was increased by a factor of ∼ 5 with respect to common practice. These changes permitted to obtain a large enough magnetization to produce effective radiation damping and trigger sustained masers for negative polarization of the spins. Such modifications happened to have significant qualitative consequences on the observed maser, that now exhibited features not seen previously, such as the onset of non-attenuated 102 maser pulses, separated by delays of seconds, for minutes or more, under microwave irradiation.

These maser pulses had a duration on the order of 200 ms, and the associated spectrum gave a 10 Hz resonance line.

In order to better understand these and the previous observations, we included in our model the presence of the dipolar field in the sample. We were able to observe the direct manifestations of the distant dipolar field, and numerically investigated the magnetization dynamics under the combined effect of radiation damping and dipolar effect in this experimental context of DNP at cryogenic temperature, that is on a hyperpolarized spin system with a broad resonance line, on the order of ∼ 25 kHz.

Results of the numerical simulations allowed us to qualitatively reproduce the present and previous observations, thereby providing further insight into these non-linear spin dynamics. In particular, the origin of the asymmetry of the maser pulse envelope, as well as the presence of multiple maser bursts within a pulse of the maser was found to be the manifestation of the distant dipolar field. In addition, our simulations illustrated the way radiation damping, by extending the coherence of the spins and by increasing the duration of the induction signals once the spins are polarized negatively, revealed the presence of the distant dipolar field.

A second objective of the thesis was to implement an electronic control device of radiation damping to control the onset of these DNP-NMR masers. The device itself, based on previous work, was designed and built by Alain Louis-Joseph from the Laboratoire de physique de la matière condensée at the Ecole Polytechnique. It was tested on a solution state 400 MHz NMR spectrometer in ambient conditions. Its implementation required to solve a number of technicalities and was eventually successful. We were able to calibrate both the phase and the gain so as to enhance and suppress radiation damping. We were also able to create a (single pulse) two-mode maser in a mixture of water and methanol. These preliminary data need to be further investigated and developed. However, this thesis has suffered from various unfavorable circumstances (the instrumentation problems mentioned above) and the radiation damping control unit could not be implemented on the polarizer in the course of this work.

Indeed, due to repeated failures of the microwave doubler, then of the 94 GHz microwave source itself, a sufficient level of polarization could not be achieved to generate spontaneous masers, a condition required for the implementation, and more precisely, the calibration, of this new device. Nevertheless, we have obtained promising results that will serve as a firm basis for the pursuit of the initial goals of this work.

ρ = 1 N N ∑ i=1 |Ψ i ⟩ ⟨Ψ i | = ⎛ ⎜ ⎝ ρ aa † ρ ab † ρ ba † ρ bb † ⎞ ⎟ ⎠ (A.3)
Where

ρ a i a † i and ρ b i b † i are the population of state | 1 2 , 1 2 ⟩ and | 1 2 , - 1
2 ⟩, and off diagonal terms are called coherence.

The polarization [1] of the spin ensemble is defined as

P = 1 |S| T r{ρS ˆ} T r{ρ} (A.4)
Where P is the polarization vector (P x , P z , P z ) and S ˆ= (S ˆx, S ˆy, S ˆz) is the spin operator defined as,

S ˆx = h 2 ⎛ ⎜ ⎝ 0 1 1 0 ⎞ ⎟ ⎠ , S ˆy = h 2 ⎛ ⎜ ⎝ 0 -i i 0 ⎞ ⎟ ⎠ , S ˆz = h 2 ⎛ ⎜ ⎝ 1 0 0 -1 ⎞ ⎟ ⎠ (A.5)
Therefore the z polarization for spin half particle is defined by,

P z = ρ aa † -ρ bb † ρ aa † + ρ bb † = e γhBo/2K B T -e -γhBo/2K B T e γhBo/2K B T + e -γhBo/2K B T = T an ( γhB o 2K B T ) ≈ γhB o 2K B T , K B T >> γhB o (A.6)
For spin ensemble at 6.7T and 1.2K, the electron polarization, P S z = 0.99(99%) and proton polarization, P I z = 0.0057(0.57%).

3(m ⃗ • r ˆ)r ˆ-m ⃗ = ( 3 2 m x sin 2 (θ) -m x )x ˆ+ ( 3 2 m y sin 2 (θ) -m y )y ˆ+ (3m z cos 2 (θ) -m z )z = m x 2 (1 -3 cos 2 (θ))x ˆ+ m y 2 (1 -3 cos 2 (θ))y ˆ-m z (1 -3 cos 2 (θ))z = (1 -3 cos 2 (θ)) 2 (m x x ˆ+ m y y ˆ-2m z z ˆ) (B.5)
Now substitute Equation B.5 in Equation B.1 we get:

B ⃗ dip (r ⃗ i ) = µ 0 4π ∫ d 3 r ′ 1 -3 cos 2 θ ij 2 |r ij | 3 × [ 3m z (r ⃗ ′ )z ˆ-m ⃗ (r ⃗ ′ ) ] (B.6)
where θ ij is the angle between z ˆand the position vector r ˆij .

Using equation C.1 and C.2, we can write delta function as:

δ 3 (r ⃗) = 1/(2π) 3 ∫ d 3 k exp ( ik ⃗ • r ⃗ ) (C.4)
Thus the Fourier transform of Dirac delta function can be written as:

δ 3 (k ⃗ ) = 1/(2π) 3 (C.5)

Fourier transform of derivatives

From any standard mathematical physics books [2], we can show the fourier transform of a derivative as:

F.T.{∂ i ∂ j F (r ⃗)} = (jk i )(jk j )F ˜(k ⃗ ) (C.6)

Fourier transform of Coulomb potential

From any electrodynamics book [3], we can write the Coulomb potential at r ⃗ due to a point charge at origin can be written as 1/4πϵ o r. It satisfies the Poisson's equation [3]:

∇ 2 1 4πϵ o r = - 1 ϵ o δ 3 (r ⃗) (C.7)
where ∇ 2 = ∂ 2 x +∂ 2 y +∂ 2 z . This lead to the famous equation equally important in electrostatics and magnetostatics:

∇ 2 1 r = -4πδ 3 (r ⃗) (C.8)
Using equation C.6, we can easily write the Fourier transform of Poisson equation as:

F.T.{ 1 r } = 1 2π 2 k 2 (C.9)

Fourier transform of convolution

Consider three functions F 1 , F 2 and F 3 given by

F 3 = ∫ d 3 r ′ F 1 (r -r ′ )F 2 (r ′ ) (C.10)
Then F 3 is said to be the convolution [2] of function F 1 and F 2 . The Fourier transform the equation C.10 can be easily shown as:

F ˜3(k) = (2π) 3 F ˜1(k)F ˜2(k) (C.11)

Secular distant dipolar field equation

The equation 3.3 can be written integral form as:

B ⃗ dip (r ⃗ i ) = µ 0 4π ∫ d 3 r ′ 1 -3 cos 2 θ ij 2 |r -r ′ | 3 × [ 3M z (r ⃗ ′ )z ˆ-M ⃗ (r ⃗ ′ ) ] (C.12)
It is clear that dipolar field is a convolution of

F 1 = 1 -3 cos 2 θ ij 2 |r -r ′ | 3 F 2 = µ o 8π [ 3M z (r ⃗ ′ )z ˆ-M ⃗ (r ⃗ ′ ) ] (C.13)
Thus the Fourier tranceform of equation C.13 can be written using equation C.11 as:

B ⃗ ˜(k ⃗ ) = (2π) 3 F ˜1 µ o 8π [ 3M z (k ⃗ )z ˆ-M ⃗ (k ⃗ ) ] (C.14)

Hence the proof

It is straight forward to see the derivative of 1/r,

∂ i 1 r = - x i r 3 (C.15)
Taking second derivative we get,

∂ j ∂ i 1 r = 3x i x j -δ ij r 2 r 5 (C.16)
This equation is not valid at r = 0 and for the case i = j it violates the equation C.8. So the correct form of equation C.16 [3] is,

∂ j ∂ i 1 r = 3x i x j -δ ij r 2 r 5 - 4π 3 δ ij δ 3 (r) (C.17)
For the case i = j = z, the equation C.17 can be written as,

∂ 2 z 1 r = 3 cos 2 θ -1 r 3 - 4π 3 δ 3 (r) (C.18)
Now applying Fourier transform on equation C.18, we get

(jk z ) 2 1 2π 2 k 2 = -F ˜1(k) - 4π 3 1 (2π) 3 F ˜1(k) = 1 6π 2 ( 3 k z k -1 ) (C.19)
This from equations C.19 and C.14 we can finally write: After application of short pulse of order of microseconds, the duplexer switch close the transmitter-probe path and open the receiver-probe path. Since the induction signal is order of micro volts, the signal is amplified using a fixed gain low noise pre-amplifier, figure F.6 before feeding to the receiver. The receiver demodulate the incoming radio frequency signal to audio frequency using the reference frequency ω RF [4] after a short delay to neglect transient effect of probe. The demodulated analog signal is digitized using analog to digital converter (ADC) and stored in computer in binary format for doing signal processing.

B ⃗ dip (k ⃗ ) = µ 0 6 [ 3 ( k ˆ• z ˆ)2 -1 ] × [ 3M z (k)z ˆ-M ⃗ (k ⃗ ) ] (

Radio frequency leakage

It was observed a radio frequency leakage between port 3 (connected to the output of radiation left). The q-factor is calculated by:

Q = F o ∆ F W HM (K.1)

Introduction xxxiv 1 A 1 1. 1 A 1 1. 2 9 1. 2 . 1 19 1. 3 . 1 19 1. 3 . 2 20 1. 3 . 3 1 . 3 1 . 4

 111129211931193220331314 classical description of NMR beyond the Bloch equations short review of basic concepts of Nuclear Magnetic Resonance . . . . . . . . . The Origin of Non-Linearity in the dynamics of a spin ensemble . . . . . . . . . Radiation Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.2 Dipolar Interactions and the distant dipolar field . . . . . . . . . . . . . 17 1.3 A short description of Dynamic Nuclear Polarization . . . . . . . . . . . . . . . Signal to noise ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Overhauser Effect and the Solid Effect: . . . . . . . . . . . . . . . . The Cross Effect and Thermal Mixing . . . . . . . . . . . . . . . . . . . 23 2 DNP NMR maser: experimental observations 36 2.1 A DNP NMR MASER at ENS . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3 Polarization buildup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 vi Largeur de raie et déplacement du signal de résonance au cours d'une expérience de polarisation. A gauche : les largeurs de raie sont respectivement 16100 Hz, 11300 Hz, 5700 Hz et 4330 Hz de bas en haut. À droite: premiers instants expérimentaux de la raie M 1 en fonction de l'intensité de la raie. Les symboles pleins indiquent les valeurs extraites des spectres expérimentaux; la ligne droite indique l'ajustement linéaire des données. . . . . . . . . . . . . . . . . . . . . . xxvii 3 Simulations de la dynamique d'aimantation pour le cas A (dimension du réseau est 21 × 21 × 7). L'enveloppe de l'aimantation transversale est figurée en rouge et la composante z en bleu. γ rd /γ 2 = 3, 10 et γ 2 = 650 Hz ; ω dip /γ rd = 44, 45 , m o = 400, 0 A/m ; γ rd = 2, 0 kHz, de γ n,ee = 20, 0 kHz ; ω dip /γ rd = 44, 45 . . . . . . . . . . . . . . . .xxviii 4 Schéma de l'unité de contrôle de rétroaction électronique . . . . . . . . . . . . . xxix 5 Signal d'induction correspondant à l'angle de retournement des spins du proton 180 o lorsque l'unité de contrôle d'amortissement du rayonnement est allumée et éteinte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx 1.1 An ideal (L,R,C) detection circuit, V s (t) is the electromotive force induced in the circuit by a rotating transverse magnetization and V RF (t) is the voltage generated by the radio frequency generator. . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Visualization of the action of the radiation damping field, B rd on the magnetization in the rotating frame (B rd = B 1 in the absence of an applied radio frequency field). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 ix x Simulation of equation 1.31 for τ RD = 0.01s and ϕ f lip = 179.9 o . . . . . . . . . . Radiation damping in the presence of transverse relaxation ([49] 1.4(a), 1.4(b) Transverse and longitudinal magnetization for different initial flip angle ( τ ∞ RD τ RD = 1.0002). 1.4(c), 1.4(d) Transverse and longitudinal magnetization for different τ ∞ RD τ RD (cos(θ(t = 0)) = 179.999 o ). As τ ∞ RD τ RD approaches unity the maser pulse becomes broader therefore, the coherence persist for longer times and the longitudinal magnetization remains close to M o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Overhauser effect: Energy diagram of electron-nucleus system. Γ o and Γ 2 are zero and double quantum cross relaxation rates. . . . . . . . . . . . . . . . . . . . . 1.6 Solid effect: Energy diagram of electron-nucleus system. Saturating double quantum transition with microwave leading to a positive enhancement. . . . . . . . . . .2.1 DNP-NMR maser at 1.2K (from Ref.[14]).

3 . 6 67 3. 7 2 . 67 3. 8 71 3. 9 72 3. 10 73 xiv 3 . 11 75 3. 12 78 4. 1 85 4. 2 86 4. 3 87 4. 4 89 4. 6 92 4. 8 92 4. 9 F. 2 F. 3 FF. 5 FF. 6

 36677267871972107331175127818528638748969289292356 Figure 2.1(b) shows multiple maser pulses during the first 100 ms of the acquisition signal; 2.1(c) the detailed structure of the maser bursts shows the asymmetric nature of the pulses and the presence of multiple bursts within the first maser pulse. . . . . . . . . . . . . . . . . . . 2.2 2.2(a) Noise triggered maser when microwave was turned off. 2.2(b) The induction signal with non zero transverse magnetization at end of acquisition, unconventional in the case of solid state sample. 2.2(c) shows the maser burst in the first 100ms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Schematics of Bruker 6.7T polarizer (reproduced from Ref. [17]) (Figure from the thesis: Application and development of dynamic nuclear polarization with dissolution (D-DNP) methods to study various aspects of cellular metabolism, David Guarin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Overview of the different elements connected to the probe. 1: Microwave Source (ELVA), 2: Microwave doubler, 3: waveguide, 4: coaxial, 5: Tuning and matching box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 2.5(a) Sample Cup and 2.5(b) sample holder . . . . . . . . . . . . . . . . . . . . xi 2.6 2.6(a) Reflected power vs. frequency before polarizing and 2.6(b) Reflected power vs. frequency after polarizing. 2.6(c) and 2.6(d) show the impedance of a LC parallel resonator (ω LC = 285 MHz), respectively uncoupled and coupled to the nuclear spins (see appendix D.1 for simulation details). . . . . . . . . . . . . . . 2.7 NMR spectra for different polarization buildup times. 2.7(a) Red (spectra is scaled × 6.5), buildup time = 16.6 min and blue, buildup time = 29.2 min . 2.7(b) Red, buildup time = 29.2 min (spectra is scaled × 16.62) and blue, buildup time = 43.752 min. The distorted spectra in figure 2.7(b) is the manifestation of the radiation damping when the sample are initially negatively polarized (line narrowing) and of the magnetization-dependent shift of the Larmor frequency due to the dipolar field, which becomes significant when the polarization is large [19, 20] 2.8 Line narrowing and line shifts upon DNP buildup. Left: the resonance line widths are respectively 16100 Hz, 11300 Hz, 5700 Hz and 4330 Hz from bottom to top. Right: experimental first moments of the line M 1 vs line intensity. Filled symbols denote values extracted from experimental spectra; the straight line indicates the linear fit of the data (see text for details) . . . . . . . . . . . . . . . . . . . . . 2.9 Lorentzian fit of an experimental spectra . . . . . . . . . . . . . . . . . . . . . . 2.10 Pulse sequence used for pulsed maser experiment. The maser is initiated by a single pulse before the acquisition starts (see text for details). . . . . . . . . . . 2.11 Sustained maser upon continuous µw irradiation. 2.11(a),2.11(b) and 2.11(c) are the 1 st ,2 nd and 19 th , 20 s extracts of a single 200 × 20s induction signal collected as a pseudo-2D experiment. A single pulse was applied at the beginning of the experiment. For each extract of the acquisition signal shown here, time is labelled from 0 to 20 s. Most of the FIDs show an evenly spaced succession of maser bursts ( 2.11(a),2.11(b)). However, on few instances, spurious maser pulses were also observed (2.11(c)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 The induction signal of Fig. 2.11(a) remains significantly above the noise level between consecutive maser pulses, indicating the persistence of magnetization coherence throughout the experiment. . . . . . . . . . . . . . . . . . . . . . . . xii 2.13 2.13(a) and 2.13(b) are the first and second maser bursts of the experiment depicted in Figure 2.11(a). In 2.13(a), the asymmetry of the maser is apparent. In 2.13(c): the Fourier transform of the decaying part of the maser in 2.13(b) shows a line width reduction to ∼ 10 Hz . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 Resonance frequency shifts due to the dipolar field -Extracts from the first (blue), and three subsequent maser bursts of the induction signal depicted in 2.11(a) are shown. Superposition of the signals clearly shows that during the same time interval, the first signal exhibits 7 periods and the subsequent ones only 6 (the first maser has been scaled down for visualization). . . . . . . . . . . . . . . . . 2.15 Maser "Square" chirped pulse observed in a DNP-NMR maser experiments. In (2.15(a)): a ∼ 18 ms induction signal with practically constant intensity and abrupt decay to zero. The increasing period of oscillation is apparent on the induction signal; (2.15(b)): time-dependent Fourier transform analysis of the induction signal in (2.15(a)). The time-frequency analysis was performed using the STFT (Short Time Fourier Transform) method [29]. The spectrogram shown was obtained through the Scipy implementation of the STFT method (scipy.signal.spectrogram). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Ellipsoidal Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Pictorial representation of the Provotorov equations 3.11. The symbols eZ, ee, 1 H and 2 H indicate the electron Zeeman, electron dipole-dipole, and both nuclear Zeeman reservoirs. The protons are coupled to the LCR circuit of the probe. . 3.3 A 2D diagram of 3D model used in the simulation. Below, a unit cell consist of ellipsoidal sample inside the dark square (21 × 21 × 7 grid points for disk and 11 × 11 × 21 grid points for elongated cylinder) and zero padding at outside. Top, Periodic repetition of the unit cell, periodicity induced by the fast Fourier transform. (adapted from [3]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Thin disk: 21 × 21 × 7 grid points . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Elongated cylinder: 11 × 11 × 21 grid points . . . . . . . . . . . . . . . . . . . xiii Transverse magnetization for different zero padding values, the paddings were inserted xy, yz and zx faces with directions +x ˆ, +y ˆand +z ˆof the 3D lattice. 3.6(a) Zero padding for disk lattice. 3.6(b) Zero padding for elongated cylinder. . . . . . . . . . . . . . . Integration flow chart for Bloch-Maxwell-Provotrov equations. For any given time t, B ⃗ rd , B ⃗ dip and M th oz is computed from M ⃗ (r ⃗, t) and the values are given to the BMP equation and solved to give M ⃗ (r ⃗, t + dt). In figure a = 1-3(k ˆ•z ˆ)2 (adapted from [4]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulations of the effect of the DDF on the magnetization dynamics for Model (B). The envelope of the transverse magnetization is depicted in red and the z component in blue. The strength of the dipolar field, measured by the frequency ω dip is controlled by the magnetization density M o . 3.8(a), 3.8(b) γ rd /γ 2 = 3.10, ω dip /γ rd = 44.45, M o = 400.0 A/m. 3.8(c), 3.8(d) γ rd /γ 2 = 3.10, ω dip /γ rd = 22.22, M o = 200.0 A/m 3.8(e) 3.8(f) γ rd /γ 2 = 3.10, and the distant dipolar interactions are absent. All simulations were performed with fixed values of the radiation damping time constant γ rd = 2.0k Hz, decay rate γ 2 = 650 Hz, and electron-nuclei equilibration rate γ n,ee = 20.0 Hz. Simulations were performed on a disk-shaped sample. . . . . . . . . . . . . . . . . . Simulations of the magnetization dynamics for different values of the relaxation rate γ 2 and ratio γ rd /γ 2 (Model B). 3.9(a), 3.9(b) γ rd /γ 2 = 3.10 and γ 2 = 650 Hz; 3.9(c), 3.9(d) γ rd /γ 2 = 2.0 and γ 2 = 1.0 kHz Simulations were performed for a disk-shaped sample and fixed values of the radiation damping time constant γ rd = 2.0 kHz, γ n,ee = 20.0 kHz, M o = 400.0A/m and for a ratio ω dip /γ rd = 44.45. The envelope of the transverse magnetization is depicted in red and the z component in blue. . . . . . . . . . . . . Magnetization dynamics for DNP model (A) The envelope of the transverse magnetization is depicted in red and the z component in blue. 3.10(a) 3.10(b) γ rd /γ 2 = 3.10, ω dip /γ rd = 22.22 ,M o = 200.0A/m, γ 2 = 650Hz, γ rd = 2.0 kHz, γ n,ee = 20.0 Hz and γ st = 0.030 Hz. Calculations were performed for a disk-shaped sample. . . . . . . . . Average and standard deviations of the dipolar field (z component). Simulations were performed for model B, with fixed values of the radiation damping time constants γ rd = 2.0 kHz and γ n,ee = 20.0 Hz. Simulations were performed on a cylindrical-shaped sample. In 3.11(a), the envelope of the transverse magnetization is depicted in red and the z component in blue and the average ⟨ω d,z ⟩ in orange. Simulations were performed for γ rd /γ 2 = 3.10, ω dip /γ rd = 22.22, γ 2 = 650Hz. In 3.11(b) the average ⟨ω d,z ⟩ (orange), the standard deviation of ω d,z (green), and the transverse magnetization are shown (the latter in arbitrary units). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dependence of a single maser time duration on the γ ∞ rd /γ 2 ratio, for a fixed flip angle θ = 179.999 o . The curves were obtained from Ref. [22] with γ 2 = 25 kHz. For ratios approaching unity, the duration increases roughly a tenfold. . . . . . . . . . . . . . . 77 3.13 Magnetization dynamics for DNP model (A) computed with γ 2 = 650Hz, γ rd = 2.0kHz (γ rd /γ 2 = 3.10), γ n,ee = 151.5Hz and γ st = 1.0 Hz, for 3.13(a) 3.13(b) an elongated cylinder with ω dip /γ rd = 11.11; and 3.13(c) 3.13(d) a disk-shaped sample ω dip /γ rd = 22.22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detailed schematics of radiation damping control unit . . . . . . . . . . . . . . Radiation damping control unit installed on 300MHz spectrometer at École Polytechnique for preliminary test before implementing on 400 MHz spectrometer at ENS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connection of the directional coupler to the NMR probe. . . . . . . . . . . . . . Directional Coupler (EME Precision-UHF-VHF): Ports 1: 1 H channel; port 2: to radiofrequency input of pre-amplifier; port 3: radiofrequency output of the radiation damping control unit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 xv 4.5 4.5(a) Radio frequency input of Receiver unit (RXAD/2 1000, port:RF_IN J1). 4.5(b) Directional coupler (Pulsar directional coupler C5-08-411) and low noise amplifier(Mini-circuits, ZFL_1000LN+) inside a aluminium box to prevent RF interference, Port 1 connect RF output of pre-amplifier and input of directional coupler, Port 2 connect output of directional coupler and RF_IN J1 of receiver unit, and Port 3 connect output of low noise amplifier to input of radiation damping control unit. 4.5(c) Source of local oscillator. Port 1: local oscillator (BLAX1000 E). Port 2: Output of proton power amplifier. . . . . . . . . . . . . Port 1: Radio frequency (RF) input of radiation damping control unit (RDCU), Port 3 is connected to port 3 of 4.5(b) of stage 1, Directional coupler (Pulsar directional coupler C5-08-411) between port 1 and 3 is used to monitor input signal, Port 4: Local oscillator input of RDCU, Port 5 is connected to source of local oscillator (RF signal generator or SGU of the console) 10 dB attenuator between port 5 and local oscillator not shown, Port 6: monitoring port, Potentiometer 7 and 8 are used to reduce the dc (direct current) offset at the output of low-pass filters located at in-phase and quadrature channel inside the unit in the absence of RF input, Port 9: RF output of RDCU, Noise gate (10) is used reduce the low amplitude noise and Port 11 going to the next stage (Stage 3) 91 4.7 4.7(a) Low power amplifier. 4.7(b) Variable attenuator . . . . . . . . . . . . . . Graphical display of pulse program, zgcz_trig . . . . . . . . . . . . . . . . . . . Induction signal corresponding to proton spins flip angle 180 o when radiation damping control unit is on and off. Red: RDCU Off, Green: RDCU enhanced the radiation damping, Blue: RDCU suppressed the radiation damping . . . . . 93 4.10 Induction signal corresponding to proton spins flip angle 90 o when radiation damping control unit is on and off. Red: RDCU Off, Green: RDCU enhanced the radiation damping, Blue: RDCU suppressed the radiation damping . . . . . . . 94 4.11 Proton resonance spectrum for flip angle 90 o when radiation damping control unit is on and off. Red: RDCU Off, Green: RDCU enhanced the radiation damping, Blue: RDCU suppressed the radiation damping . . . . . . . . . . . . . . . . . . 95 xvi 4.12 Proton spectrum for different pulse durations, 0-40 µs for radiation damping control unit, on (blue, suppression) and off (red) . . . . . . . . . . . . . . . . . . . 4.13 Maser observed in 400MHz spectrometer with radiation damping control unit. . 4.14 Maser for different strength of the feedback field. The gain of the radiation damping control unit was varied between 4-5.5 dB . . . . . . . . . . . . . . . . . . . . 4.15 Magnetic resonance spectrum of mixture of H 2 O and CH 3 OH. The leftmost peak on the corresponds to protons of H 2 O (offset frequency = 1846.84 Hz) and the peak on the right corresponds to protons of CH 3 of methanol offset frequency = 1276.51 Hz. The spectrum was recorded when the radiation damping control unit was off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 Radiation damping enhanced induction signal from water and methanol using RDCU. Orange and blue corresponds to the case where the RDCU was off/on. Orange signal: single mode maser corresponds to the induction signal from proton of methanol (CH 3 ). Blue signal: two mode maser, with a high frequency (f1) and low frequency (f2) pulses corresponding to water and methanol respectively. . . . D.1 D.1(a) and D.1(b) shows impedance of a LC parallel resonator (ω LC = 285 MHz) uncoupled and coupled to the nuclear spins respectively. . . . . . . . . . . . . . D.2 Simulation of 'Wobb' curve, when there is coupling between resonator and nuclear spins. D.2(a) Experimental and D.2(b) Simulation . . . . . . . . . . . . . . . . F.1 F.1(a) Bruker Polarizer, 6.7 T. Bruker Spectrometer 18.8 T . . . . . . . . . . . Parallel resonant circuit with series matching. F.2(a) Schematics of probe. F.2(b) Fine tuning and matching variable capacitors . . . . . . . . . . . . . . . . . . . .3(a) Bruker solid state probe. F.3(b) Bruker liquid state probe. F.3(c) Bruker D-DNP probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.4 D-DNP resonator. F.4(a) Fixed resonator ω LCt ≈ 285 MHz. F.4(b) Resonator with coupling coils F.4(c) Resonator removed . . . . . . . . . . . . . . . . . . . xvii .5(a) Bruker acquisition system (Bruker AQS/3+), DRU-E: Digital reciever unit, RXAD/2 1000: Radio frequency receiver unit, REF/3 1200: Reference generator, SGU/3 1200: Radio frequency generator. F.5(b) Bruker power amplifier . . . . Bruker fixed gain low noise pre-amplifier. F.6(a) Front, radio frequency input from probe. F.6(b) Back, radio frequency output to receiver unit. . . . . . . . . G.1 Graphical representation of pulse program zgcw_trig . . . . . . . . . . . . . . . G.2 Turn off radiation damping control unit after 40 ms of acquisition. Input and output of RDCU is shown in green and red respectively. . . . . . . . . . . . . . . G.3 Graphical representation of pulse program zgadc_trig . . . . . . . . . . . . . . G.4 Radio frequency output of radiation damping control unit in the absence input signal. G.4(a) Before optimization of the dc offset. G.4(b) After reducing the dc offset to less than 2 mV. G.4(c) Potentiometers for adjusting the dc offset of the unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.5 Graphical representation of pulse program zgcw_cal . . . . . . . . . . . . . . . . G.6 Graphical representation of pulse program DNP_buildup . . . . . . . . . . . . . G.7 Graphical representation of pulse program SolidState_maser . . . . . . . . . . . H.1 Directional Coupler (EME Precision-UHF-VHF) . . . . . . . . . . . . . . . . . . I.1 I.1(a) Local oscillator frequency equals . I.1(b) Local oscillator detuned from proton Larmor frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2 I.2(a) Fine phase control. I.2(b) Coarse phase control. . . . . . . . . . . . . . . I.3 Phase difference between input (red) and output (green) of radiation damping control unit for different coarse steps. I.3(a) Step 1: 0.0 o , I.3(b) Step 2: 90.0 o ,I.3(a) Step 3: -90.0 o and I.3(a) Step 4: 180.0 o . . . . . . . . . . . . . . . . . . . . . . I.4 Fine phase control, phase difference between input and output of radiation damping control unit as function of voltage . . . . . . . . . . . . . . . . . . . . . . . xviii I.5 Saturation of output of the radiation damping control unit. I.5(a) RDCU input, I.5(b) RDCU output with out attenuator in the input, I.5(c) RDCU output with 20 dB attenuator in the input and I.5(d) RDCU output with 30 dB attenuator in the input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.6 Radio frequency leakage. I.6(a) Local oscillator frequency, O1 = 1919.0 Hz. I.6(a) Local oscillator frequency, O1 = 2000.0 Hz. . . . . . . . . . . . . . . . . . . . . K.1 Q-factor measurement setup: Spectrum analyser (RIGOL DSA815), radio frequency splitter (ZFSC-2-4-S+), external tuning -matching box and D-DNP probe K.2 Spectrum corresponding to a tuned and matched probe . . . . . . . . . . . . . .

  This is unusual in solid state NMR, where dipolar interactions between nuclear spins essentially result in rapid decoherence, and thus in significant line broadening with line widths of several tens of kHz. In contrast, possible effects of distant, much weaker dipole interactions require the persistence of coherent magnetization on a much longer time scale. The effects of the dipole field have been studied in the particular context of solid 3He in a seminal paper by Deville et al. In typical samples the short decoherence times prevent the development of dynamic remote dipole field (DDF) effects, and to our knowledge, no study of the combined collective effects of radiation damping and remote dipole field has been carried out in this context. In this work, we explore the dynamics of a magnetization subjected to these combined effects in the context of DNP hyperpolarized spins at liquid helium temperature. We present observations illustrating the presence of the dipole field for long times and show that the details of the observed maser pulses can be related to the combined effect of the dipole field and radiation damping.Les expériences de polarisation nucléaire dynamique (DNP) à des températures cryogéniques peuvent conduire à des polarisations extrêmement élevées qui donnent lieu à des effets non linéaires qui ne se produisent pas dans la RMN conventionnelle à l'état solide. Pour des polarisations aussi élevées, le couplage de la grande aimantation avec le circuit de détection devient extrêmement intense et donne lieu à un amortissement par rayonnement cohérent (ou "radiation damping" (RD) en Anglais), qui tend à ramener l'aimantation vers sa direction d'équilibre. Dans certaines expériences DNP, lorsque les spins sont polarisés négativement, cela peut conduire à des impulsions M/RASER entretenues. Nous avons récemment observé de telles impulsions apparaissant pour des temps courts (100 ms) suivies après leur disparition par un signal persistant pendant plusieurs dizaines de secondes dans des échantillons DNP-hyperpolarisés deutérés ; mais aussi à type de séries d'impulsions maser séparées de plusieurs secondes et d'intensités quasi-constantes.. Ce comportement peut être attribué à la combinaison de deux mécanismes concurrents, à savoir la perte de l'aimantation nucléaire du proton par "radiation damping" et la repolarisation des spins du proton à partir, soit des spins du deutérium dans l'échantillon par l'intermédiaire de leurs interactions avec les spins électroniques, soit des seuls spins électroniques, xxii l'absence de deutérieum dans l'échantillon. Ces observations peuvent être qualitativement interprétées à l'aide des équations de Bloch-Maxwell pour le "radiation damping" couplées aux équations de Provotrov (équations BMP) pour le modèle de mélange thermique (TM) de la polarisation nucléaire dynamique (PND) qui rendent compte des flux de températures de spin entre les réservoirs Zeeman nucléaires et non-Zeeman électroniques. Cependant, ce modèle ne parvient pas à expliquer certains détails des observations. En particulier, dans les expériences réalisées précédemment, les impulsions maser présentent un profil asymétrique. De plus, on observe des sortes d'«échos» impulsions maser les plus intenses, qui ne peuvent être reproduits par les équations BMP et exigent une description plus complète du système de spin. Dans ce contexte, l'intensité de l'aimantation de ces échantillons fortement polarisés s'accompagne de champs dipolaires importants qui contribuent à la dynamique de l'aimantation. Ceci est inhabituel en RMN du solide, où les interactions dipolaires entre spins nucléaires ont essentiellement pour effet qu'une décohérence rapide, donc un élargissement de raie important avec des largeurs de raie de plusieurs dizaines de kHz. En revanche, des effets possibles d'interactions dipolaires distantes, beaucoup plus faibles, nécessitent la persistance d'une aimantation cohérente sur une échelle de temps beaucoup plus longue. Les effets du champ dipolaire ont été étudiés dans le contexte particulier de l'3He solide dans un article fondateur de Deville et al. Dans des échantillons typiques les temps de décohérence courts empêchent le développement d'effets dynamiques du champ dipolaire distant (DDF), et à notre connaissance, aucune étude des effets collectifs combinés du "radiation damping" et du champ dipolaire distant n'a été étudiée dans ce contexte. Nous explorons dans ce travail la dynamique d'une aimantation soumise à ces effets combinés dans le contexte de spins hyperpolarisés par DNP à la température de l'hélium liquide. Nous présentons des observations illustrant la présence du champ dipolaire pour pour des temps longs et montrons que les détails des impulsions maser observées peuvent être reliées à l'effet combiné du champ dipolaire et du "radiation damping".

  efficace. Nous avons pu voir le décalage de la raie de résonance et son affinement lors d'une expérience de polarisation par DNP (figure2). Ce décalage de fréquence, proportionnel à la composante longitudinale de l'aimantation et provoqué par la présence d'un fort champ dipolaire xxv

Figure 1 :

 1 Figure 1: Maser entretenu lors d'une irradiation µw continue. 1(a) et 1(b) sont les 1 st et 2 nd , extraits de 20 s d'un seul signal d'induction de 200 × 20 s. Une impulsion unique a été appliquée au début de l'expérience.

Figure 2 :

 2 Figure 2: Largeur de raie et déplacement du signal de résonance au cours d'une expérience de polarisation. A gauche : les largeurs de raie sont respectivement 16100 Hz, 11300 Hz, 5700 Hz et 4330 Hz de bas en haut. À droite: premiers instants expérimentaux de la raie M 1 en fonction de l'intensité de la raie. Les symboles pleins indiquent les valeurs extraites des spectres expérimentaux; la ligne droite indique l'ajustement linéaire des données.

Figure 3 : 45 xxixFigure 4 :

 3454 Figure 3: Simulations de la dynamique d'aimantation pour le cas A (dimension du réseau est 21 × 21 × 7). L'enveloppe de l'aimantation transversale est figurée en rouge et la composante z en bleu. γ rd /γ 2 = 3, 10 et γ 2 = 650 Hz ; ω dip /γ rd = 44, 45 , m o = 400, 0 A/m ; γ rd = 2, 0 kHz, de γ n,ee = 20, 0 kHz ; ω dip /γ rd = 44, 45

γ g-factor m Electron - 1 .

 1 7608 × 10 11 rads -1

  h 2 γB o and h 2 γB o respectively. The energy difference between states |α⟩ and |β⟩ is given by ∆E = hγB o = hω o , where ω o = -γB o is the Larmor frequency (in angular units). From Equations 1.4 and 1.7 we can rewrite the population of states as:

  in Stanford extended the work of Rabi independently. Purcell observed the absorption of a continuous radio-frequency wave by protons in a sample of solid paraffin in a 0.71 T (30 MHz) static field orthogonal to the applied oscillating magnetic field. The power of radiofrequency used was weak in order to prevent saturation of the protons. Bloch[11] placed water at room temperature in a B o field of 0.1826 T (7.77 MHz) , and positioned two radio frequency coils for transmission of a B 1 x ˆfield, in z-y plane and receiver in z-x plane to receive the weak induction signal due to the precession of the nuclear spins around B o z ˆfield. The same year Bloch[12] came up with the principle of continuous wave nuclear magnetic resonance. Bloch modeled the spin system as ensemble of classical non-interacting dipoles with an angular momentum. The dynamics of a single classical dipole placed in an external magnetic field B ⃗ thus obeys the classical motion equation of angular momentum:

Figure 1 . 1 :

 11 Figure 1.1: An ideal (L,R,C) detection circuit, V s (t) is the electromotive force induced in the circuit by a rotating transverse magnetization and V RF (t) is the voltage generated by the radio frequency generator.

Figure 1 . 2 :

 12 Figure 1.2: Visualization of the action of the radiation damping field, B rd on the magnetization in the rotating frame (B rd = B 1 in the absence of an applied radio frequency field).

Figure 1 . 3 :

 13 Figure 1.3: Simulation of equation 1.31 for τ RD = 0.01s and ϕ f lip = 179.9 o

  Such an effect was observed by Hwang et. al. [48]. In the absence of detuning, the circuit impedance |Z(ω o )| and phase Ψ(ω o ) become ω o L/Q and 0 respectively. Thus, the radiation damping rate simplifies to ηQγµ o M o /2, and equation 1.30 can be rewritten as: ω x,RD = ζM avg y and ω y,RD = -ζM avg x (1.35)

Figure 1 . 4 :

 14 Figure 1.4: Radiation damping in the presence of transverse relaxation ([49] 1.4(a), 1.4(b) Transverse and longitudinal magnetization for different initial flip angle ( τ ∞ RD τ RD = 1.0002). 1.4(c), 1.4(d) Transverse and longitudinal magnetization for different τ ∞ RD τ RD (cos(θ(t = 0)) = 179.999 o ). As τ ∞ RD τ RD approaches unity the maser pulse becomes broader therefore, the coherence persist for longer times and the longitudinal magnetization remains close to M o .

Figure 1 . 5 :

 15 Figure 1.5: Overhauser effect: Energy diagram of electron-nucleus system. Γ o and Γ 2 are zero and double quantum cross relaxation rates.

4 ⟩

 4 , Unlike the Overhauser effect the solid effect requires saturation of forbidden transition (double or zero quantum). A microwave irradiation off resonance with respect to ω o,e , either at ω M W = ω o,e -ω e,n , inducing transitions between the perturbed states of the hamiltonian of Equation 1.45 |Ψ ′ 2 ⟩ ↔ |Ψ ′ 3 ⟩; or at ω M W = ω o,e + ω e,n , for transitions |Ψ ′ 1 ⟩ ↔ |Ψ ′ leading to positive or negative enhancement, as shown in figure 1.6. These perturbed eigenstates are |Ψ

  ) is met. Saturating one of either electron Larmor frequencies, ω o,e1 and ω o,e2 , (ω o,e1 > ω o,e2 ) induces triple spin flip-flop-flip between two electron spins and the nuclear spin |α e1 ⟩ |β e2 ⟩ |α n ⟩ → |β e1 ⟩ |α e2 ⟩ |β n ⟩ or |β e1 ⟩ |α e2 ⟩ |β n ⟩ → |α e1 ⟩ |β e2 ⟩ |α n ⟩, to create respectively negative or positive

2 DNP

 2 the DNP theory thermal mixing[START_REF] Borghini | Spin-temperature model of nuclear dynamic polarization using free radicals[END_REF][START_REF] Abragam | Chapter viii dynamic polarization of nuclear targets[END_REF][START_REF] Abragam | Principles of dynamic nuclear polarisation[END_REF]. Irradiating the system with microwave at frequency ω O,e1 can saturates a part of EPR spectrum which can be distributed across the spectrum by fast electronic spectral diffusion. If the nuclear Larmor frequency ω O,e1 falls in this saturated range, ω o,n = ω o,e2 -ω o,e1 , electron spins can transfer energy to nuclei via triple spin flip. This process is efficient when δ epr > ω o,n , since the homogeneous width depends on the mutual dipolar interaction among the electron spins. Thus thermal mixing mechanism is stronger when the concentration of paramagnetic impurity is large. As a result of the Provotorov theory, it is possible to describe the time evolution of the inverse spin temperatures of the different nuclear Zeeman and electron-electron dipole energy reservoirs as a system of kinetic equations with rates characterizing the temperature transfers between the various reservoirs. These will be presented in Chapter 3.Chapter NMR maser: experimental observations NMR masers have been observed in various contexts in the past decades. Such manifestations of radiation damping have been reported in hyperpolarized gas ([1, 2]

Figure 2 . 1 :

 21 Figure 2.1: DNP-NMR maser at 1.2K (from Ref.[14]).Figure 2.1(b) shows multiple maser pulses during the first 100 ms of the acquisition signal; 2.1(c) the detailed structure of the maser bursts shows the asymmetric nature of the pulses and the presence of multiple bursts within the first maser pulse.

Figure 2 .

 2 Figure 2.1: DNP-NMR maser at 1.2K (from Ref.[14]).Figure 2.1(b) shows multiple maser pulses during the first 100 ms of the acquisition signal; 2.1(c) the detailed structure of the maser bursts shows the asymmetric nature of the pulses and the presence of multiple bursts within the first maser pulse.

Figure 2 . 2 : 1 :Figure 2 . 1 .

 22121 Figure 2.2: 2.2(a) Noise triggered maser when microwave was turned off. 2.2(b) The induction signal with non zero transverse magnetization at end of acquisition, unconventional in the case of solid state sample. 2.2(c) shows the maser burst in the first 100ms

(

  asymmetric time envelopes, bursts within burst -see Figures 2.1(b) and 2.1(c)) that cannot be explained by the Bloch-Maxwell-Provotorov equations alone. Understanding these details of the DNP NMR maser is one of the purposes of this work. In this chapter, we describe the instrumentation used in this work and present experimental observations of the DNP NMR maser.

Figure 2 . 3 :

 23 Figure 2.3: Schematics of Bruker 6.7T polarizer (reproduced from Ref. [17]) (Figure from the thesis: Application and development of dynamic nuclear polarization with dissolution (D-DNP) methods to study various aspects of cellular metabolism, David Guarin.

Figure 2 . 4 :

 24 Figure 2.4: Overview of the different elements connected to the probe. 1: Microwave Source (ELVA), 2: Microwave doubler, 3: waveguide, 4: coaxial, 5: Tuning and matching box

Figure 2 . 5 :

 25 Figure 2.5: 2.5(a) Sample Cup and 2.5(b) sample holder

Figure 2

 2 Figure 2.6: 2.6(a) Reflected power vs. frequency before polarizing and 2.6(b) Reflected power vs. frequency after polarizing. 2.6(c) and 2.6(d) show the impedance of a LC parallel resonator (ω LC = 285 MHz), respectively uncoupled and coupled to the nuclear spins (see appendix D.1 for simulation details).

Figure 2 .Figure 2 . 7 :

 227 Figure 2.7: NMR spectra for different polarization buildup times. 2.7(a) Red (spectra is scaled × 6.5), buildup time = 16.6 min and blue, buildup time = 29.2 min . 2.7(b) Red, buildup time = 29.2 min (spectra is scaled × 16.62) and blue, buildup time = 43.752 min. The distorted spectra in figure 2.7(b) is the manifestation of the radiation damping when the sample are initially negatively polarized (line narrowing) and of the magnetization-dependent shift of theLarmor frequency due to the dipolar field, which becomes significant when the polarization is large[19, 20]

. 1 )Figure 2 . 8 :

 128 Figure 2.8: Line narrowing and line shifts upon DNP buildup. Left: the resonance line widths are respectively 16100 Hz, 11300 Hz, 5700 Hz and 4330 Hz from bottom to top. Right: experimental first moments of the line M 1 vs line intensity. Filled symbols denote values extracted from experimental spectra; the straight line indicates the linear fit of the data (see text for details)

Figure 2 .Figure 2 . 10 :

 2210 Figure 2.10 (see also pulse program SolidState_maser in Appendix: G). In order to save and

Figure 2 . 11 :

 211 Figure 2.11: Sustained maser upon continuous µw irradiation. 2.11(a),2.11(b) and 2.11(c) are the 1 st ,2 nd and 19 th , 20 s extracts of a single 200 × 20s induction signal collected as a pseudo-2D experiment. A single pulse was applied at the beginning of the experiment. For each extract of the acquisition signal shown here, time is labelled from 0 to 20 s. Most of the FIDs show an evenly spaced succession of maser bursts ( 2.11(a),2.11(b)). However, on few instances, spurious maser pulses were also observed (2.11(c)).

Figure 2 . 12 :

 212 Figure 2.12:The induction signal of Fig.2.11(a) remains significantly above the noise level between consecutive maser pulses, indicating the persistence of magnetization coherence throughout the experiment.

Figure 2 .

 2 Figure 2.13: 2.13(a) and 2.13(b) are the first and second maser bursts of the experiment depicted in Figure 2.11(a). In 2.13(a), the asymmetry of the maser is apparent. In 2.13(c): the Fourier transform of the decaying part of the maser in 2.13(b) shows a line width reduction to ∼ 10 Hz

Figure 2 .

 2 Figure 2.11(a), shows a shorter period during the first maser pulse than during the next ones.

Figure 2 . 14 :

 214 Figure 2.14: Resonance frequency shifts due to the dipolar field -Extracts from the first (blue), and three subsequent maser bursts of the induction signal depicted in 2.11(a) are shown. Superposition of the signals clearly shows that during the same time interval, the first signal exhibits 7 periods and the subsequent ones only 6 (the first maser has been scaled down for visualization).

Figure 2 . 15 : 2 . 1 .

 21521 Figure 2.15: Maser "Square" chirped pulse observed in a DNP-NMR maser experiments. In (2.15(a)): a ∼ 18 ms induction signal with practically constant intensity and abrupt decay to zero. The increasing period of oscillation is apparent on the induction signal; (2.15(b)): time-dependent Fourier transform analysis of the induction signal in (2.15(a)). The timefrequency analysis was performed using the STFT (Short Time Fourier Transform) method [29]. The spectrogram shown was obtained through the Scipy implementation of the STFT method (scipy.signal.spectrogram).

Figure 3 . 1 :

 31 Figure 3.1: Ellipsoidal Sample

Figure 3 . 2 :

 32 Figure 3.2: Pictorial representation of the Provotorov equations 3.11. The symbols eZ, ee, 1 H and 2 H indicate the electron Zeeman, electron dipole-dipole, and both nuclear Zeeman reservoirs. The protons are coupled to the LCR circuit of the probe.

Figure 3 . 4 :Figure 3 . 5 :

 3435 Figure 3.4: Thin disk: 21 × 21 × 7 grid points

Figure 3 . 6 :

 36 Figure 3.6: Transverse magnetization for different zero padding values, the paddings were inserted xy, yz and zx faces with directions +x ˆ, +y ˆand +z ˆof the 3D lattice. 3.6(a) Zero padding for disk lattice. 3.6(b) Zero padding for elongated cylinder.

Figure 3 . 7 :

 37 Figure 3.7: Integration flow chart for Bloch-Maxwell-Provotrov equations. For any given time t, B ⃗ rd , B ⃗ dip and M th oz is computed from M ⃗ (r ⃗, t) and the values are given to the BMP equation and solved to give M ⃗ (r ⃗, t + dt). In figure a = 1-3(k ˆ•z ˆ)2 2

Figure 3 . 8 :

 38 Figure 3.8: Simulations of the effect of the DDF on the magnetization dynamics for Model (B). The envelope of the transverse magnetization is depicted in red and the z component in blue. The strength of the dipolar field, measured by the frequency ω dip is controlled by the magnetization density M o . 3.8(a), 3.8(b) γ rd /γ 2 = 3.10, ω dip /γ rd = 44.45, M o = 400.0 A/m. 3.8(c), 3.8(d) γ rd /γ 2 = 3.10, ω dip /γ rd = 22.22, M o = 200.0 A/m 3.8(e) 3.8(f) γ rd /γ 2 = 3.10, and the distant dipolar interactions are absent. All simulations were performed with fixed values of the radiation damping time constant γ rd = 2.0k Hz, decay rate γ 2 = 650 Hz, and electron-nuclei equilibration rate γ n,ee = 20.0 Hz. Simulations were performed on a disk-shaped sample.

Figure 3 . 9 :

 39 Figure 3.9: Simulations of the magnetization dynamics for different values of the relaxation rate γ 2 and ratio γ rd /γ 2 (Model B). 3.9(a), 3.9(b) γ rd /γ 2 = 3.10 and γ 2 = 650 Hz; 3.9(c), 3.9(d) γ rd /γ 2 = 2.0 and γ 2 = 1.0 kHz Simulations were performed for a disk-shaped sample and fixed values of the radiation damping time constant γ rd = 2.0 kHz, γ n,ee = 20.0 kHz, M o = 400.0A/m and for a ratio ω dip /γ rd = 44.45. The envelope of the transverse magnetization is depicted in red and the z component in blue.

Figure 3 . 10 :

 310 Figure 3.10: Magnetization dynamics for DNP model (A) The envelope of the transverse magnetization is depicted in red and the z component in blue. 3.10(a) 3.10(b) γ rd /γ 2 = 3.10, ω dip /γ rd = 22.22 ,M o = 200.0A/m, γ 2 = 650Hz, γ rd = 2.0 kHz, γ n,ee = 20.0 Hz and γ st = 0.030 Hz. Calculations were performed for a disk-shaped sample.

  (a)) was similar to the observed one and depicted in Figures 2.11(a)-2.11(b), that is, a "steady state" of regularly spaced maser bursts. It is also noteworthy that in the absence of dipolar field, the model gives qualitatively the same predictions as the {3+1}-dimensional simplified Bloch-Maxwell-Provotrov's (BMP) model introduced in Reference[1]. This transition is illustrated in Figure 3.8. It is also interesting that we were not able to reproduce the kind of long term evolution either with the simplified BMP model, or with the present lattice model without DDF.

Figure 3 . 11 :

 311 Figure 3.11: Average and standard deviations of the dipolar field (z component). Simulations were performed for model B, with fixed values of the radiation damping time constants γ rd = 2.0 kHz and γ n,ee = 20.0 Hz. Simulations were performed on a cylindrical-shaped sample. In 3.11(a), the envelope of the transverse magnetization is depicted in red and the z component in blue and the average ⟨ω d,z ⟩ in orange. Simulations were performed for γ rd /γ 2 = 3.10, ω dip /γ rd = 22.22, γ 2 = 650Hz. In 3.11(b) the average ⟨ω d,z ⟩ (orange), the standard deviation of ω d,z (green), and the transverse magnetization are shown (the latter in arbitrary units).

Figure 3 . 13 :

 313 Figure 3.13: Magnetization dynamics for DNP model (A) computed with γ 2 = 650Hz, γ rd = 2.0kHz (γ rd /γ 2 = 3.10), γ n,ee = 151.5Hz and γ st = 1.0 Hz, for 3.13(a) 3.13(b) an elongated cylinder with ω dip /γ rd = 11.11; and 3.13(c) 3.13(d) a disk-shaped sample ω dip /γ rd = 22.22.

Figure 4 . 1 . 1 )

 411 Figure 4.1 shows a schematics of radiation damping control unit (RDCU). The radiation damping control unit is divided into four stages, described below and indicated by different colors in Figure 4.1.

Figure 4 . 1 :Figure 4 . 2 :

 4142 Figure 4.1: Detailed schematics of radiation damping control unit

Figure 4 . 3 :

 43 Figure 4.3: Connection of the directional coupler to the NMR probe.

4. 3 . 1

 31 Radio frequency signal pathway

Figure 4 .

 4 Figure 4.2 shows the radiation damping control unit installed in 400 MHz spectrometer. The entire radio frequency signal pathway is follows:

Figure 4 . 4 :

 44 Figure 4.4: Directional Coupler (EME Precision-UHF-VHF): Ports 1: 1 H channel; port 2: to radiofrequency input of pre-amplifier; port 3: radiofrequency output of the radiation damping control unit.

Figure 4

 4 Figure 4.5: 4.5(a) Radio frequency input of Receiver unit (RXAD/2 1000, port:RF_IN J1). 4.5(b) Directional coupler (Pulsar directional coupler C5-08-411) and low noise amplifier(Minicircuits, ZFL_1000LN+) inside a aluminium box to prevent RF interference, Port 1 connect RF output of pre-amplifier and input of directional coupler, Port 2 connect output of directional coupler and RF_IN J1 of receiver unit, and Port 3 connect output of low noise amplifier to input of radiation damping control unit. 4.5(c) Source of local oscillator. Port 1: local oscillator (BLAX1000 E). Port 2: Output of proton power amplifier.

Figure 4 . 6 :

 46 Figure 4.6: Port 1: Radio frequency (RF) input of radiation damping control unit (RDCU), Port 3 is connected to port 3 of 4.5(b) of stage 1, Directional coupler (Pulsar directional coupler C5-08-411) between port 1 and 3 is used to monitor input signal, Port 4: Local oscillator input of RDCU, Port 5 is connected to source of local oscillator (RF signal generator or SGU of the console) 10 dB attenuator between port 5 and local oscillator not shown, Port 6: monitoring port, Potentiometer 7 and 8 are used to reduce the dc (direct current) offset at the output of low-pass filters located at in-phase and quadrature channel inside the unit in the absence of RF input, Port 9: RF output of RDCU, Noise gate (10) is used reduce the low amplitude noise and Port 11 going to the next stage (Stage 3)

Figure 4 .

 4 Figure 4.7: 4.7(a) Low power amplifier. 4.7(b) Variable attenuator

Figure 4 . 8 :

 48 Figure 4.8: Graphical display of pulse program, zgcz_trig

Figure 4 . 9 : 1 :Figure 4 . 10 :

 491410 Figure 4.9: Induction signal corresponding to proton spins flip angle 180 o when radiation damping control unit is on and off. Red: RDCU Off, Green: RDCU enhanced the radiation damping, Blue: RDCU suppressed the radiation damping

Figure 4 .

 4 Figure 4.12 shows a set of proton spectra for pulse calibration, with increasing flip angles (pulse durations is increased from 0-40 µs) while radiation damping control was on (enhancement and suppression) and off. In figure4.12 it is evident that the effect of radiation damping field on nutation of magnetization. For flip angle less than 180 o the nutation of proton magnetization is slow down and for flip greater than 180 o is speed up by radiation damping. This is due to the fact that radiation damping field always rotate the magnetization towards the +z ˆ. When the radiation damping field is suppressed by the feedback field with phase -90 o , the proton spectrum for different flip angle was sinusoidal as expected (M o sin(ϕ f lip )), and the pulse length for 90 o and 180 o was found to be 12µs and 24µs respectively.

Figure 4 . 11 :

 411 Figure 4.11: Proton resonance spectrum for flip angle 90 o when radiation damping control unit is on and off. Red: RDCU Off, Green: RDCU enhanced the radiation damping, Blue: RDCU suppressed the radiation damping

Figure 4 . 12 :Figure 4 . 13 :

 412413 Figure 4.12: Proton spectrum for different pulse durations, 0-40 µs for radiation damping control unit, on (blue, suppression) and off (red)

4. 4

 4 Single Mode Liquid State Maser We were able to observe sustained maser pulses from the proton nuclear spin (400 µL D 2 O, 100 µL H 2 O and 10 µL 200 mM CuSO 4 ) in a 400MHz spectrometer by using the radiation damping control unit. We initiated the maser by by flipping the nuclear spins by approximately 180 0 from thermal equilibrium using the pulse program zgcw_trig (appendix G.1). When the phase and gain of the device were set to -156 o and 5 dB, we observed the sustained maser as show in the figure 4.13. The maser was recorded for 30 seconds showing regularly spaced pulses of width approximately 2 seconds, with same intensity except for the first. This phenomenon can be ascribed to two competing torques acting on the magnetization in opposite direction due to the radiation damping field and feedback field, pointing in the opposite direction and with greater intensity. The effect of strength of the feedback field on the maser was studied by varying the gain of the feedback unit, and the induction signals are shown in figure 4.14. It was observed that the width of the maser pulses decreased thus increasing the repetition frequency of pulses with the strength of the feedback field This illustrates the possibility to control the onset of the maser.

Figure 4 . 14 :

 414 Figure 4.14: Maser for different strength of the feedback field. The gain of the radiation damping control unit was varied between 4-5.5 dB

Figure 4 .

 4 15 shows the proton spectrum from 225µL H 2 O, 225 µL CH 3 OH and 50µL deuterated DMSO. The peaks at 1846.84 Hz (f1) and 1276.51 Hz (f2) corresponds to water and methanol (CH 3 ), respectively. In the absence of radiation damping control, one

Figure 4 . 15 :

 415 Figure 4.15: Magnetic resonance spectrum of mixture of H 2 O and CH 3 OH. The leftmost peak on the corresponds to protons of H 2 O (offset frequency = 1846.84 Hz) and the peak on the right corresponds to protons of CH 3 of methanol offset frequency = 1276.51 Hz. The spectrum was recorded when the radiation damping control unit was off

Figure 4 . 16 :

 416 Figure 4.16: Radiation damping enhanced induction signal from water and methanol using RDCU. Orange and blue corresponds to the case where the RDCU was off/on. Orange signal: single mode maser corresponds to the induction signal from proton of methanol (CH 3 ). Blue signal: two mode maser, with a high frequency (f1) and low frequency (f2) pulses corresponding to water and methanol respectively.

  From equation 1.8, we can write equlibrium magnetisation as function of temperature, (m ⃗ • r ˆ)r ˆ= (m x sin 2 (θ) cos 2 (ϕ) + m y sin 2 (θ) sin(ϕ) cos(ϕ) + m z cos(θ) sin(θ) cos(ϕ))x + (m x sin 2 (θ) cos(ϕ) sin(ϕ) + m y sin 2 (θ) sin 2 (ϕ) + m z cos(θ) sin(θ) sin(ϕ))y + (m x sin(θ) cos(θ) cos(ϕ) + m y sin(θ) cos(θ) sin(ϕ) + m z cos 2 (θ))y ˆ(B.4) where ϕ = ω o t. Now taking a time average on 3(m ⃗ • r ˆ)r ˆ-m ⃗ gives (< sin(ϕ) >=< cos(ϕ) >= 0 and < sin 2 (ϕ) >=< cos 2 (ϕ) >= 1/2):

C. 20 )Figure

 20 fig = plt . figure ( co ns tra in ed _la yo ut = True , figsize =(10 , 5) ) spec = fig . add_gridspec (1 , 1)

Figure D. 2 :

 2 Figure D.2: Simulation of 'Wobb' curve, when there is coupling between resonator and nuclear spins. D.2(a) Experimental and D.2(b) Simulation

  plt . figure ( no , figsize =(25 ,10) ) for (x ,y , style , labeling ) in zip ( xlist , ylist , stylelist , labellist ) : plt . plot (x ,y , style , linewidth =1.0 , markersize =5 , label = labeling ) plt . grid ( True ) plt . axhline (0 , color = ' black ' , linewidth =1) plt . axvline (0 , color = ' black ' , linewidth =1) plt . xticks ( weight = ' bold ' , size =20) plt . yticks ( weight = ' bold ' , size =20) # plt . xlim (0.0 ,0.0005) # plt . ylim ( -20 e -22 ,20 e -22) plt . legend ( fontsize =20 , frameon = False ) plt . xlabel ( xlabel , fontsize =20 , color = ' black ' , fontweight = ' bold ') plt . ylabel ( ylabel , fontsize =20 , color = ' black ' , fontweight = ' bold ') plt . title ( title , fontsize =20 , color = ' black ' , fontweight = ' bold ') permeability /3.0) * 2.0 * np . average ( M [2: nx * ny * nz *3:3]) else : Wdp [0: nx * ny * nz *3:3] = Wdp [0: nx * ny * nz *3:3] + ( -Gamma * permeability /6.0) * np . average ( -M [0: nx * ny * nz *3:3]) Wdp [1: nx * ny * nz *3:3] = Wdp [1: nx * ny * nz *3:3] + ( -Gamma * permeability /6.0) * np . average ( -M [1: nx * ny * nz *3:3]) Wdp [2: nx * ny * nz *3:3] = Wdp [2: nx * ny * nz *3:3] + ( -Gamma * permeability /6.0) * 2.0 *

"#

  save_file . write ( " \ nDirectory = " + foldername ) save_file . close () # section : lattice unit_no = nx * ny * nz unit_vol = Vol / unit_no spacing = (( Mo ) /200.0) **(1.0/3.0) # float ( sys . argv [1]) # unit_vol **(1.0/3.0) # Default 10.0 e -10 if mask : xp = np . linspace ( -( nx -1) /2 ,( nx -1) /2 , nx , endpoint = True ) yp = np . linspace ( -( ny -1) /2 ,( ny -1) /2 , ny , endpoint = True ) zp = np . linspace ( -( nz -1) /2 ,( nz -1) /2 , nz , endpoint = True ) xpr , ypr , zpr = np . meshgrid ( xp , yp , zp , sparse = False , indexing = ' ij ') xprc , yprc , zprc = np . meshgrid ( xp , yp , zp , sparse = False , indexing = ' ij ') if maskType == " Circle " : maskc = xpr **2/( a **2) + ypr **2/( b **2) + zpr **2/( c **2) <= 1 if maskType == " Cylinder " : maskc = xpr **2/( a **2) + ypr **2/( b **2) <= 1 if maskType == " Both " : maskc1 = xpr **2/( a **2) + ypr **2/( b **2) <= 1 maskc = np . full ( xprc . shape , False ) for i in range (400) : maskc2 = ( xpr -random . randrange ( -a , a ) ) **2/( r **2) + ( yprrandom . randrange ( -b , b ) ) **2/( r **2) + ( zpr -random . randrange ( -c , c ) ) **2. linspace ( -a * inhomo_width , a * inhomo_width , nz ) dWinho = Winho [1] -Winho [0] Winho_Gaussian = np . exp ( -Winho **2/(2* inhomo_width **2) ) Winho_Lorn = ( np . pi * inhomo_width /2) *( inhomo_width /(2* np . pi ) ) /(( Winho ) **2+( inhomo_width /2) **2) Winho_Dist1d = Winho_Lorn plt . figure (6 , figsize =(25 ,10) ) plt . plot ( Winho , Winho_Gaussian , 'o -' , linewidth =1.0 , markersize =5 , label = r " Gaussian " ) plt . plot ( Winho , Winho_Lorn , '* -' , linewidth =1.0 , markersize =5 , label = r " Lorentzian " ) # plt . grid ( True ) plt . axhline (0 , color = ' black ' , linewidth =1) plt . axvline (0 , color = ' black ' , linewidth =1) plt . xticks ( weight = ' bold ' , size =20) plt . yticks ( weight = ' bold ' , size =20) # plt . xlim (0.0 ,0.0005) # plt . ylim ( -20 e -22 ,20 e -22) plt . legend ( fontsize =20 , frameon = False ) plt . xlabel ( " Frequency " , fontsize =20 , color = ' black ' , fontweight = ' bold ') plt . ylabel ( " Distribution " , fontsize =20 , color = ' black ' , fontweight = ' bold ') plt . title ( " Bo Inhomogenety " , fontsize =20 , color = ' black ' , fontweight = ' bold ') if save : plt . savefig ( foldername + '/ Bo_inhomo . png ' , transparent = True , dpi =300 , bbox_inches = ' tight ') bold ') plt . title ( " Bo Inhomogenety " , fontsize =20 , color = ' black ' , fontweight = ' bold ') if save : plt . savefig ( foldername + '/ Bo_inhomo . png ' , transparent = True , dpi =300 , bbox_inches = ' tight ') # section : radiation damping Xi = Gamma * permeability * etaQ /2 RDrate = Xi * Mo A = np . sin ( Psi * np . pi /180.0) * ( Xi / Vol ) * unit_vol B = np . cos ( Psi * np . pi /180.0) * ( Xi / Vol ) * unit_vol # section : boltzman if boltzmann : theta = np . linspace (0 ,180.0 ,10000 , endpoint = True ) # Polar angle E = -Bo * mp * np . cos ( theta * np . pi /180.0) # Energy of magnetic dipole pro_theta = np . exp ( -E /( kb * T ) ) # Probabilty pro_theta = pro_theta / np . sum ( pro_theta ) # Probabilty normalized # print (" Sum of all probability =" , np . sum ( pro_theta ) ) Theta = np . random . choice ( theta , size =( nx , ny , nz ) , p = pro_theta ) # Polar angle Phi = np . random . uniform ( low =0.0 , high =360.0 , size =( nx , ny , nz ) ) # Azimuthal angle # Phi = 0.0 * np . ones (( nx , ny , nz ) ) plt . figure (1 , figsize =(25 ,10) ) plt . subplot (121) plt . plot ( theta , pro_theta , ' -. r ' , linewidth =3.0) plt . grid ( True ) plt . xticks ( weight = ' bold ' , size =20) plt . yticks ( weight = ' bold ' , size =20) plt . xlabel ( " Angle ( deg ) " , fontsize =20 , color = ' black ' , fontweight = ' bold ' ) plt . ylabel ( " Probabilty " , fontsize =20 , color = ' black ' , fontweight = ' bold ') plt . title ( " Boltzmann Distribution " , fontsize =20 , color = ' black ' , fontweight = ' bold ') plt . subplot (122) plt . plot ( theta , E /( kb * T ) , ' -. b ' , linewidth =3.0) plt . grid ( True ) plt . xticks ( weight = ' bold ' , size =20) plt . yticks ( weight = ' bold ' , size =20) plt . xlabel ( " Angle ( deg ) " , fontsize =20 , color = ' black ' , fontweight = ' bold ' ) plt . ylabel ( " E /( kb * T ) " , fontsize =20 , color = ' black ' , fontweight = ' bold ') plt . title ( " Boltzmann Distribution " , fontsize =20 , color = ' black ' , fontweight = ' bold ') plt . savefig ( foldername + '/ Boltmann . png ' , transparent = True , dpi =300 , bbox_inches = ' tight ') plt . figure (2 , figsize =(25 ,17) ) plt . subplot (211) plt . hist ( np . reshape ( Theta ,( nx * ny * nz ,) ) , bins = 180 , density = False ) plt . grid ( True ) plt . xticks ( weight = ' bold ' , size =20) plt . yticks ( weight = ' bold ' , size =20) plt . xlabel ( " Angle ( deg ) " , fontsize =20 , color = ' black ' , fontweight = ' bold ' ) plt . ylabel ( " Count " , fontsize =20 , color = ' black ' , fontweight = ' bold ') plt . title ( " Histogram Of Randomly generated Theta ( polar angle ) " , fontsize =20 , color = ' black ' , fontweight = ' bold ') plt . subplot (212) plt . hist ( np . reshape ( Phi ,( nx * ny * nz ,) ) , bins = 360 , density = False ) plt . grid ( True ) plt . xticks ( weight = ' bold ' , size =20) plt . yticks ( weight = ' bold ' , size =20) plt . xlabel ( " Angle ( deg ) " , fontsize =20 , color = ' black ' , fontweight = ' bold ' ) plt . ylabel ( " Count " , fontsize =20 , color = ' black ' , fontweight = ' bold ') plt . title ( " Histogram Of Randomly generated Phi ( azimuthal angle ) " , fontsize =20 , color = ' black ' , fontweight = ' bold ') plt . savefig ( foldername + '/ Boltmann_Angles . png ' , transparent = True , dpi =300 , bbox_inches = ' tight ') else : Theta = 0.0 * np . ones (( nx , ny , nz ) ) Phi = 0.0 * np . ones (( nx , ny , nz ) ) # section : initial distribution Mtemp = np . zeros (( nx , ny , nz ,3) ) M = np . zeros (( nx , ny , nz ,3) ) Mtemp [: ,: ,: ,0] = Mo * np . sin ( Theta * np . pi /180.0) * np . cos ( Phi * np . pi /180.0) * maskxyz Mtemp [: ,: ,: ,1] = Mo * np . sin ( Theta * np . pi /180.0) * np . sin ( Phi * np . pi /180.0) * maskxyz Mtemp [: ,: ,: ,2] = Mo * np . cos ( Theta * np . pi /180.0) * maskxyz M [: ,: ,: ,1] = Mtemp [: ,: ,: ,1] M [: ,: ,: ,1] = Mtemp [: ,: ,: ,1] * np . cos ( PhiFLIP * np . pi /180.0) -Mtemp [: ,: ,: ,2] * np . sin ( PhiFLIP * np . pi /180.0) M [: ,: ,: ,2] = Mtemp [: ,: ,: ,1] * np . sin ( PhiFLIP * np . pi /180.0) + Mtemp [: ,: ,: ,2] * np . cos ( PhiFLIP * np . pi /180.0) M = np . reshape (M ,( nx * ny * nz *3 ,) ) if True : M [0: nx * ny * nz *3:3] = M [0: nx * ny * nz *3:3] * Winho_Dist M [1: nx * ny * nz *3:3] = M [1: nx * ny * nz *3:3] * Winho_Dist M [2: nx * ny * nz *3:3] = M [2: nx * ny * nz *3:3] * Winho_Dist else : # For continuing simulation from previous simulation . Last M value end_time = time . time () timetaken = end_time -start_time print ( " Total time = % s seconds " % ( timetaken ) ) save_file = open ( foldername + " / info . txt " , " a " ) save_file . write ( " \ nTotal time = % f seconds " % timetaken ) . sum ( Mr [0:3* nx * ny * nz :3 ,:] , axis =0) My = np . sum ( Mr [1:3* nx * ny * nz :3 ,:] , axis =0) Mz = np . sum ( Mr [2:3* nx * ny * nz :3 ,:] , axis =0) Mabs = np . sqrt ( Mx **2 + My **2) np . savetxt ( foldername + '/ datat . txt ' ,t , fmt = ' %1.10 e ') np . savetxt ( foldername + '/ dataMx . txt ' ,Mx , fmt = ' %1.10 e ') np . savetxt ( foldername + '/ dataMy . txt ' ,My , fmt = ' %1.10 e ') np . savetxt ( foldername + '/ dataMz . txt ' ,Mz , fmt = ' %1.10 e ') np . savetxt ( foldername + '/ dataMxall . txt ' , Mr [0:3* nx * ny * nz :3 ,:] , fmt = ' %1.10 e ') np . savetxt ( foldername + '/ dataMyall . txt ' , Mr [1:3* nx * ny * nz :3 ,:] , fmt = ' %1.10 e ') np . savetxt ( foldername + '/ dataMzall . txt ' , Mr [2:3* nx * ny * nz :3 ,:] , fmt = ' %1.10 e ') # section : real data if real_data : dat = np . fromfile ( data_file , dtype = np . int32 ) t_r = np . linspace (0 , AcqTime_r , dat . shape [0]//2) t_r = np . linspace (0 , AcqTime_r , dat . shape [0]//2) Dt = t_r [1] -t_r [0] inter = interp1d (t , Mabs , kind = ' cubic ' , fill_value = " extrapolate " ) t_inter = np . arange (0 , tf , Dt ) x_r = dat [0::2] + 1 j * dat [1::2] Mx_r = x_r [0: t_inter . shape [0]]. real My_r = x_r [0: t_inter . shape [0]]. imag Ma_r = np . sqrt ( Mx_r **2 + My_r **2) # section : plotting sub_fun1 ( save , no =3 , xlist =[ t ,t , t ] , ylist =[ Mabs , Mx , My ] , stylelist =[ "o -" ," * " ," + " ] , labellist =[ " Mt " ," Mx " ," My " ] , xlabel = " Time " , ylabel = " $M_T$ " , title = " FID " , folder = foldername , name = " simu_data1 " ) sub_fun1 ( save , no =4 , xlist =[ t , t ] , ylist =[ Mabs , Mz ] , stylelist =[ "o -" ," . " ] , labellist =[ " Mt " ," Mz " ] , xlabel = " Time " , ylabel = " $M_T$ " , title = " FID " , folder = foldername , name = If you want to input parameter from terminal when you run prgram , like python BMP_simulation . py parameter_1 parameter_2 ... parameter_i , use int ( sys . argv [ i ]) for integer and float ( sys . argv [1]) for flot model_B_or_A = True # True model B and False Model A save = True # save into file boltzmann = False # True : Boltzmann distribution of protons ; False : We consider a bulk proton sample coarse -grained and all unit cells has same inital equlibrium magnetization inhomog = False # Bo Inhomogenety

Figure F. 2 :

 2 Figure F.1: F.1(a) Bruker Polarizer, 6.7 T. Bruker Spectrometer 18.8 T

Figure F. 3 :

 3 Figure F.3: F.3(a) Bruker solid state probe. F.3(b) Bruker liquid state probe. F.3(c) Bruker D-DNP probe .

Figure

  Figure F.4: D-DNP resonator. F.4(a) Fixed resonator ω LCt ≈ 285 MHz. F.4(b) Resonator with coupling coils F.4(c) Resonator removed

Figure G. 1 :Figure G. 3 :

 13 Figure G.1: Graphical representation of pulse program zgcw_trig

Figure G. 4 :

 4 Figure G.4: Radio frequency output of radiation damping control unit in the absence input signal. G.4(a) Before optimization of the dc offset. G.4(b) After reducing the dc offset to less than 2 mV. G.4(c) Potentiometers for adjusting the dc offset of the unit

Figure I. 3 :

 3 Figure I.3: Phase difference between input (red) and output (green) of radiation damping control unit for different coarse steps. I.3(a) Step 1: 0.0 o , I.3(b) Step 2: 90.0 o ,I.3(a) Step 3: -90.0 o and I.3(a) Step 4: 180.0 o

2 Figure K. 2 :

 22 figure K.2

  1.1 gyromagnetic ratio and g-factor of electron and proton . . . . . . . . . . . . . . 1.2 Proton polarization for different magnetic field and temperature . . . . . . . . . 2.1 Microwave frequency for positive and negative polarization . . . . . . . . . . . .

3.1 Values of k αα for different ellipsoidal shapes . . . . . . . . . . . . . . . . . . . . 4.1 Phase and gain calibration of radiation damping control unit . . . . . . . . . . . H.1 Directional Coupler: Power attenuation measurement done by vector network analyzer (Hewlett Packard 8711C) at 400.1818 MHz . . . . . . . . . . . . . . . . J.1 RDCU, phase modulator: Relation between potentiometer, potential difference and phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Abstract Dynamic nuclear polarization (DNP) experiments at cryogenic temperatures can lead to extremely high polarizations that give rise to nonlinear effects that do not occur in conventional solid-state NMR. At such high polarizations, the coupling of the large magnetization to the detection circuit becomes extremely intense and gives rise to radiation damping (RD), which tends

  De plus, on observe des sortes d'«échos» impulsions maser les plus intenses, qui ne peuvent être reproduits par les équations BMP et exigent une description plus complète du système de spin. Dans ce contexte, l'intensité de l'aimantation de ces échantillons fortement polarisés s'accompagne de champs dipolaires importants[8] qui contribuent à la dynamique de l'aimantation. Ceci est inhabituel en RMN du solide, où les interactions dipolaires entre spins nucléaires ont essentiellement pour effet qu'une décohérence rapide, donc un élargissement de raie important avec des largeurs de raie de plusieurs dizaines de kHz. En revanche, des effets possibles d'interactions dipolaires distantes, beaucoup plus faibles, nécessitent la persistance d'une aimantation cohérente sur une échelle de temps beaucoup plus longue. Les effets du champ

ticulier, dans les expériences réalisées précédemment, les impulsions maser présentent un profil xxiii xxiv asymétrique. dipolaire ont été étudiés dans le contexte particulier de l'3He solide dans un article fondateur de Deville et al [9]. Dans des échantillons typiques les temps de décohérence courts empêchent le développement d'effets dynamiques du champ dipolaire distant (DDF), et à notre connaissance, aucune étude des effets collectifs combinés du "radiation damping" et du champ dipolaire distant n'a été étudiée dans ce contexte. Nous explorons dans ce travail la dynamique d'une aimantation soumise à ces effets combinés dans le contexte de spins hyperpolarisés par DNP à la température de l'hélium liquide. Nous présentons des observations illustrant la présence du champ dipolaire pour pour des temps longs et montrons que les détails des impulsions maser observées peuvent être reliées à l'effet combiné du champ dipolaire et du "radiation damping".

  Des simulations numériques ont été effectuées sur un ensemble de N (= n x × n y × n z ) spins situés sur un réseau régulier de n x × n y × n z points. Chaque noeud du réseau correspond un "cluster" de spins d'aimantation M ⃗ i avec une vitesse de relaxation transversale, T 2 due à

	l'interaction dipolaire dans les clusters. Ceux-ci snt couplés par le champ dipolaire qui s'écrit
	dans le référentiel tournant:[8]

  Signal d'induction correspondant à l'angle de retournement des spins du proton 180 o lorsque l'unité de contrôle d'amortissement du rayonnement est allumée et éteinte rayonnement dans un échantillon de 90% H 2 O et 10% D 2 O comme le montre la figure 5 et avons pu créer un maser à deux modes avec cet appareil.

	xxx
	Figure 5:
	a développé une unité de contrôle d'amortissement de rayonnement à
	rétroaction électronique (RDCU) afin de contrôler, supprimer ou amplifier le champ de "radiation
	damping" en modulant en phase une partie du signal d'induction et en le renvoyant à la sonde
	après amplification. Un schéma du RDCU est montré dans la figure 4. Cette unité est destinée à
	être installée sur le polariseur DNP pour contrôler l'amortissement du rayonnement afin d'étudier
	en détail la dynamique non linéaire de l'aimantation provenant de spins hyperpolarisés DNP à
	l'état solide. Ce dispositif a été testé sur des spectromètres 300 MHz et 400 MHz, respectivement
	à l'Ecole polytechnique et à l'ENS. Nous avons pu supprimer et améliorer l'amortissement du

Table 1 .

 1 T -1 -2.002319 9.109 × 10 -31 kg Proton 2.675 × 10 8 rads -1 T -1 5.58569 1.67262 × 10 -27 kg

1: gyromagnetic ratio and g-factor of electron and proton population of a state |S, m⟩ is given by:

Table 3 .

 3

1: Values of k αα for different ellipsoidal shapes

Acknowledgements

I believe, a teacher has an important role in a student's life to find their potential and pursue their goal by guiding and motivating. I remember and thanks all those teachers who came into

During the buildup experiment leading to the spectra of Figure 2.8, the electron polarization is assumed to be in a steady state, so that the shift of the resonance line, measured by the first moment, varies linearly with the nuclear ( 1 H) longitudinal magnetization. In the sample, the resonance shift of the nuclear first moment is due to the local dipolar fields generated by both the nuclei and the paramagnetic impurities [24,26]. Thus, m 1 is the sum of the respective contributions of the nuclear and electrons, m nn 1 = 2 3 µ 0 Iγ 2 h hξN h P h and m ne 1 = µ 0 Sγ h γ e hξN e P e [27,24,26], where I, S are the nucleus and electron spins, N e and N h their densities, and P h and P e their concentrations. The shape factor, ξ = 3

can be calculated for different sample shapes, and takes the simple values (ξ = 0, -1/2 for a sphere or an infinitely long cylinder). [26] In contrast, the computation of the second moment cannot be related to the line narrowing observed upon polarization, and shown in Figure 2.8. In this experiment, the line width drops from ∼ 16 kHz to ∼ 4300 Hz. Indeed, in this experiment, the width and shape of the resonance line strongly depend on the radiation feedback from the probe that maintains a phase coherence between the spins. Therefore, the conventional expressions derived in the literature for the second

Radiation damping control unit

The amplified radio frequency signal from stage 1 is fed into the input of radiation damping control unit. A 26 dB attenuator was placed between the port 3 (output of stage 1) of figure As discussed in section 4.2 the input radio frequency signal is then demodulated with the reference signal and modulated back after filtering the higher frequency component using a low pass filter. The phase of the re-modulated radio frequency signal can be changed between 0-360 o using a phase modulator. The Phase modulator consist of voltage controlled (VC) phase shifter, the voltage (0-30 V) given to the phase shifter set the phase difference 0 -360 o between the input and output of phase modulator. The voltage given to the phase shifter can be set between 0-30 V using a potentiometer, which has 10 turns and each turns are divided into 100 divisions

Appendix A Polarization

Consider an ensemble of N spin half particle say electron (S) or protons (I), state of an i th spin is given by

Therefore the projection operator can be written as,

The ensemble average or density matrix is defined as, 106

Appendix B Secular Approximation

Consider an ensemble of spins on a rigid lattice with moment m i , the distant dipolar field acting on this moment m i can be written as:

where

Definition of Fourier Transform

Fourier transform F ˜(k ⃗ ) of a three dimensional function F (r ⃗) defined as [2,3]:

and its inverse is defined as:

Fourier transform of Dirac delta function

By definition Drirac delta function is,

Appendix D

Probe-Nuclear Spin Coupling: Simulation

Program

Impedance of a parallel LC circuit is given by [4],

The Impedance of the resonance circuit modify when there is a coupling between nuclear spins and the resonator and is given by [5]: D.1) and coupled ( Eq. D.2) to the on resonance nuclear spins respectively.

Reflection coefficient [4] of the resonator is determined the total impedance of the resonance circuit with respect to impedance, Z o of the source (say spectrum analyzer or Bruker Console).

The relation between S 11 parameter [6] and the reflection coefficient is given by, S 11 = Γ.

Nuclear Magnetic Resonance

Instrumentation

The magnetic resonance of a nuclear spin in detected by a sophisticated instrument called spectrometer [7]. It consist of a superconductor magnetic or permanent magnetic to generate a constant static magnetic field B o along z ˆ, electronic units, transmitter to generate radio frequency field B 1 and receiver to demodulate the radio frequency induction signal from probe generated by precession of bulk magnetization and different types of amplifiers with fixed and variable gain to enhance the radio frequency signal.

Magnet

There are many commercial magnets with different fields available in market today, a few provider are Bruker, Jeol. In ENS, Paris we have five magents (Bruker) with different field capable for doing liquid, solid state and low temperature magic angle spinning dynamic nuclear polarization experiments. In addition there are two polarizers (6.7T Bruker and 9.4T cryo-free magnet, Cryogenic) for performing dynamic nuclear polarization experiments at liquid helium temperat-

Remarks

The radiation damping control unit with correct phase and gain was able to suppress (near cancellation) and enhance the radiation damping field. In this section we discuss about some precautions and limiting factors of current radiation damping control unit.

Off-resonance artifacts The local oscillator frequency of radiation damping control unit must be always equals to

Non-linearity of phase modulator

The phase modulator consist of voltage controlled phase shifter. The phase can be adjusted The calibration was done using Rohde and Schwarz SMB100A, radio frequency generator as the local oscillator of the radiation damping control unit. The output power of local oscillator was set to 14.7dBm and frequency to 400.1818807 MHz. A small power was drawn from local oscillator to input of radiation damping control unit using a directional coupler. The input and output was monitored using an oscilloscope (Lecroy WaveSurfer 10M Oscilloscopes 1 GHz).

The period, T of the radio frequency was found to be 2.