H Gle

Brigi Insa

Renée El

Mel Blaise

Pascal Age

R Gourdon

M Lagard

J Y Toussain Hie

ELECTROT Lhem N E

Havgoudouk

Kian Technique

Sation

M Jean

M Gérar

Gerard Sc

Mme Gud

Mme Em

Mme Sylv

M Jean-Y

M Philip

ALE

Keywords: Nuclear Magnetic Resonance, Low field, portable permanent Magnet, Halbach, Mandhalas, Shim magnets, Homogeneity simulation

I would never have been able to finish my dissertation

Synthèse

Les travaux menés au cours de cette thèse ont porté essentiellement sur le développement d'un aimant pour système de Résonance Magnétique Nucléaire (RMN) portable : conception, modélisation et simulation puis réalisation et validation. A l'arrêt de l'impulsion radiofréquence, les spins continuent leur précession autour d'un axe perpendiculaire au champ B 0 , puis l'aimantation globale (résultante de tous les moments magnétiques) revient à son état d'équilibre. Le temps de retour à l'équilibre est appelé temps de relaxation longitudinale T 1 . Il est caractéristique des tissus et dépend des molécules en présence.

Il existe aussi un temps de relaxation dit T 2 , temps de relaxation transversale correspondant à la constante de temps de décroissance de la tension électrique correspondante à l'onde restituée, lors de la détection, par les spins. T 2 dépend essentiellement de la nature de l'atome et de son environnement magnétique immédiat.

A l'état macroscopique le phénomène de précession et de relaxation est décrit par les équations de Bloch :

∧ 0 0
La RMN portable trouve un intérêt dans de nombreuses applications y compris dans des milieux extrêmes. Un exemple significatif est les systèmes de la société Magritek (Nouvelle-Zélande), pour l'analyse des changements des propriétés mécaniques des glaces arctiques avec le réchauffement climatique.

Le développement de la technologie des aimants permanents tels que des aimants de « terre rare » a permis aux chercheurs de les utiliser pour des applications RMN. Ces aimants peuvent produire un champ fort et homogène, deux critères importants pour les exigences expérimentales en RMN.

Autres caractéristiques attrayantes d'aimants permanents sont évidemment l'absence de la source d'énergie électrique extérieure et pas de maintenance. Ainsi, ils remplissent les conditions pour la conception de dispositifs portables RMN.

Les dispositifs RMN portables sont divisés en deux groupes en raison des applications visées. Un premier groupe appelé ex-situ avec des échantillons d'étude placés à l'extérieur de l'aimant. Un second groupe appelé in situ où les échantillons sont analysés à l'intérieur de l'aimant.

Les appareils de RMN portables ex-situ ont une configuration simple avec le volume sensible (région d'intérêt: ROI) à proximité de leur surface. Ces systèmes sont appropriés pour des investigations de surface. Ainsi, ils peuvent être utilisés pour des objets aux dimensions illimitées.

Les aimants in situ ont, quant à eux, la particularité de renforcer le champ en leur centre et par conséquent nul en dehors du champ de la structure. Dans ce cas, leur champ magnétique est homogène à l'intérieur de la structure en comparaison avec aimants ex situ. Un autre avantage d'aimants in situ, il est moins sensible aux influences environnementales externes en raison du confinement du champ.

Dans ce chapitre, nous nous sommes intéressés aussi aux techniques d'optimisation de l'homogénéité du champ magnétique et leurs applications aux aimants portables.

Fondamentalement, deux méthodes répandues ont été utilisées pour augmenter l'homogénéité du champ magnétique. La première méthode est l'optimisation de l'écart entre les deux anneaux de la configuration. Cependant, cette méthode n'améliore pas suffisamment l'homogénéité pour spectroscopie. Le « Shim » est la seconde méthode c'est la plus utilisée pour augmenter l'homogénéité du champ magnétique des appareils de RMN en général et les RMN portables en particulier. La technique du « Shim » est non seulement utilisée pour augmenter l'homogénéité du champ magnétique, mais aussi utilisé pour compenser l'hétérogénéité causée par les imperfections du matériau, les tolérances de fabrication et les imprécisions d'assemblage.

Chapitre 3 : Aimant pour RMN Portable: Conception et Modélisation

Ce chapitre est consacré à la discussion sur les propriétés des matériaux magnétiques et les méthodes numériques.

Les propriétés des matériaux magnétiques ont été discutées dans la première section de ce chapitre. Nous avons examiné différentes familles de matériaux magnétiques les plus utilisés ces quinze dernières années afin de choisir le matériau le plus approprié pour notre application.

Les méthodes numériques de simulation mises en oeuvre ont également été détaillées dans ce chapitre. Nous présentons le calcul de champ magnétique sur la base de la méthode des éléments finis, sous-jacente au logiciel ANSYS (nous avons utilisé ce logiciel pour les calculs et la simulation). En outre, les matériaux de l'élément de la bibliothèque d'éléments d'ANSYS sont décrits dans notre procédure de simulation.

Les méthodes numériques et la simulation jouent un rôle important à la conception de l'aimant et le processus d'optimisation. La géométrie de l'instrument doit être définie par les utilisateurs et les paramètres spécifiques peuvent être alors optimisés. Les stratégies analytiques fournissent des solutions précises pour la distribution du champ magnétique dans la région d'intérêt (ROI) et le profil de champ magnétique. Dans la littérature, il existe différentes méthodes numériques que nous avons décrites aussi dans ce chapitre.

Les éléments utilisés pour la simulation dans la bibliothèque de logiciel ANSYS ont été décrits.

Selon le problème à résoudre, les utilisateurs peuvent choisir les éléments appropriés. Par exemple, l'élément PLANE53 est utilisé pour la simulation 2-D tandis que l'élément SOLID98 est considéré pour des solutions 3-D. Dans ce travail, deux éléments sont utilisés pour la simulation.

L'élément PLANE13 et SOLID97 est respectivement utilisé pour la 2-D et 3-D simulation.

Nous nous sommes intéressés aussi aux matériaux permanents, matériaux qui maintiennent une aimantation en l'absence de champ extérieur. Ces matériaux sont souvent anisotropes (ils admettent un axe d'aimantation) et suivent une courbe d'hystérésis après leur première aimantation. Dans ce chapitre, nous discutons, par ailleurs, certains de leurs paramètres essentiels à ce travail. Ainsi plusieurs propriétés magnétiques initiales d'un matériau peuvent être déterminées:

-La rémanence d'un matériau (Br) mesurée en Tesla (T) est sa capacité à maintenir une magnétisation lorsqu'un champ magnétique externe est retiré après avoir atteint la saturation, avec Br = μ0Mr où Mr est l'aimantation rémanente.

-Le coercivité d'un matériau (Hc) mesurée en Oersted (Oe) ou ampères/mètre (A/m) est l'intensité du champ magnétique extérieur H opposé à l'aimantation rémanente, nécessaire pour que l'induction magnétique dans le matériau soit annulée (

0 B H M         ).
Il existe une variété de matériaux pour aimants permanents, nous avons donné un bref résumé sur des matériaux de qualité généralement utilisés dans la fabrication d'aimants, leurs avantages et désavantages respectifs. Plus particulièrement les aimants permanents utilisant un matériau moderne tel que le NdFeB, que nous avons retenu pour sa meilleure rémanence et coercivité ainsi que son prix bas.

Chapitre 4 : Conception, construction de poids léger RMN portables de type Halbach de barres identiques aimants

Ce chapitre présentera une conception simple pour dispositif de RMN portable. La configuration est basée sur le type Halbach. Il a deux anneaux principaux alignés pour compenser le champ magnétique à l'extérieur des anneaux. Chaque anneau est composé de 12 aimants identiques de forme cylindrique. Pour prédire les propriétés de cette configuration, l'intensité du champ magnétique et l'homogénéité sont calculées et simulées par deux logiciels qui sont ANSYS et RADIA. Ces deux méthodes sont complémentaire: RADIA est utilisé pour l'optimisation et de simulation. ANSYS est utilisé pour la vérification des résultats obtenus par RADIA. Les résultats obtenus par les deux logiciels sont en bonne corrélation à la fois pour la simulation et l'optimisation. La valeur maximale de B 0 calculée avec RADIA est 0,103 T alors que la valeur correspondante dérivée de l'analyse ANSYS est de 0,11 T. La différence de calcul entre RADIA et ANSYS est 6,79%. Cette différence a été discutée dans ce chapitre. Une nouvelle méthode de shim pour augmenter l'homogénéité et corriger les imperfections du champ B 0 a été aussi introduite. La position des aimants de shim a fait l'objet d'une optimisation avec le logiciel RADIA et vérifié par le logiciel ANSYS.

Sur la base de ces résultats, nous avons réalisé un nouveau prototype. Ses propriétés ont été vérifiées par simulation et mesurée également. Ce prototype est constitué de deux anneaux de 12 aimants. Ces aimants sont placés en cercle de 30 mm de rayon et insérés dans douze trous de deux couronnes en aluminium. Les deux anneaux du prototype, fixés par des vis sur les couronnes en aluminium, peuvent coulisser sur trois tiges pour obtenir la position souhaitée des aimants.

Le champ magnétique est mesuré à l'aide d'un gaussmètre numérique Hirst GM08 avec une limite de sensibilité de 10-4 T dans une gamme variant de 0 à 0,299 T. Le Micro-positionneur Signatone S-926 est utilisé pour contrôler le mouvement de la sonde dans les trois directions. La résolution est de 254 µm « per knob révolution ». Les différences constatées entre la mesure et la simulation sont expliquées à la fin de ce chapitre. Un autre aimant basé sur la configuration Mandhalas (Magnet Arrangement for Novel Discrete Halbach Layout) a fait l'objet d'une étude comparative, portant sur deux configurations utilisant des aimants de formes circulaires et de formes carrés, effectuée par simulation 2D (sur la base de trois critères: la masse, l'homogénéité et l'intensité du champ magnétique). Les Mandhalas fabriqués à partir d'aimants circulaires permettent d'avoir de meilleurs résultats (0.32 T, 178 ppm). Sur la base du résultat de la simulation 2D, la configuration avec 16 aimants permanents de formes circulaires a été choisie pour la construction de notre prototype. La simulation 3D a permis d'évaluer le système dans sa globalité.

Tenant compte des résultats obtenus, un système de shim passif a été aussi utilisé dans ce cas et a permis l'augmentation de la zone d'homogénéité de manière significative. Les résultats de l'optimisation indiquent que dans l'axe longitudinal, la région homogène s'étend sur 2,5 fois en comparaison de celle sans le système de shim. L'homogénéité du champ magnétique dans un volume cylindrique de diamètre de 40 mm et de 50 mm de longueur est de 178 ppm au lieu de 638 ppm dans le cas de la configuration sans les aimants de shim, soit une valeur d'homogénéité supérieure d'un facteur 3,5.

Partant des résultats de simulation, nous avons conçu un prototype d'environ 20 kg de poids total.

Cet aimant de table génère un champ magnétique de 0,32 T et offre une homogénéité de 178 ppm dans un volume sensible de 40 mm de diamètre et de 50 mm de longueur.

Conclusion et perspective

En raison de nos objectifs d'applications, nous avons choisi un aimant in situ de configuration de type Halbach pour son homogénéité et sa capacité à confiner le champ à l'intérieur. Les avantages et les inconvénients des matériaux magnétiques ont été analysés. Sur la base de ces analyses, nous avons constaté que la famille des aimants NdFeB est adaptée à nos travaux. Ce matériau magnétique a des propriétés similaires au matériau « terre rare », Samarium Cobalt (SmCo) non retenu en raison de son prix élevé (30% à 40% plus cher que le NdFeB).

Les méthodes numériques existantes ont été également examinées afin de définir la meilleure solution pour nos travaux de simulation et d'optimisation. Nous avons choisi la méthode des éléments finis, procédure sous-jacente de nombreux logiciels.

Les calculs et les simulations sont principalement basés sur ANSYS, dont les résultats ont été confrontés et vérifiés par les résultats obtenus avec le logiciel RADIA. Le principe de calcul ANSYS est basé sur la méthode des éléments finis tandis que celui de RADIA repose sur des méthodes d'intégrales aux limites. Les résultats sont en parfaite concordance aussi bien pour l'homogénéité que pour l'optimisation. Cependant, l'intensité du champ magnétique est différente entre les deux logiciels en raison du maillage choisi. Nous avons également utilisé Matlab pour les différents tracés.

Les mesures ont été effectuées avec l'gaussmètre Hirst GM08 avec une limite de sensibilité 10-4T.

Le micro positionneur Signatone S-926 est utilisé pour contrôler le mouvement de la sonde dans les trois directions de l'espace.

A partir des résultats de la simulation, nous avons réalisé un prototype compact et léger avec 24 aimants « tige » identiques. Notre prototype a deux anneaux; chaque anneau est composé de 12 aimants. Chaque aimant a une longueur de 50 mm et 8 mm de diamètre. Le prototype génère un champ B 0 dans le plan transversal à 0.12T d'intensité. Son homogénéité est 4230 ppm sur un volume de 7x8x20 mm 3 . Il est bien évident que cette homogénéité n'est pas suffisante pour avoir une grande résolution. Pour l'améliorer, nous avons proposé un système de shim constitué de huit petits aimants placés dans l'alésage du prototype. En ajustant le positionnement de ces aimants, nous avons obtenu une configuration optimisée. Les résultats montrent une amélioration significative de l'homogénéité. Ainsi avec ce système de shim, l'homogénéité est 18 fois meilleure : soit 230 ppm en comparaison avec 4230 ppm sur le même volume 7x8x20 mm 3 . Ces résultats de simulation ont été aussi vérifiés par mesure.

Les valeurs de mesure du champ magnétique variant dans l'axe longitudinal sont en bonne corrélation avec les résultats de simulation. Cependant, l'homogénéité obtenue par mesure est différente de celle simulée en raison des caractéristiques réelles du matériau utilisé.

Nous avons également proposé une autre configuration pour des applications biomédicales et/ou agroalimentaires : la configuration Mandhalas (Magnetic Arrangement for Novel Discrete Halbach Layout). Pour parfaire notre choix, deux configurations Halbach à aimants permanents de forme cubique et cylindrique, respectivement, ont été comparés en termes de force moyenne du champ

Résume en Français

Les travaux menés au cours de cette thèse portent sur le développement d'un aimant pour les systèmes de RMN portable. Une homogénéité élevée a été recherchée tout en maintenant le champ magnétique statique B 0 aussi élevé que possible (100 ppm, 0.12T). Les dimensions de l'aimant sont prédéfinies ainsi que celles de la zone d'intérêt en fonction de la taille des aimants permanents utilisés. Ce type de système est dédié à la recherche biomédicale et agroalimentaire.

Les travaux présentés ont consisté, à discuter dans un premier temps un certain nombre des paramètres des matériaux magnétiques essentiels à la construction d'aimants de RMN portables.

Plus particulièrement le choix des aimants permanents à base de NdFeB, qui a été justifié.

Une combinaison entre portabilité, prix et sensibilité a abouti à la conception d'un prototype d'aimant portable à partir d'un système simple d'arrangement de 24 aimants permanents. Le champ magnétique et l'homogénéité de ce système ont été calculés et simulés à l'aide du logiciel ANSYS puis les résultats obtenus ont été vérifiés avec le logiciel RADIA. Une nouvelle méthode de shim pour augmenter l'homogénéité et corriger les imperfections du champ B 0 a été aussi introduite. La position des aimants de shim a fait l'objet d'une optimisation modélisée et simulée sous RADIA. Sur la base de ces résultats, un prototype a été réalisé. Les résultats des mesures de champ magnétique et de l'homogénéité sont en bonne corrélation avec les résultats obtenus par simulation. Les erreurs de mesure ont été estimées et une précision suffisante a été atteinte compte tenu des tolérances portant à la fois sur les caractéristiques des aimants et sur leur fabrication.

Un autre aimant basé sur la structure Mandhalas (Magnet Arrangement for Novel Discrete Halbach Layout) a fait l'objet d'une étude comparative, portant sur deux configurations utilisant respectivement des aimants de formes circulaires et de formes carrés. Les simulations 2D ont été effectuées sur la base de trois critères : la masse, l'homogénéité et l'intensité du champ magnétique. Il est à noter que les Mandhalas fabriqués à partir d'aimants circulaires permettent d'avoir de meilleurs résultats (0.32 T, 178 ppm). D'autre part, la simulation 3D a été faite afin d'évaluer la totalité du système. A partir des résultats obtenus, un système de shim passif a été aussi utilisé dans ce cas et a permis l'augmentation de la zone d'homogénéité de manière significative.

MOTS CLÉS:

Résonance magnétique nucléaire, bas champ, homogénéité, aimant permanent portable, configuration Halbach, configuration Mandhalas, shims, méthodes numériques.

Résume en Anglais

This thesis focuses on the development of a magnet system for NMR applications with high homogeneity while maintaining the static magnetic field B 0 as high as possible (100 ppm, 0.12T).

Due to the application targets, the magnet dimensions are predefined as well as those of the region of interest according to the size of the used permanent magnets. Such system is dedicated to biomedical and agroalimentary applications.

The goal of this research has been firstly, the discussion of parameters of magnetic materials which are essential to the construction of portable NMR magnets, and then the choice of the permanent magnet material the "NdFeB" that was explained.

A compromise between the portability, price and the sensitivity has led to the design of a prototype of portable NMR magnet with a simple system of arrangement of 24 permanent magnets. The magnetic field and the homogeneity of the system were calculated and simulated by using ANSYS software and these results were correlated to those obtained by the RADIA software. A new shim method has been used to increase the homogeneity and correct the field B 0 imperfection.

Based on these results, a prototype was realized. The results of the magnetic field strength and homogeneity obtained by measurements are in good correlation with the results obtained by simulation. Sufficient accuracy was reached to take into account and correct errors due to manufacturing tolerances of the magnets.

Another magnet system based on Mandhalas configuration (Magnet Arrangement for Novel Discrete Halbach Layout) was studied. The comparison between two configurations made from circle and square magnets was performed by 2D simulation (using three criteria: mass, homogeneity and the magnetic field strength). The Mandhalas made from circle magnets give

Introduction

Since its discovery in 1945, Nuclear Magnetic Resonance (NMR) has become an analytical tool in physics, chemistry, biology and medicine. The devices for this phenomenon are also quickly developed in order to adapt to the diversity of applications. Usually, these devices are large, heavy and used inside. They are expensive, consume a lot of energy and need a high cost of maintenance. The portable NMR devices have been developed in order to overcome these difficulties. However, the drawbacks of portable NMR devices are its low field and poor homogeneity. To achieve the high homogeneity while maintaining the field and reducing material consumption are the challenge for researchers until now. Thus, increase of homogeneity and miniature the magnet sizes must achieved when designing a portable NMR device.

The objective of this thesis is to develop a magnet for NMR system with the homogeneity as high as possible while maintaining the magnetic field. Due to the objective of application, the dimensions of magnets are first defined. The mission of this PhD is to extend the region of interest (the region where the samples are detected), as large as possible for a given magnet size.

This thesis is organized into five chapters. compensate the magnetic field outside of the rings. Each ring consists of 12 identical magnets of cylindrical shape. In order to predict the properties of this configuration, its magnetic field strength and homogeneity are calculated and simulated respectively by ANSYS and RADIA softwares.

The results obtained from both softwares are in good correlation for simulation as well as for optimization. Based on these results, we realized a new prototype. The properties of prototype have been also verified by measurement. To increase the homogeneity, we use two shim rings placed inside the bore of this configuration. The magnetic field homogeneity has been significantly improved by optimizing the positions of these rings. In that case we also designed, realized and characterized a new prototype. Carr in his PhD thesis [1]. Much later, in 1973, Paul Lauterbur and his team expanded the Carr's technique and introduced the method of using magnetic field gradients to obtain the images of objects in two dimensions and three dimensions [2]. Today, it is a powerful tool for the diagnostic of human diseases and other medical applications.

Chapter 1

Basic of NMR

The NMR spectroscopy is a technique that exploits the magnetic properties of certain atomic nuclear such as 1 As this work is concerned with the NMR spectroscopy, it is necessary to figure out the NMR phenomenon. In this chapter, we will briefly describe the principle of NMR and the techniques used to measure the NMR signal.

Principle of NMR and MRI

The nucleus of an atom consists of two particles, protons and neutrons. These particles spin about their axis. These motions produce an angular momentum. Because a proton has a mass, a positive charge and spins, it produces a small magnetic field like a tiny bar magnet (Figure 1

Signal to noise ratio

The concept of signal to noise ratio (SNR) for NMR is first proposed by Ernst et al [7] in 1966 and applied to Micro-NMR by Odelblad [8] in the same year . The SNR is defined as the ratio of the amplitude of the signal to the average value of the noise. It is an important criteria to analyze the sensitivity of detection systems [9][10] in NMR experiments. Therefore, it is necessary to compute the SNR for evaluation of the quality of the detection of the NMR signal.

This evaluation involves both ability of the system to detect the signal and the influence of the noise on this system.

The signal

There are numerous detection devices of NMR signal proposed by previous researchers.

These devices have been used in the specific experimental conditions. The Superconducting Quantum Interference Device (SQUID) [11][12] and the mixed-sensor [13][14] is only operated in a low magnetic field condition. Another method is Cantilever (force detection) [15][16] which is suitable for the samples with nano size. The most popular device used to detect NMR signal is conductive coil due to the simple use and good intrinsic sensitivity. D. I. Hoult and R. E. Richards [9] proposed an approach to calculate the signal to noise ratio of the conductive coil using the principle of reciprocity. This method considers an arbitrary loop S where a unit current circulates and an arbitrary point P locates in the free space. The unit current creates a magnetic field B1 in the loop. They are placed in the static magnetic field B 0 which is perpendicular to B 1 as shown in . . .

MF) induced

.

(1-12)
We substitute the equation 1-9 to the equation 1-12. We have:

.

(1-13)
This result is called the principle of reciprocity. The equation 1-13 was validated for the intermediate and radiation zone fields [17]. This equation is applied to calculate the signal induced by spins precession [9][10].

After the 90° pulse, the net magnetization M 0 of the sample tips to transverse plane and rotate away from the longitudinal axis at an angular velocity (Figure 1-8) given by:

(1-14)

The rotation of the net magnetization M on the transverse plane produces an alternating magnetic field at the loop of conductor S (Figure 1-8-a). Using the principle of reciprocity, the EMF induced in the loop S is:

.

(

Where:

is the electromotive force induced in the loop S by elementary volume of the sample.

-B 1 is the magnetic field created by unit current i.

Thus the electromotive force EMF induced in the loop S by the volume V of sample is:

∭ .

(1-16)

The behavior of the net magnetization M 0 is described by the well-known Bloch equation [18][19]:

(1-17)

Where:

-, , : are the unit vectors in Ox, Oy and Oz directions of respectively.

-T 1 , T 2 are the longitudinal and transversal relaxation time respectively.

The magnetic field B 1 produced by the sinusoidal current in the loop S is:

(1-18)

The NMR signal is only received on the transverse plane, thus the component is ignored. Hence the equation 1-18 can be written as:

(1-19)
Where:

is the unit vector of .

is the magnitude of .

We substitute the equations 1-17 and 1-19 into the equation 1-16, then, we have:

∭ .

(1-20)

And *

(1-21)

.

(

.

(

Where:

-M 0 and * are given by the equations 1-1 and 1-7 respectively.

Substitute the equations 1-21, 1-22 and 1-23 into the equation 1-20, the integral simplifies to:

* (1-24)
If the tip angle θ = 90° and B 1 is homogeneous over the sample volume. The amplitude of NMR signal of a pulse RF is:

(1-25)

The noise

In NMR experiments, the noise of the detection systems mostly comes from the thermal noise associated to the resistance of the coil while other noises such as radiation, sample are negligible [20]. The thermal noise is given by formula:

∆ (1-26)
Where:

-1.38 * 10 is Boltmann constant.

-T p (K) is the temperature of the coil.

-R is the electrical resistance of the coil.

-∆ is the detection bandwidth.

The sensitivity

The sensitivity of detection systems refer to the signal to noise ratio (SNR). It is defined as the ratio of the NMR signal (equation 1-25) to the noise (equation 1-26).

∆ (1-27) √ (1-28)
The equation 1-28 is used to evaluate the sensitivity of the antennas and optimize the configuration of antennas [21].

T 1 and T 2 measurements

T 2 measurement

Both T 1 and T 2 are important in NMR experiment. However, the measurement of T 2 relaxation is more practical because it can be obtained quicker than T 1 . There are two well-known techniques which are used to measure T 2 .

Firstly, Spin echoes or Hahn's echo, this technique was first discovered by Hahn in 1950 [22]. The echo is a signal which has first vanished with time and then reappears some time later.

The Hahn echo consists of two pulses, the first one is 90° pulse and then followed by 180° after the delay time τ as shown in Figure 123456789. 3)

(1-34) and

(1-35)

The amplitude of the spin echo is weaker than the initial amplitude by a factor of .

The echo amplitude at the time 2τ is:

(1-36)

When using a pulse train and observing the various echoes, the amplitude of the n th echo at the time 2nτ is:

(1-37)

Or else, as a function of time 2 , the equation 1-37 can be written:

(

This is an exponential function of time, both D and T 2 are possible to retrieve with the rate constant that depends on the interval 2τ between the pulses:

(1-39)

Influence of magnetic field homogeneity on NMR spectroscopy

The homogeneity of magnetic field is an important criteria which influents on the NMR signals. Therefore, it is necessary to emphasize the extremely homogeneous fields for NMR spectroscopy (MRS) when design a portable NMR device. NMR is the result of Zeeman Effect on the nucleus. The external part of the spin Hamiltonian which describes the interaction of the spins with magnetic environment is given by the equation 1-40 [18]:

, (1-40) ∑ (1-41) , ∑ , (1-42)
∑

(1-43)

Where:

is the interaction of each spin I j with the external static field B 0 .

-, is the interaction of each spin I j with the gradient field B grad .

is the interaction of each spin I j with the RF field B RF generated by the RF coil.

Some of these interactions are proportional to the external field (chemical shift), some are not (dipolar coupling), but always remain in the order of the part per million (ppm) of the main Zeeman Hamiltonian. The small changes of the static field B 0 in each point of the sample will affect on the Larmor frequency. Hence, MRS requires sub-ppm over the volume of sample in order to be useful [START_REF] Cédric | Aimants permanents pour la RMN et l'IRM[END_REF], for example, ∆ 10 across the sample for good spectroscopic resolution [START_REF] Blumich | NMR Imaging of Materials[END_REF].

Chapter 2

Portable NMR system

Summary:

- The objective of this chapter is to introduce the development of portable NMR devices from the early time until now. In addition, we will present the different kinds of them and their diversity applications as state of the art.

The need of portable NMR devices

In 1952, one dimensional MRI image is reported by Herman Carr in his PhD thesis [1].

Much later on, in 1973, Paul Lauterbur and his team expanded the Carr's technique and introduced the way using magnetic field gradients to obtain the images of objects in two dimensions and three dimensions [2]. The major drawback of NMR is its lack of sensitivity. This lack of sensitivity makes difficult the study of low γ nuclei in NMR spectroscopy by increasing the experiment time.

The low sensitivity is also a limitation of MRI resolution [START_REF] Cédric | Aimants permanents pour la RMN et l'IRM[END_REF]. One solution to solve this difficulty is to increase the external magnetic field in order to improve the SNR. Thus, the idea of the increase of magnetic field motivated many researchers performing experiments with high magnetic field.

Rabi and his team were performed the first experiment at a field of 0.6T (about 25.6 MHz) while Lauterbur's experiment was done at a field 1.4T (60 MHz). In 1964, a superconducting magnet at 200 MHz was introduced by Nelson and Weaver [START_REF] Nelson | Nuclear Magnetic Resonance Spectroscopy in Superconducting Magnetic Fields[END_REF] for the NMR spectroscopy Basically, f a ferroma ure [START_REF] Halbach | Design of permanent multipole magnets with oriented rare earth cobalt material[END_REF].

Yo

The yoked Where: The conce m has reaso In 2014, based on the works of [START_REF] Raich | Design and construction of a dipolar Halbach array with a homogeneous field from identical bar magnets: NMR Mandhalas[END_REF][79], Qiaoyan Chen et al [START_REF] Chen | Simulation and Optimization of an Octagonal Halbach Permanent Magnet Array for Small-Size NMR[END_REF] compare a cube and an octagonal Halbach permanent magnet array in equal quantity of the material consumption with Maxwell software simulation. They found that the octagonal shape magnet has better performance than the cube one. By analysis of magnetic field strength, the best homogeneity in 5 mm DSV (Diameter of Spherical Volume) with 133.4 ppm was achieved when the height of magnetic blocks is 16.25 mm. However, the prototype has not been constructed yet.

-B 0 is th -B r is re -r 1 , r 2 ar
To conclude, although the Halbach structure is less homogeneity, they indicate that they are tremendous cheaper than homogeneous standard NMR spectrometers.

Techniques are used to improve the magnetic field homogeneity

One of the drawbacks of portable NMR devices is their inhomogeneity for high resolution spectroscopy. In order to overcome this difficulty, there are some techniques have been used to increase the homogeneity. Basically, two popular methods have been used.

Optimize the gap between two rings

In the Halbach structure, the homogeneity is homogeneous with the infinite magnets (2D).

However, with the finite height magnets 3D), the magnetic field homogeneity is much smaller than the 2D case. In the 3D case, the homogeneity is homogeneous at the center of configuration and becomes worse towards at the end. Thus, in order to compensate this distortion, the Halbach structure is divided into many rings. By optimizing the gaps between these rings, the homogeneity of the magnet along the cylinder axis is improved. Based on this concept, Anferova et al [START_REF] Anferova | Improved Halbach sensor for NMR scanning of drill cores[END_REF] splits the sensor made by [START_REF] Anferova | A mobile NMR device for measurements of porosity and pore size distributions of drilled core samples[END_REF] into the stack of six magic rings. These six magic rings are combined into two identical arrays, each of them consisting of three magic rings. The magnetic field profile along the cylinder axis becomes more homogeneous as the distance between the two arrays increases from zero as shown in the Figure 2-30. Although this method is easy to perform, it does not improve too much the homogeneity for high resolution spectroscopy. The field dis achieved 0. ong [START_REF] Danieli | Mobile sensor for high resolution NMR spectroscopy and imaging[END_REF]. duplexer. The duplexer is a circuit which consists of two diodes to ensure a good isolation when a signal is applied to the generation coil. The information of FID, acquisition time, sequence repetition time are transferred to PC via USB port.

Motivation for this work

The homogeneity of magnetic field is an important criterion in NMR experiments. Thus, the increase of homogeneity is necessary in the design of portable NMR devices. Most of the portable NMR devices do not provide the sufficient homogeneity for NMR spectroscopy and highresolution MRI over a macroscopic volume. Some previous design achieved such homogeneity on a very small sample size (5x5x0.5mm [START_REF] Perlo | Ex Situ NMR in Highly Homogeneous Fields: 1H Spectroscopy[END_REF] or 21 nanoliter [START_REF] Mcdowell | Ultracompact NMR: 1H Spectroscopy in a Subkilogram Magnet[END_REF]). This is difficult to precisely position the sample and limit the signal to noise ratio. Moreover, the magnetic field is perfectly homogeneous for a perfect Halbach geometry and infinite long magnets. However, the magnets must have a finite length in reality, thus, the strength and homogeneity of magnetic field gradually decreases from the center to the edge of the cylinder. In order to get an enough homogeneous magnetic field region for NMR experiments, the magnet length must be extended. This results in the increase of magnetic materials, the weight of device and the cost of production. This work is devoted to maximize the region of interest (ROI) as large as possible with the given magnet dimensions (due to the objective of applications) while maintaining the high magnetic field. Based on the need of applications, the in-situ configuration has been chosen for design the prototypes.

The calculations and simulation are performed with the help of two different software packages for the magnetic field modeling brought similar results, then; these results are verified by measurement with gaussmeter. The results shows that the prototypes generate the sufficient homogeneity and field strength for biomedical and agroalimentary applications.

Chapter 3: Magnetic materials and numerical methods

Introduction

In magnetic resonance, there are two sources of magnetic field required for. The first source can be produced by a current through a superconducting or resistive wire, typically in a solenoid shape. The benefits of using a superconductor are easy to achieve the homogeneity of magnetic field and the high field strength. Its field strength is not influenced by external environment. The superconductors is not compact, thus limit the widely applications. In addition, they need too much energy to operate and high cost for maintenance. Unlike the first source, the second source is produced by permanent magnets. They are the most suitable material for compact MR instruments. The drawbacks of permanent magnets are low field, inhomogeneity and their field is instable with external environment such as temperature. However, they are compact, low cost and no maintenance. Their field strength and homogeneity depend on the quality and kind of magnetic materials. Therefore, in order to proceed with magnet design, it is necessary to bear in mind some elementary aspects of the materials used.

In order to achieve the optimum configuration, numerical methods and simulation play an important role to magnet design and optimization process. The geometry of instrument needs to be defined by the users and specific parameters can be then optimized. Analytical strategies provide accurate solutions for the magnetic field distribution in the region of interest (ROI) and magnetic field profile. There are various numerical methods have been published. The dipole approach was proposed by Soltner et al [START_REF] Soltner | Dipolar Halbach magnet stacks made from identically shaped permanent magnets for magnetic resonance[END_REF]. This method is used to optimize the gap between two rings of magnet stack in order to increase the homogeneity in longitudinal axis of instrument. Hugon et al [START_REF] Hugon | Design, fabrication and evaluation of a low-cost homogeneous portable permanent magnet for NMR and MRI[END_REF][72] bases on the spherical harmonics expansion of the field to design homogeneous magnets systems for NMR and MRI. This approach for magnet design gives a method to control magnetic field of arbitrarily magnet systems. A method of designing 2D magnets based on scalar potential proposed by Marble et al in 2005 [START_REF] Marble | An analytical methodology for magnetic field control in unilateral NMR[END_REF]. Based on this method, they built a prototype with a constant gradient field near its surface [START_REF] Marble | A constant gradient unilateral magnet for near-surface MRI profiling[END_REF]. They also developed a method of designing 3D magnets with the scalar potential approach [START_REF] Marble | Designing Static Fields for Unilateral Magnetic Resonance by a Scalar Potential Approach[END_REF]. However, no prototype has been built with this method. Sometime, some designs and optimization are based on software packages such as COMSOL [START_REF] Zhang | Design, construction and NMR testing of a 1 tesla Halbach Permanent Magnet for Magnetic Resonance[END_REF], RADIA [START_REF] Hugon | Design, fabrication and evaluation of a low-cost homogeneous portable permanent magnet for NMR and MRI[END_REF].

This chapter gives a short description of magnetic materials, their properties. We also describe the numerical method based on the finite element method (FEM) which is the underlying procedure of ANSYS (the software we used for simulation in this project). Finally, we give an overview and the procedures of magnetic field simulations of ANSYS software [100].

Units

There are three systems of units that are commonly used in the magnetism. These are the CGS or Gaussian system, and two MKS or SI systems that are referred to as the Kennelly and Sommerfeld conventions, respectively. The CGS system is based on the fundamental units of Centimeter, Gram and Second, and the SI systems are based on the MKSA units of Meter, Kilogram, Second, and Ampere. The units for the SI (Sommerfeld) and CGS systems are as follows the

Magnetic materials

The fundamental element in magnetism is the magnetic dipole. A magnetic dipole has a magnetic dipole moment m measured in A.m 2 . The magnetization M measured in A.m -1 is a measure of the net magnetic dipole moment per unit volume. It is given by equation 3-1

[101] [START_REF] Coey | Magnetism and Magnetic Materials[END_REF].

∆ → ∑ ∆ (3-1)
Where the ∑ is a vector sum of the dipole moments contained in the elemental volume ∆ .

From the relationship between magnetic moment and flux, the relationship between B and M can be found. A bar magnet with flux density at the center, the dipole length l and with the cross sectional area A has a magnetic moment m given by . [START_REF] Jiles | Introduction to Magnetism and Magnetic Materials[END_REF]. The magnetization can be written as equation 3-2:

. .

(3-2)

The equation 3-2 hence:

(3-3) Where 4 . 10

. is the permeability of free space.

Another indispensable auxiliary field when dealing with magnetism is H field known as the magnetic field strength. The distinction between B and H is trivial in free space. Their relationship is described by equation 3-4:

(3-4)

The magnetic induction consists of two contributions: one from the magnetic field as equation 3-4, the other from the magnetization as equation 3-3. The magnetic induction is then simply the vector sum of these.

(3-5)

B is the magnetic induction, expressed in Tesla (T). H is the magnetic field, expressed in Amperes per meter (A/m).

The magnetic polarization or intensity (in Tesla) might also be used. In the Kennelly [START_REF] Furlani | Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications[END_REF] convention, the constitutive relation is:

(3-6)
Where is called the magnetic polarization.

The magnetization is defined by a constitutive equation of material. This constitutive equation simplifies for linear, homogeneous and isotropic media. For such materials, both B and M are proportional to H. The permeability and susceptibility of the material μ and respectively can be defined as:

(3-7) (3-8)
These coefficients are related to one another. From the equations 3-5, 3-7 and 3-8, the permeability of material is:

(3-9)
The relative permeability μ is a dimensionless quantity defined as:

(3-10)

In inhomogeneous materials, the permeability and susceptibility are function of coordinate variables, μ μ x, y, z , χ χ x, y, z . A material is said to be isotropic if μ depends on direction.

In this case, the equation 2-7 can be written:

(

The expansion for equation 2-8 is similar. Permeability is usually discussed in relation to soft ferromagnetic materials, where μ can take very large values, up to 10 4 or more [START_REF] Coey | Magnetism and Magnetic Materials[END_REF]. The quantities discussed above are very important when using magnetic materials. There are two types of materials are pertinent to our discussion: Soft and hard magnetic materials. They are classified as either soft or hard depending on their coercivity H c . The coercivity of a material is its capacity to maintain its magnetization in an opposing external magnetic field.

-Soft magnetic materials are characterized by a high permeability and a low coercivity (Hc < 1000 A/m), which makes them easy to magnetize and demagnetize. However, the soft -The saturation magnetization corresponds to the maximum magnetization achieved when increasing the external magnetic field. In this status, the magnetic moments in material are aligned with the external field (the value at point S on the hysteresis loop).

-The remanence of a material (B r ) measured in Tesla (T) is its ability to maintain a magnetization when an external magnetic field is removed after achieving saturation. The relationship between the remanence induction and the remanence magnetization is described by equation 3-14:

(3-14)

-The coercivity of a material (H c ) measured in Oersted or Ampere/meter is the intensity of applied magnetic field that is required to reduce the magnetization of the material to zero.

All the properties of magnetic material are highly dependent on temperature. Each of magnetic materials has the temperature called Curie temperature or Curie point where a material's permanent magnetism changes to induced magnetism. Below the Curie temperature, the magnetic moments of magnetic material align in a ferromagnet in the absence of an applied magnetic field.

On the other hand, above its Curie temperature T c , a ferromagnet becomes paramagnetic [START_REF] Cullity | Introduction to Magnetic Materials[END_REF]. It means that the magnetic moments are randomly aligned in a paramagnet and the magnetic material does not maintain its magnetization when the external magnetic field is removed. The Curie temperature is an important quantity must be considered when designing portable NMR devices for out of laboratory applications. The material should be selected as to be far from the Curie temperature.

There exists a variety of materials for permanent magnets. We will give the brief summary of top grade materials typically used in magnet manufacturing and their respective advantages and drawbacks.

Lodestones are known as the first magnetic material. Lodestones contain magnetite, an oxide iron. They are a naturally occurring magnet having the composition Fe 3 O 4 . Lodestone has low magnetic field, but high resistance to demagnetization.

Magnetic carbon steels were developed in the 18 th century. They have high magnetic saturation, but are prone to demagnetization, thus can be used in limited designs [START_REF] Demas | Compact magnets for magnetic resonance[END_REF].

Ferrite magnets are the most common commercial magnets due to their low cost. They have excellent corrosion resistance. They are very hard and brittle. Thus, this makes them difficult to machine and drill holes. Ferrite magnets have high coercivity corresponding to resistance to demagnetization which makes them suitable for use in complex shapes. Ferrite magnets are used in electronic sensing devices, electric motors, loudspeakers and lifting devices….

Alnico magnets were developed in the 1930 s . They are alloys of iron, cobalt, nickel and aluminum (Al) along with lesser amount of other metals such as copper (Cu). Alnico magnets can be either isotropic or anisotropic, depending on whether or not the magnetic particles are oriented during their formation. Alnicos are hard as well as too brittle for cold working. They provide considerable magnetic hardness based on shape anisotropy. Alnico's remanence (B r ) may exceed 12.000 gauss. They have an excellent Curie temperature which is approximate 850°C. Alnico magnets have impressive magnetic properties. Unfortunately, they also have poor physical properties. They are extremely hard and brittle. Thus, in order to achieve a finished magnet with tight tolerances, it requires tedious and costly machining [START_REF] Furlani | Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications[END_REF].

Samarium cobalt magnets (SmCo) were developed in the late 1960 s . They are a type of rareearth magnet and strong permanent magnet made from an alloy of samarium and cobalt. The two most common SmCo materials are SmCo 5 and Sm 2 Co 17 . They are generally ranked similarly in strength to Neodymium magnets, but have higher Curie temperature. The maximum operating temperatures for cobalt magnets are between 250 and 550°C; Curie temperatures range from 700 to 800°C. SmCo materials offer high coercivity up to 10.500 Oersted. These materials have good thermal stability and high energy product make them popular specifically for MR applications, despite their high production cost. In addition, a samarium cobalt magnet is less subject to corrosion than a neodymium magnet, and usually does not need coating and plating. Samarium cobalt magnets are widely used in high temperature and poor working conditions.

Neodymium iron boron magnets (NdFeB) was developed in the 1980 s . They are another type of rare-earth magnet made from an alloy of neodymium, iron and boron. This material has similar properties as the samarium cobalt except that it does not have the same temperature resistance.

NdFeB offers the highest energy product approaching 50MGOe. They have the highest magnetic field strength and have a higher coercivity in comparison to samarium cobalt magnets. However, their Curie temperature (310°C) is lower than the Curie temperature of samarium cobalt magnets.

This prohibits them from being used in high temperature applications. While today neodymium magnets with higher quality grades have eliminated an oxidization chain reaction in the surface of the magnets, a protective coating is still necessary to protect them from the atmosphere. In addition, NdFeB is lighter and less brittle than SmCo.

The mass density of NdFeB is about 7g/cm -3 while the correspondent value of SmCo is about 8.

Numerical methods and simulation

In order to analyze the magnetic field of portable NMR devices and their homogeneity, a variety of numerical and simulation approaches can be used. The popular method is dipole approach. In this approach, each magnet is considered as a simple dipole. The magnetic field of a magnetic dipole can be calculated by formula 3-15.

, , .

Thus, the flux density in the bore of the Halbach cylinder can be written as the formula 3-16. The Figure 3-5 shows the component definitions of this formula.

∑ .

(3-16)

Where:

-m i are the magnetic moment vectors of the n th magnets, for i = 0,1,…,n-1.

(

-P is the measurement field point.

(3-18)

-M p is the positions of magnets on the array.

(

-n i are unit vectors from the field points.

(3-20)

-R is the distance between the measurement field point P and the dipole as shown on -N is the number of magnets. Substitute the parameters m i , n i , M p , R and P into the equation 3-16, the flux density can be written as:

(3-23)

Where:

sin cos P x z y β i m i o r p r R α i
The equation 3-16 is very useful for calculating the flux density in the bore of a Halbach cylinder from the given B r of the magnet materials. It is a good approximation method (because in this method, the size of the magnets is neglected) to model first the element of a design by dipoles and then move toward a specific geometry. Based on this method, Soltner and Blümler [START_REF] Soltner | Dipolar Halbach magnet stacks made from identically shaped permanent magnets for magnetic resonance[END_REF] proposed formulas and numerical values to calculate the flux density of single rings and composed 3D system.

Another method based on the spherical harmonic expansions (SHE) of the scalar potential and the magnetic field. Based on this method, Hugon et al [START_REF] Hugon | Design, fabrication and evaluation of a low-cost homogeneous portable permanent magnet for NMR and MRI[END_REF] [START_REF] Cédric | Aimants permanents pour la RMN et l'IRM[END_REF] proposed an approach to design an axisymmetric magnet achieving a specific homogeneity ΔB 0 in a given sphere of diameter R 0 with the homogeneity condition:

∆ (3-24)
Marble et al [START_REF] Marble | A compact permanent magnet array with a remote homogeneous field[END_REF] proposed the 2D calculation of magnetic field for a bar magnet. Based on this calculation, they were built a prototype with the remote homogeneous field. Their calculation considered the permanent magnet as two sheets of thin wires. However, this approach neglected inhomogeneity in the magnetization or saturation effects from the magnets in close proximity. Thus, it is not a solution for the finite magnet.

Finite element methods (FEM) are currently the most popular method for the solution of electromagnetic and magnetostatic problems because of two reasons. Firstly, FEM is a powerful method widely used in a variety of applications. Secondly, There exists numerous commercially FEM software packages such as ANSYS, COMSOL…These software packages enable users to solve complex problems without developing their own application specific algorithms. For these reasons, we chose the finite element methods which are underlying principles of ANSYS (the commercial software package we use for simulation and optimization in this work).

The calculation of ANSYS for electromagnetic and magnetostatic problems are based on the Maxwell's equations. Electromagnetic fields governed by Maxwell's equations can be written as:

(3-25) .

(3-27) functio the sca for ele

Where:

is In ANSYS, the values of H and B can be obtained by two solutions, which are the magnetic scalar potential and the magnetic vector potential. The second solution is described here because it is implemented in PLANE13, SOLID97 and SOLID117. The elements PLANE13 and SOLID97 are respectively used for 2D and 3D simulation in this work. The detail of the first solution can be found in the reference [106].

- . is di - is - is t - is - is - is - is - is tim - is - is - is
Unlike the scalar potential formulations are restricted to static field analysis, the vector potential formulation can be applied to both static and dynamic fields with partial orthotropy nonlinear permeability. The basic equation to be solved is [106]:

(3-33)

Degrees of freedom:

(

Where:

is magnetic vector potential (AX, AY, AZ).

is time integrated electric scalar potential .

Coefficient matrices:

(

(3-37)

. . Where:

is the magnetic flux density.

is the curl operator.

is the shape functions.

is the nodal magnetic vector potential.

From the flux density results, the magnetic field intensity is then calculated by equation 3-52.

(3-52)

Where:

is the magnetic field intensity.

is the reluctivity matrix.

ANSYS

ANSYS multiphysics [100] is the general purpose finite element analysis software. It can solve complex problems which concern with variety of engineering field such as heat transfer, fluid problems, magnetic field…It is a powerful tool for simulation and optimization. The analysis process of ANSYS consists of three steps: pre-process, FEM analysis and post-process.

Pre-process is the processes of geometry modeling, setting up material properties, meshing and setting boundary condition. The ANSYS program has a large library of element types.

Depending on the solved problem, the users can choose the suitable elements. For example, the element PLANE53 is used for 2-D simulation while the element SOLID98 is considered for 3-D solutions. In this work, two elements are used for simulation. The element PLANE13 and SOLID97 is respectively used for 2-D and 3-D simulation.

The element PLANE13 models (planar and axis-symmetric) 2-D static magnetic field analysis. It is defined by four nodes with up to four degrees of freedom per node. This element has nonlinear capability for modeling B-H curves or permanent magnet demagnetization curves.

PLANE13 is based on the magnetic vector potential formulation. In this element, the MGXX, MGYY represent vector components of the coercive force for permanent magnet materials in two directions X and Y. The magnitude of the coercive force is the square root of the sum of the squares of the components. The direction of polarization of permanent magnets is assigned in whether X or Y direction through the command MP [106].

The element SOLID97 models 3-D magnetic fields. The element is defined by eight nodes and has up to five degrees of freedom per node. SOLID97 is also based on the magnetic vector potential formulation with the Coulomb gauge and it is applicable to the low-frequency magnetic field analysis such as magnetostatic, Eddy current… This element has nonlinear capability for modeling B-H curves or permanent magnet demagnetization curves. In this element, the MGXX, MGYY and MGZZ represent vector components of the coercive force for permanent magnet materials in three directions X, Y and Z. The magnitude of the coercive force is the square root of the sum of the squares of the components. The direction of polarization of permanent magnets is assigned in the same command with the element PLANE13 [106].

Post-process is the processes of display results and data extraction after building the model and obtaining the solution. Via the post processor of ANSYS, the results can be plotted or visualized. Data extraction is an important process for our work. In our work, the homogeneity of magnetic field in the region of interest is the most important criteria need to be calculated. Thus, the magnetic field nodes in this region of interest are extracted and treated by Matlab. The region of interest is first defined in the pre-process step.

Conclusions

The desired properties of the magnetic materials used for portable NMR devices are highly required such as high remanence, high coercivity… In order to achieve the highest magnetic field possible. There exists a variety of magnetic material families which have been discovered through history. Each of magnetic material has different advantages and drawbacks. The users can choose the suitable magnetic material depend on their objective of applications. In addition, the properties of magnetic materials are influenced by the external environment such as temperature. Therefore, it is necessary to figure out the properties of magnetic materials in order to exactly choose for applications. In this chapter, we briefly described the magnetic material properties which is the desired materials used in recent years. From these descriptions, we have chosen the best material for our applications.

The simulation plays an important role to the design of portable NMR devices. It is a tool to predict the properties of magnet systems. From the results of simulation, the designers can give the solutions to improve or optimize them. In this chapter, we have examined the numerical and simulation methods of previous researchers and chosen the finite element method which is the underlying procedure of ANSYS for our calculation and simulation. Finally, we also examined the field calculation based on the magnetic vector potential and gave the short description about the two elements used in our work. and perpendicular, to the external magnetic field. In order to provide the sufficient field homogeneity for NMR experiments, a popular method to improve this inhomogeneity is shimming. The concept of movable permanent magnets in the shim unit of a Halbach array is reported by Ernesto Danieli et al [START_REF] Danieli | Mobile sensor for high resolution NMR spectroscopy and imaging[END_REF]. Another method of shimming based on the spherical harmonic expansion proposed a complete procedure for permanent magnet design, fabrication, and characterization [START_REF] Hugon | Design of arbitrarily homogeneous permanent magnet systems for NMR and MRI: Theory and experimental developments of a simple portable magnet[END_REF]. In this chapter, we introduce the light weight (3kg), low cost (200euros) and simple design NMR apparatus which consists of 24 identical rod magnets arranged in Halbach array. This apparatus generates a B 0 field strength about 0.12T. Its characteristics are described by simulating and calculating the field strength and homogeneity in three dimensions (3D). These simulations and calculation were performed by two softwares RADIA and ANSYS Multiphisics.

The results of simulation which refer to the field strength B 0 and homogeneity of device were compared to those obtained with a digital gaussmeter. The results of verification show that the homogeneity in the longitudinal axis of apparatus and the field strength B 0 were found to be similar. However, the homogeneity in transverse plane differs from simulation and measurement because of the quality of the magnets. To overcome this difficulty, we propose a new shim method to compensate the imperfection of the magnets. The homogeneity of magnetic field significantly improves with 18 times in comparison to one without shim magnets.

Modeling and simulation

Modeling

In order to design and construct the prototype, we chose the Halbach type. This prototype has two rings placed alignment to compensate the magnetic field outside of the rings, each ring consist of 12 identical magnets in cylindrical shape. The direction of magnetization of static field B 0 is defined parallel to the Ox axis. The homogeneous region is expected in the middle length of the prototype. The geometric parameters are displayed in Figure 4-1 and Table 4-1.

The Table 4-1 shows the geometric parameters of apparatus, their definitions and dimensions. -a is the diameter of magnets.

-r inner is the inner radius of the ring.

-r outer is the outer radius of the ring.

-r is the radius of the ring.

i is the angle between the Oy axis and the i th magnet. -β i is the angle between the Ox axis and the direction of magnetization of i th magnet.

The i th magnet is placed on a circle at an angle

n i i . 2  
and its magnetization is defined by an angle β i = 2α i . Where, n is the number of magnets, i = 0, 1, 2… n-1. In our case, the apparatus has 12 magnets placed on a circle of radius r 0 = 30 mm. As shown on Figure 4-1-a, each magnet is placed at an angle 6    i and its magnetization is rotated by 3

   i .

Simulation and results

Simulation

In order to calculate and simulate the magnetic field and the homogeneity, two softwares RADIA and ANSYS Multiphisics were used. They are two different methods of calculation.

RADIA was developed to design the Insertion Devices for Synchrotron light sources. It uses boundary integral methods. Each volume created to represent the magnets is subdivided in a number of sub elements to solve the general problem in terms of magnetization. The solution is performed by building a large matrix in memory which represents mutual interactions between the object. RADIA does not need to mesh the vacuum. On the other hand, ANSYS is a multiphysics software and it uses FEM (Finite Element Modeling) modeling. Each volume is sub divided with some elements. Even the air between and around the magnets has to be meshed. Flux conditions have to be placed outside the global volume in order to apply parallel or normal condition. These two methods are used complementary: RADIA is used for optimization and simulation. ANSYS is used for verification of the results obtained by RADIA The size of the meshing is reduced until the simulation results do not change.

The properties of the material used for simulation that represent magnets were chosen to represent magnets from "HCKM MAGNETS from STOCK" [107]. The magnet material is Neodinium NdFeB with the following characteristics. In order to calculate the homogeneity, the values of the magnetic field (mesh nodes) are selected in the homogeneous region, then exported to and treated by Matlab Software. The homogeneity is calculated by the formula 4-1.

∑ (4-1)

Where:

-B i is the value of magnetic field at the i th position in the homogeneous region.

-B is the value of magnetic field at the center of homogeneous region.

-m is the number of mesh nodes in the homogeneous region.

Results

A NMR device constitute with 24 magnets placed as shown in the Figure 4-1 is considered during the simulation with ANSYS and RADIA. The magnetic field B 0 is oriented along Ox axis.

First, esp = 0 is considered. The maximal value of B 0 calculated with RADIA is 0.103 T while the correspondent value derived from ANSYS Analysis is 0.11 T. The difference of calculation between RADIA and ANSYS is 6.79 %. This difference can be accounted for by the problem of mesh size convergence. It means that the results of fine meshes are higher than coarse meshes. This difference can be accounted for by the problem of mesh size convergence. It means that the results of fine meshes are higher than coarse meshes. In order to mesh the volumes, the command SMRTSIZE, n was used in this simulation. Where, n is the mesh size level ranging from 1 (fine mesh) to 10 (coarse mesh). We used the default mesh size level with n=6 for this simulation. Furthermore, the difference can be acceptable. Although the magnetic field strength calculated by two softwares is slightly different, the magnetic field distribution in the homogeneous region is similar as shown in Figure 4-4-a,b. This results in the similarity of the magnetic field homogeneity.

The Figure 4-3-a displays the variation of magnetic field B 0 along Ox axis. The value of magnetic field is almost constant for 1.8mm < x < 4.2mm or in the distance of 2.4mm at the center of configuration. The Figure 4 The variation of the magnetic field Bx profile along Oz axis depends on the gap esp between the two rings. The best homogeneity is achieved at a certain distance. In order to optimize this gap, we increased the value of esp by the steps of 0.1mm. The Figure 4-5 shows the B x profiles versus the Oz axis for four values of esp. When esp = 0, the magnetic field outside one ring does not compensate the magnetic field for the other ring. For esp = 0.9 mm, the compensation is optimum and the magnetic field at the center almost constant. The magnetic field at the center of apparatus significantly decreases at the gap esp = 1.3mm. The useful volume for NMR sample is determined from the coordinates (x,y,z) of the point where Bx is maximal. Then, the volume is calculated with the 3 coordinates that generate a variation of 0 0 B B  of 100 ppm. The Figure 456shows that the volume of the homogeneous region is a function of the esp. The optimal value of esp determined by RADIA is around 0.77 mm and the useful volume is about 2640 mm 3 . The volume is increased by a ratio of around 80 when the space between the two rings is improved. This is caused by the decrease of the magnetic field outside one ring is similar to the other ring. There's an optimum space between the two rings where the sum of the variations of the magnetic field outside the rings are canceled.

Optimization

Although the homogeneity of magnetic field increases by adjusting the gap between the two rings, the inhomogeneity of magnetic field also comes from magnetic material (dispersion of The results of optimization give a great improvement of homogeneity, as can be seen in the Each shim magnet shown on the Figure 4-15 is glued in a nonmagnetic cylinder. These cylinders can rotate, move along the longitudinal axis and slide along the radius of the prototype in order to find the optimal position of the shim magnets. These cylinders are placed in the holes of sliding-blocks that slide on the four apertures of an aluminum frame which is concentric with the prototype.

Experimental set up

The magnetic field is measured by the digital gaussmeters Hirst GM08 with the sensibility limit of 10 -4 T in the range 0 -0.299 T. The micropositioner Signatone S-926 is used to control the probe movement in three directions as shown in the Figure 4-16. The resolution is 254 µm per knob revolution. Matlab software carries out plotting the measurable values.

Measurement results

Before shimming measurements

To optimize the gap esp between two rings, at the beginning, it was set at 0mm, and was progressively increased by turning the screws on the frame of the device (Figure 45678910111213141516).

The Figure 4-17 shows the magnetic field profiles in OZ direction for four different gaps esp between the rings. The optimal gap is displayed in Figure 4

Discussion

The difference between the simulated and measured values of the magnetic field homogeneity is due to the poor quality of the magnets. There is a large dispersion of the magnets properties. For our prototype we selected 24 magnets among 27 having a similar magnetic field strength and the magnets 5, 22 and 23 was rejected. This was done with the measurement of the magnetic field on each tip of the magnet cylinder. In the discussion part, we mentioned about the influence of misalignment of magnetic directions on the homogeneity. In order to correct this error, we divide the long magnet (50mm)

into five small magnets. The diameter and length of each magnet are respectively 8 and 10mm.

These magnets are placed in an aluminum cylinder to constitute a magnet with 8mm of diameter and length of 50mm. Such magnets can be rotated in cylinder so that their magnetic directions are in alignment with those of adjacent magnets as in the Figure 4-25.

Figure 4-25: The five small magnets are placed in an aluminum cylinder to constitute a long magnet

With the new configuration, the magnetic field strength and homogeneity need to be re- The Figure 4-28 shows the variation of magnetic field in XOZ plane. In the region of 5 x 6.4mm, the homogeneity is 230ppm while the case of long magnets is 200ppm. The magnetic field distribution is found to be similar shape to the Figure 4-4. The Figure 4-29 shows the dependence of magnetic field homogeneity on the gap esp. In this case, the optimal gap is also 0.9mm as displayed in Figure 4

After shimming

In this case, we also use eight small shim magnets with the same characteristics as the shim magnets used in the configuration with the long magnets. This configuration with shim magnets is shown on the Figure 4 The homogeneity of magnetic field in the volume of 7 x 8 x 20 mm 3 after shimming is 254ppm while the correspondent value of configuration with long magnets is 230ppm. The Figure 4-32 shows the distribution of magnetic field in this volume.

Evaluation of the magnetic force between the magnets

Because the long magnets are divided into five small magnets, the magnetic force of these magnets causes the difficulty for the assembly process. Therefore, it is necessary to determine the magnitude and direction of this force. The long magnet with 8mm of diameter and length of 50mm divided into five magnets with length of 10mm for each one is modeled by ANSYS (Figure 4-33). The magnetic direction of these magnets is assigned in Y direction.

Figure 4-33: Long magnet is divided into five magnets

The result of simulation shows that the magnitude of magnetic force distribution in the magnets is around 0.03kN. The magnetic force direction of magnets turns about the longitudinal of magnets (Z direction) as shown in the Figure 4

Conclusion

The study presented herein depicts two methods of simulation and the main measurements results of a light weight NMR portable Halbach type. We described the optimization process of this permanent magnet designed with two rings of 12 magnets each one, that provide a magnetic field B 0 around 0.1 T. The simulation results have been published in [START_REF] Poulichet | Simulation and optimisation of homogeneous permanent magnet for portable NMR applications[END_REF]. The study describes the used based on the RADIA software process, for calculate and simulate the magnetic field B 0 and its homogeneity. We verified also those results with the finite element software ANSYS multiphysics. The obtained results with the two softwares are in good agreement. Based on the software analysis, we simulated the homogeneity of magnetic field and optimized the gap esp of the two consecutive rings to increase the size of the homogeneous region. The optimum gap length is around esp= 0.8 mm. The measurement of the magnetic field profile for different values of the gap esp between the two rings give a similar value of the optimal gap.

To compensate for the magnetic field inhomogeneity caused by the errors of fabrication process and the dispersion of the magnetic properties of the magnets, we used eight small shim magnets placed at the center of the device. By optimizing their position, the homogeneity had been significantly improved. The results of optimization show that the homogeneity for a given volume (7 x 8 x 20 mm 3 ) is improved 18 times in comparison to the same configuration without shim magnets. Thus the value of the homogeneity decreases from 4320 ppm to 230 ppm.

For a given volume of 6 x 7 x 20 mm 3 , the measurement of the magnetic field variation,

shows the same homogeneity improvement, using the shim magnets. Thus the homogeneity is of 1335 ppm while it was of 4415 ppm for the case without shim magnets. The magnetic field homogeneity was enhanced of a 3.3 factor. However, there is still a difference between the simulation and the measurement, which could be explained by the poor quality of the magnets. For each used magnets for the NMR device design, the magnetic field on the tip of the cylindrical magnet and the misalignment angle of the radial magnetic field were measured. The misalignment angle could be as high as 17 degrees. The simulations with some misalignment angle error of 3 degree on four magnets were performed and the same shape of the magnetic field distribution was obtained. Thus we attribute the difference between the simulation and the measurement to the misalignment angle of the magnets.

We also propose a solution to correct the misalignment of magnetic directions. The long magnets are divided into five magnets placed in an aluminum cylinder. These five magnets can rotate in the cylinder so that their magnetic directions are parallel to each other. The magnetic field strength and homogeneity have been re-simulated. The results of simulation show that the magnetic field strength does not change. The magnetic field homogeneity slightly changes in the same volume of 7 x 8 x 20mm 3 with 230ppm and 254ppm respectively. The difference is 10.4%. It can be accepted. The magnetic force of magnets for the assembly process has been simulated. The magnitude of magnetic force is determined approximately 0.03N and its direction turns about the longitudinal of the magnets. Despite these results, there's a good agreement between the simulation results and the measurement.

Introduction

In 1979, Klaus Halbach proposed a novel type of permanent magnet [START_REF] Halbach | Design of permanent multipole magnets with oriented rare earth cobalt material[END_REF] constructed entirely by rare earth permanent magnet material. The initial purpose of this type called Halbach ring was not used for the NMR applications. The Halbach ring reinforces the field in its center and cancels the field outside of its structure. This structure provides a generous volume for sample positioning (large bore/magnet size ration) and generates the field in transverse plane which allows the use of sensitive solenoid radio frequency coil to detect the NMR signal. Due to these prominent features, it is widely used for the NMR applications. However, the ideal Halbach ring is a continuously polarized material as shown in the Figure 5-1-a. It is hard to polarize exactly the direction of magnetization of material in practice in comparison to theory. In order to overcome this difficulty, the Halbach ring is divided into the same magnet blocks or replaced by the identical magnets in number n. To achieve the highest homogeneity of magnetic field in the center, the angle between each block is and the magnetizing direction of the next adjacent magnet is set by an angle as the principle of Halbach ring.

Based on this idea, the Halbach structure with discrete magnets abbreviated Mandhalas (Magnet Arrangement for Novel Discrete Halbach Layout) configuration (Figure 5-1-b, Figure 5-1c) was proposed by Raich and Blümler [START_REF] Raich | Design and construction of a dipolar Halbach array with a homogeneous field from identical bar magnets: NMR Mandhalas[END_REF]. It is based on an arrangement of identical bar magnets, described by the analytical equations reported in literature [START_REF] Soltner | Dipolar Halbach magnet stacks made from identically shaped permanent magnets for magnetic resonance[END_REF]. These magnets arranged on a restricted space on a circle to obtain the strongest magnetic field and the highest homogeneity in the center. There are some prototypes of Mandhalas have been published in different magnet shapes such as squares [START_REF] Raich | Design and construction of a dipolar Halbach array with a homogeneous field from identical bar magnets: NMR Mandhalas[END_REF][109], polygons [79][80]. These types of configuration are not only inherits the advantages of Halbalch ring but are also easy to assembly, cheap to produce because they are constructed from identical bar magnets. The Table 5-2 shows the summarization of 2D simulation results. The 2D simulation results obtained from Mandhalas made by square magnets are quite similar with those of reported by Blümer et al [START_REF] Raich | Design and construction of a dipolar Halbach array with a homogeneous field from identical bar magnets: NMR Mandhalas[END_REF] * : The homogeneity is calculated in the region of radius 25mm.

The weights of Mandhalas are calculated by formula 5-3:

. .

(5-3)

Where:

-V is the volume of magnet (the length L=300mm).

-m is the mass density (mass density of material 3 cm g 7,52   ) -I is the number of magnet.

To compare the performance of Mandhalas in circle and square magnets, three coefficients proposed by H. Raich and P. Blümler [START_REF] Raich | Design and construction of a dipolar Halbach array with a homogeneous field from identical bar magnets: NMR Mandhalas[END_REF] are defined. Then, the field strength related to homogeneity, mass and NMR sensitivity are calculated by formulas 5-4, 5-5 and 5-6 respectively.

∆

(5-4) ∆ .

(5-5) ∆ .

∆ .

(5-6)

Where:

-B is the mean magnetic field strength value in ROI.

-B  is the homogeneity in ROI.

- is the gyromagnetic ratio.

-A is total areas of magnets.

Based on these factors, the Mandhalas made from circle magnets is better than that made from square magnets, except the case of n = 4. Although the case of n = 32 have a good homogeneity and light weight, its static magnetic field is quite small. On the other hand, n = 4 achieves a high magnetic field, its homogeneity and weight are unacceptable. Thus, the configurations of n = 16 cylindrical magnets were chosen for building the prototype. The In theory, the magnetic field generated by an infinitely long magnet (2D) is perfectly homogeneous. However, in the 3D case, the field homogeneity for the finite-height magnet array is much smaller, apart from the fact that the absolute values are reduced by a factor of about 8 compared to the 2D case [START_REF] Soltner | Dipolar Halbach magnet stacks made from identically shaped permanent magnets for magnetic resonance[END_REF]. Therefore, to fully evaluate the properties of configuration, 3D With the gap G = 1mm, the field variation is stable in the distance of nearly 50 mm along configuration and 40 mm in OY direction. This allows us to take into account the chosen sensitive volume of 40mm diameter and 50 mm length for calculation of magnetic field and homogeneity.

In this region, the homogeneity is 638 ppm and the mean value of field strength is 0.32 T. The

Improvement of magnetic field homogeneity

Although adjustment of the center gap is optimized in order to improve the homogeneity in the longitudinal axis, the homogeneity of magnetic field is not sufficient for high resolution NMR experiments. There are other affords to increase the homogeneity of magnetic field. The first method is that the Mandhalas are divided into many rings. These rings are stacked onto each other.

By optimizing the gaps between these rings, the homogeneity of magnetic field in the longitudinal axis of configuration becomes more homogeneous [START_REF] Soltner | Dipolar Halbach magnet stacks made from identically shaped permanent magnets for magnetic resonance[END_REF]. However, this approach is too cumbersome. The second method is the use of shim magnets [77][81]. This approach is easier, faster, robust and more practical. The results of simulation show that the magnetic field is homogeneous at the center of the configuration and becomes worse toward at the end of configuration as in the Figure 5-12. In order to compensate these distortions, we use the shimming methods described in [77][81]. Two shim rings are placed in the bore of configuration. Each ring consists of eight small magnets as shown in Figure 5-13. "By setting the polarization of the shim unit opposite to that of the main field, the inhomogeneities of the last are corrected while the total field strength is maintained at an acceptable magnitude" [START_REF] Danieli | Mobile sensor for high resolution NMR spectroscopy and imaging[END_REF]. Two shim rings are arranged on the circle with a radius of 40 mm. The gap between R outer of the shim rings and R inner of the main rings is 7 mm. This space is used for the cover of them. The geometric parameters with the shim magnets are displayed on Table 5-3. There are two variables that need to be optimized which are G and d. In order to keep the usable bore diameter as large as possible, the positions of shim magnets are optimized in one direction. Another reason is that this prototype consumes more magnetic material In the longitudinal axis, the homogeneous region extends over 2.5 times in comparison to that without shim magnets. This is confirmed by the magnetic field profile and field maps in the

Prototype design

We choose the commercial magnets made from NdFeB with coercively H c =870kA/m.

These magnets have the dimensions 24 mm for the diameter, 150mm for the length (2x16 magnets) arranged on a circle with radius of 62 mm.

This configuration consists of two rings separated by the gap of 0.5 mm shown on the nce agin [START_REF] Moresi | Miniature permanent magnet for table-top NMR[END_REF] mler [START_REF] Raich | Design and construction of a dipolar Halbach array with a homogeneous field from identical bar magnets: NMR Mandhalas[END_REF] t al [START_REF] Jachmann | Multipole shimming of permanent magnets using harmonic corrector rings[END_REF] al [START_REF] Danieli | Mobile sensor for high resolution NMR spectroscopy and imaging[END_REF] al [START_REF] Hugon | Design of arbitrarily homogeneous permanent magnet systems for NMR and MRI: Theory and experimental developments of a simple portable magnet[END_REF] l [START_REF] Windt | A portable Halbach magnet that can be opened and closed without force: The NMR-CUFF[END_REF] chapter 4 chapter 5

Table 5- Based on the results of 3D simulation, we propose the shim method to improve the homogeneity of magnetic field. The results of shim method show that the magnetic field homogeneity significantly improves. In the volume of 40 mm diameter and 50 mm length, the homogeneity of the configuration with shim magnets is 3.5 time better than that without shim magnets with 178 ppm and 638 ppm respectively. These impressive results motivate us design the prototype with two main rings and two shim rings. It has the length of 300 mm and 70 mm diameter in use separated by a gap of 0.5 mm. The total weight is approximate 20 kg. This mobile magnet generates magnetic field of 0.32 T and offers the homogeneity in the large sensitive volume 40 mm diameters, 50 mm length with 178 ppm. It is suitable for our NMR experiments on Biomedical and Agro alimentary applications.

Conclusion and perspectives

Based on the objective of application and the examination of previous works (chapter I), we have decided to choose the in-situ magnet for our applications. Our configurations are based on the Halbach type due to its homogeneity and ability to confine the field inside it. Before the simulation, we have figured out the properties of different families of magnetic materials. The advantages and drawbacks of magnetic materials are analyzed in order to give a comparison.

Based on these analyses, we found that the family of neodymium iron boron magnets (NdFeB) is suitable for our works. This family of magnetic material has similar properties with another rareearth magnetic material samarium cobalt (S m C o ) while 30% to 40% cheaper than [START_REF] Cédric | Aimants permanents pour la RMN et l'IRM[END_REF]. The numerical and simulation methods of the existing ideas also have been examined to find the best solution for our simulation and optimization. The finite element method which is the underlying procedure of many software packages was chosen. The process of simulation based on ANSYS software has been introduced in order to show progresses to be done.

The calculation and simulation are mainly based on ANSYS. The results of ANSYS will be verified to that of RADIA software. They are two different methods. The principle calculation of ANSYS is based on the finite element method while RADIA is based on boundary integral methods. The results of verification are impressive in both homogeneity and optimization.

However, the magnetic field strength is quite different between two softwares due to the meshing.

Another software used in our works is Matlab. This software is used to calculate the homogeneity with the mesh nodes turned out from ANSYS. Matlab is also used to plot and visualize the measurable field points. The measurement is carried on with the gaussmeter Hirst GM08 with the limit sensibility of 10 -4 T. The micropositioner Signatone S-926 is used to control the movement of the probe in three dimensions.

Based on the simulation results, we realized a compact and light weight prototype with 24 identical rod magnets. Our prototype has two rings; each ring consists of 12 magnets. Each magnet has the length of 50mm and 8mm in diameter. The prototype generates the field B 0 in transverse plane with the field strength 0.12T. Its homogeneity is 4230ppm over a volume 7x8x20mm. It is obvious that this homogeneity is not enough for high resolution. In order to improve the homogeneity, we proposed the shim method with 8 small magnets placed in the bore of the prototype. By optimizing the positions of these magnets, we achieve the optimum configuration.

The results of optimization show the great improvement of homogeneity. The homogeneity achieved after shim is 18 times better than that of before shim with 230ppm in comparison to 4230ppm over a same volume 7x8x20mm. The measurement was carried out to verify the result of simulation. The magnetic field which varies in the longitudinal axis obtained by measurement is quite similar to the results of simulation. However, the homogeneity obtained by simulation and measurement is different due to the imperfection of the material.

We also proposed another configuration for biomedical and agroalimentary applications. This configuration is based on the idea of NMR Mandhalas (Magnetic Arrangement for Novel Discrete Halbach Layout) [START_REF] Raich | Design and construction of a dipolar Halbach array with a homogeneous field from identical bar magnets: NMR Mandhalas[END_REF] [START_REF] Soltner | Dipolar Halbach magnet stacks made from identically shaped permanent magnets for magnetic resonance[END_REF]. In order to choose the best configuration, two Halbach permanent magnet arrays with cube and cylindrical shape, respectively, were compared in criteria of average magnetic field strength, magnetic field homogeneity and material consumption. This comparison was performed by 2-D simulation with ANSYS software. It was found that the cylindrical shape magnet has better performance than the cube one; except for the case with n = 4.

Based on the results of 2-D simulation, we chose the NMR Mandhalas with 16 cylidrical magnet for our design. 3-D simulation was carried out in order to fully evaluate the properties of this configuration. Then the field homogeneity optimization was performed by using the passive shim method. After optimization, the magnetic field homogeneity has been significantly improved in comparison to the case before using the shim magnets. The difficulties of assembly process have also been discussed and well compared to the previous works. These results laid a foundation for the design and manufacture of the NMR Mandhalas with cylindrical magnets.

Future work will be focused on two main parts

The first part is concentrated the correction of the misalignment of magnetic direction mentioned in chapter 4. We bought the magnets from "HKCM MAGNTES from STOCK"[107].

We will assembly the magnets and measure the magnetic field strength and homogeneity. The NMR experiments will be applied after this step.

The second part is machining and assembling accuracy the NMR Mandhalas to achieve high performance. Then this configuration will be used to carry out the NMR experiments in biomedical and agroalimentary field. Along with the micro antennas developed by our group, we can envision a wide development of our magnet systems for biomedical and agroalimentary application. Emails: hung@creatis-insa-lyon.fr, p.poulichet@esiee.fr, latifa.bouchet@univ-lyon1.fr In recent years, NMR/MRI portable devices [1][2] have drawn attention of numerous researcher teams. They are used for variety of applications, from medical diagnosis [3] to archaeological analysis [4], nondestructive material testing [5], evaluation of water presence in building materials [6] and food emulsions [7]. Different magnets designs have been proposed by many groups of researchers. They can be divided into two groups: the magnets ex-situ [8][9] and the magnets insitu [10][11]. The first group has the simple configuration with the sensitive volume near their surface and the samples under test are located outside the magnets. Thus, they can be used for the experimental investigation of objects with unlimited dimensions. Although the ex-situ magnets have simple shape and are light weight, they are difficult to achieve in terms of homogeneity of the magnetic field in the sensitive volume.

T.

Abstract-A light weight, simple design NMR apparatus consists of 24 identical magnets arranged in Halbach array was designed and built. The homogeneity of the magnetic field B 0 can be improved by dividing a long magnets into several rings. The size of the useful volume depends on both the gap between each ring and some others shim magnets. Our aim is to enhance the sensitive volume and to maintain the highest magnetic static field (B 0 ). This apparatus generates a B 0 field strength of about 0.1 T. This work focuses on the magneto-static simulation of NdFeB magnets arrangement and on the comparison with the measurement of the magnetic field strength and homogeneity in three dimensions (3D). The homogeneity of the magnetic field B 0 is optimized with the help of CAD and mathematical software. Our results were also validated with a Finite Element Method (FEM). The simulation results of the strength and of the homogeneity of B

In comparison, the in-situ magnets have their static field reinforced inside their bore center and canceled outside of the structure. Thus, their magnetic field is homogeneous inside the structure.

The in-situ magnets use Halbach [11] or Aubert Configurations [17].

Starting with the proposition of Klaus Halbach in 1980 [12], the Halbach ring consists of segments of permanent magnets put together in an array. This creates a homogeneous field in the transverse plane. Based on this principle, the Halbach structure with discrete magnets for portable NMR magnet known as NMR Mandhalas was given in 2004 by Raich and Blümler [13]. It is based on an arrangement of identical bar magnets, described by the analytical equations reported in literature [14]. This concept has been widely used for building prototypes due to their easy assembly and the accessibility of their region of interest. The homogeneity of Halbach type is poor compared to traditional magnets [15]. For measurement of the relaxation times T2 and T1 or the spectrum, the inhomogeneity should not be higher than 10 ppm. To insure the sufficient field homogeneity for NMR experiments, a popular method is to add shimming magnets. The concept of movable permanent magnets in the shim unit of a Halbach array was reported by Ernesto Danieli et al [16]. Another method of shimming, based on the spherical harmonic expansion, proposes a complete procedure for permanent magnet design, fabrication, and characterization [17][18]. The advantages of Halbach structure motivated us to choose it for building our prototype.

However, increasing homogeneity while maintaining high field strength is a challenge when building NMR portable devices. In this study, we propose a light weight magnet system for NMR applications. Such system consists of two rings of 12 magnets arranged in a Halbach configuration. Its homogeneity and its magnetic field strength B 0 are simulated and calculated by RADIA and Mathematica software, and confirmed by Finite Element software 'ANSYS multiphysics'. In order to improve its homogeneity, we used eight small shim magnets placed inside its bore. By optimizing the position of these magnets, we have reached a configuration with a significant increase in the homogeneous region. Based on the results of simulations, we designed and built a prototype. The magnetic field strength and homogeneity of our prototype were also measured by a digital gaussmeter, and then compared to those obtained by simulation.

Comparison shows that homogeneity in the longitudinal axis of apparatus and field strength B 0 are similar. However, the homogeneity in transverse plane differs from results of simulation and measurement. One explanation could be the real characteristics of the used magnets and their quality. This difference has been also discussed in this study.

II. MATERIALS AND METHODS

In most of the Halbach configurations, the static field B 0 is transverse to the cylindrical axis as shown in Figure 6-1. The direction of magnetization of each magnet is defined by two angles α i and β i . The i th magnet is placed on a circle at an angle α i as

2 .i i n   
and its magnetization is defined by an angle β i as β i = 2.α i . Where n is the number of magnets (i = 0, 1, 2… n-1). Our configuration has 12 magnets placed on a circle of radius r0 = 30 mm. As shown in Figure 6-2-a, each magnet is placed at an angle

6 i i   
and its magnetization is rotated by an angle The geometric parameters are displayed in Table 6-1. To calculate the magnetic field of the magnet configuration, RADIA [19] and ANSYS [20] softwares were used. RADIA was developed to design the Insertion Devices for Synchrotron light sources. It uses boundary integral methods. Each volume created to represent the magnets is subdivided in a number of sub-elements to solve the general problem of magnetization. The solution is performed by building a large matrix with represents mutual interactions between the objects.

ANSYS is a multiphysics software using FEM modeling. Each volume is divided with subelements. Even the air between and around the magnets has to be meshed. The flux conditions have to be placed outside the global volume in order to apply parallel or normal condition. Usually, boundary integral method is considered faster than FEM. In our case, these two complementary methods are implemented: RADIA allowing a faster simulation, is used for optimization and ANSYS is used for verification and validation of results. The size of the meshing is reduced until the simulation results do not change.

The properties of the material modeled magnets during simulation were chosen to represent magnets from "HKCM MAGNETS from STOCK" [21]. The magnet material is Neodymium NdFeB with the following characteristics:

 a saturation magnetization of 1.37 T,  a coercivity Hc = 1000 kA/m,  the diameter and the length of the magnets are respectively 8 mm and 50 mm,  the maximal operating temperature is 120 °C,  the temperature coefficient is 0.11 %.°C -1 ,  and the magnetization is oriented along the diameter.

In order to calculate the homogeneity, the values for the magnetic field are selected in the homogeneous region and treated by Matlab Software. The homogeneity is calculated by the formula (1).

  6

10

n i i Abs B B B Homogeneity m    (1)
Where,  Bi is the value of magnetic field at the i th position in the homogeneous region,  B is the magnetic field value at the center of the homogeneous region,  m is the number of mesh nodes in the homogeneous region.

III. SIMULATION RESULTS

a. Optimization of the gap between two rings without shim results

Our NMR portable magnet model is constituted with 24 magnets, placed as displayed in Figure 6-7 and used for the simulations with ANSYS and RADIA. Considering the magnetic field B 0 oriented along Ox axis and the gap between two rings esp = 0, the maximum value of B 0 calculated with RADIA is about of 0.103 T and of 0.11 T obtained with ANSYS Analysis. The difference of calculation between RADIA and ANSYS is about 6.79 %. This difference can be accounted for by the problem of mesh size convergence. It means that the results obtained with fine meshes are higher than those obtained with coarse meshes. Furthermore, the difference is acceptable. The variation of profile of the magnetic field Bx along Oz axis depends on the esp gap between the two rings. To optimize this gap, we increased the value of esp by steps of 0.1mm. Figure 6-5 shows Bx profile for four values of esp. When esp = 0, the magnetic field outside one ring does not compensate exactly the one of the other ring. For esp = 0.9 mm, the compensation is optimum and the magnetic field at the center is almost constant. The "useful" volume for NMR sample is determined from the coordinates (x,y,z) of the point where Bx is maximal. Then, the volume is calculated with the coordinates (x,y,z) that generate a variation of 0 0 B B  not higher than 100 ppm. The Figure 6-6 shows that the volume of the homogeneous region is a function of the esp. The optimal value of esp determined by RADIA is around 0.77 mm and the "useful" volume is about 2640 mm 3 . When the spacing esp between the two rings is optimized, the "useful" volume is increased by a ratio of around 80. This is caused by the decrease of the magnetic field outside one ring, which is similar to the increase of the other ring. There's an optimum gap between the two rings where the sum of the variations of the magnetic field outside the rings are canceled. Although the magnetic field homogeneity increases by adjusting the gap esp between the two rings, the inhomogeneity of magnetic field also comes from magnetic material (dispersion of both the value and the orientation of the magnetization), from errors in fabrication process and positioning of the array magnets. These factors cannot be corrected only by adjustment of esp. To overcome these difficulties, the shim magnets are considered as a way to compensate for the inhomogeneity of the magnetic field [16][17] [18]. In our case, we use eight small magnets placed inside the bore of the two rings as shown in the Figure 7. The direction of magnetization of the shim magnets is defined as shown in Figure 8. There are three variables that need to be optimized: esp, r1 and dH. The optimization objective is to determine the values for esp, r1 and dH that maximize the volume for an inhomogeneity of 100 ppm. The flow chart shown in Figure 9 describes the optimization process implemented with Mathematica software and the calculation of the magnetic field with RADIA software. To avoid the superposition of the main magnets and the shim magnets, we set the range of r1 from 15 to 23 mm and the one for esp ranging from 0.1 to 0.6 mm. The optimal value for esp, considered here, is different from the value considered before because of the presence of shim magnets.

Each step of increase of r1 is 1 mm while correspondent value of esp is 0.1 mm. Each possible values of r1 is placed in a matrix. The corresponding magnetic field and then the three coordinates (x,y,z) for an homogeneity lower than 100 ppm are also determined. For each value of r1, we have a value for the homogeneous volume. The value of r1 leading to the highest value of the volume will be saved. The same process is repeated with the others parameters esp and dH. After a variation of one parameter, the variation is refined around the best value previously obtained. It's very important to choose good initial conditions and started the variation of one parameter with reliable value for the others parameters. This method was preferred to the use of Mathematica software optimization functions, as FindMaximum. The optimized parameters are presented in the Table 6-2. The optimizations results allow a great improvement of homogeneity, as it can be seen in Figure 6-10. It shows that the inhomogeneity of the magnetic field calculated in a 7 x 8 mm² region is 90 ppm after shimming while the value before shimming was 370 ppm.

The magnetic field inhomogeneity calculated by RADIA and by ANSYS is in good agreement. However, ANSYS gives always smaller useful volumes than those obtained by RADIA due to the method of calculation. This can be explained by the fact that RADIA result is the highest value at the region edge while ANSYS compute the mean value for the overall region. As shown in Figure 6-13, the inhomogeneity of magnetic field in a volume of 7 x 8 x 20 mm 3 is respectively 4320 ppm without shim magnets and of 230 ppm with the shim magnets. Each shim magnet shown on the Figure 6-14 is glued in a nonmagnetic cylinder. These cylinders can rotate, move along the longitudinal axis and slide along the radius of the prototype to find the optimal position of the shim magnets. These cylinders are placed in the holes of sliding-blocks moving on the four apertures of an aluminum frame concentric with the prototype.

b. Experimental setup

The magnetic field is measured by the digital gaussmeter Hirst GM08 with sensitivity limit of 10 V. EXPERIMENTAL RESULTS

a. Measurements before shimming

To optimize the gap esp between two rings, at the beginning, it was set at 0 mm, and was progressively increased by turning the screws on the frame of the device (Figure 6-15). The magnetic field distribution is similar to the simulation as shown in the Figure 6-4. In the region of 7 x 20 mm 2 (xOz plane), the homogeneity calculated by formula ( 1) is 894 ppm

VI. DISCUSSION

The difference between the simulated and measured values of the magnetic field homogeneity is due to the poor quality of the magnets. There is a large dispersion of the magnets properties. For our prototype we selected 24 magnets among 27 having a similar magnetic field strength and the magnets 5, 22 and 23 was rejected. This was done with the measurement of the magnetic field on each tip of the magnet cylinder. Table 6-3 shows the value of B 1 and B 2 for 27 magnets and the misalignment angle α. B 1 and B 2 are respectively the magnetic field in the vicinity of the two faces of the magnet cylinder as shown on the Figure 21. The average value of the magnetic field is around 27.3 mT. The misalignment angle represents the error of orientation of the radial direction of the magnetic field on each face. In ideal case, this angle value is zero but for some magnets this value can reach 17 degrees and the consequence, is an error of homogeneity of the magnetic field.

VII. CONCLUSION

The study presented here depicts two methods of simulation and the main measurement results of a light weight NMR portable Halbach type magnet. We described the optimization process of this permanent magnet designed with two rings of 12 magnets each one, that provide a magnetic field B 0 around 0.1 T. The simulation results have been published in [22]. The study describes the used based on the RADIA software process, for calculate and simulate the magnetic field B 0 and its homogeneity. We verified also those results with the finite element software ANSYS multiphysics. The obtained results with the two softwares are in good agreement. Based on the software analysis, we simulated the homogeneity of magnetic field and optimized the gap esp of the two consecutive rings to increase the size of the homogeneous region. The optimum gap length is around esp= 0.8 mm. The measurement of the magnetic field profile for different values of the gap esp between the two rings give a similar value of the optimal gap.

To compensate for the magnetic field inhomogeneity caused by the errors of fabrication process and the dispersion of the magnetic properties of the magnets, we used eight small shim magnets placed at the center of the device. By optimizing their position, the homogeneity had been significantly improved. The results of optimization show that the homogeneity for a given volume (7 x 8 x 20 mm 3 ) is improved 18 times in comparison to the same configuration without shim magnets. Thus the value of the homogeneity decreases from 4320 ppm to 230 ppm.

For a given volume of 6 x 7 x 20 mm 3 , the measurement of the magnetic field variation, shows the same homogeneity improvement, using the shim magnets. Thus the homogeneity is of 1335 ppm while it was of 4415 ppm for the case without shim magnets. The magnetic field homogeneity was enhanced of a 3.3 factor. However, there is still a difference between the simulation and the measurement, which could be explained by the poor quality of the magnets. For each used magnets for the NMR device design, the magnetic field on the tip of the cylindrical magnet and the misalignment angle of the radial magnetic field were measured. The misalignment angle could be as high as 17 degrees. The simulations with some misalignment angle error of 3 degree on four magnets were performed and the same shape of the magnetic field distribution was obtained. Thus we attribute the difference between the simulation and the measurement to the misalignment angle of the magnets. Despite these results, there's a good agreement between the simulation results and the measurement.

The RADIA software

The RADIA software package was designed for solving physical and technical problems The solution is performed by building a large matrix in memory which represents mutual interactions between the objects. We call this an Interaction Matrix. The final magnetization in each small object is obtained iteratively, by a sequence of multiplications of the Interaction Matrix by instant magnetizations vector, taking into account the material properties. We call this a Relaxation procedure.

Field source

Magnetic field strength

Field intergral Force Torque Vector potential

Chapitre 1 :

 1 Introduction à la RMN Nous présentons dans ce premier chapitre les bases nécessaires pour comprendre les points clés de l'expérience de RMN. Depuis sa découverte en 1945, la résonance magnétique nucléaire (RMN) est devenue un outil précieux pour la physique, la chimie, la biologie et la médecine. La RMN se décline sous forme de deux techniques d'investigation non-invasives spectroscopie de résonance magnétique nucléaire (SRM) et l'imagerie par résonance magnétique (IRM). Ces techniques exploitent le fait que certains noyaux atomiques possèdent un moment cinétique intrinsèque « le spin ». Placés dans un champ magnétique statique B 0 , les moments magnétiques associés aux spins s'orientent parallèlement et antiparallèlement au champ B 0 selon deux niveaux d'énergie tout en maintenant leurs mouvements de précession. Le phénomène de la RMN consiste à faire passer les spins d'un niveau d'énergie à l'autre. Pour ce faire on a recours à une impulsion de champ radiofréquence B 1 , un champ électromagnétique qui oscille à la même fréquence de précession des spins. Les noyaux absorbent l'énergie radiofréquence qu'ils restituent sous forme d'onde donnant naissance, selon le cas à un spectre ou une image RMN dont les caractéristiques dépendent du type de noyau et de son environnement chimique.

Chapitre 5 :

 5 Modélisation et conception de la configuration Mandhalas pour des applications biomédicales et agro-alimentaire.

Chapter V :

 : In this chapter, we propose a configuration based on the Halbach structure with discrete magnets abbreviated Mandhalas (Magnet Arrangement for Novel Discrete Halbach Layout) configuration. The properties of Mandhalas made respectively from cube and cylindrical magnets are compared by 2D simulation. Taking into account criterias as magnetic field strength, homogeneity and mass, we have chosen the Mandhalas with cylindrical magnet for modeling and 3D simulation.

  Since its discovery in 1945, the Nuclear Magnetic Resonance (NMR) has become a valuable tool in physics, chemistry, biology and medicine. The Nuclear Magnetic Resonance (MRS) and the magnetic resonance imaging (MRI) are non-invasive techniques. Both of them are based on the principle of NMR. These techniques are complementary and are used in the characterization of tissue, molecules..etc. The MRI is a medical imaging technique used to investigate inside the human body by producing high quality images. In 1952, one dimensional MRI image was reported by Herman

  H or13 C. Since NMR experiment was first described by Rabi et al[3] in 1938, the field of NMR has drawn much attention of researchers. This technique has been expanded for the use on liquids and solids by Felix Bloch and Edward Mills Purcell, for which they shared the Nobel Prize in Physics in 1952. NMR spectroscopy can analyze the chemical solutions and determine the structure, environments of various molecules. It thus becomes the standard technique in chemical analysis of materials, elucidation of protein structures, drug research and production control, …etc.

  Figure 1

Figure 1

 1 Figure 1-2 resulting in

Figure 1

 1 Figure 1-8.

Figure 1 - 7 :

 17 Figure 1-7: The current is plotted as a function of time and decay with the time T 2 * due to the dephasing of spins. Then, the FID is converted into spectrum by Fourier transform [6].

  Figure decay; sequen

  Figure[START_REF] Blümich | Mobile single-sided NMR[END_REF].

Figure 2

 2 Figure 2-8: Th

  Figur

Figure 2

 2 Figure 2-10 iron pole ca

Figure 2

 2 Figure 2-18: 2.3.2 In-While the

  Fig

Figure 2

 2 Figure 2-2 (a) The fiel and outer r The Halbac structure. T

  Figure 2

Figure 2

 2 Figure 2-2

  Figure 2-2

  Fig

Figure 2

 2 Figure 2-31 erest before

  Figur

  Figure 3-2

  Figure 3-2-b

4 g

 4 Fig

Figure 3

 3 Figure 3-5.

Figure 3 - 5 :

 35 Figure 3-5: Definition of parameters in equation 3-16.

  Fig

  Halbach type magnet has been developed extensively for use in a variety of applications because of their homogeneity and confinement of magnetic field inside the bore. Starting with the proposition of Klaus Halbach in 1980[START_REF] Halbach | Design of permanent multipole magnets with oriented rare earth cobalt material[END_REF], the Halbach ring consists of segments of permanent magnets joined together in an array to create the homogeneous field in transverse plane. This allows using the solenoid coil for NMR experiments. However, its assembly and polarization of magnets are the problems which can be affected to their homogeneity and performances. Based on the principle of Halbach ring, the Halbach structure with discrete magnets for NMR portable known as NMR Mandhalas was given by Raich and Blümler[START_REF] Raich | Design and construction of a dipolar Halbach array with a homogeneous field from identical bar magnets: NMR Mandhalas[END_REF]. It is based on an arrangement of identical bar magnets, described by the analytical equations reported in literature[START_REF] Soltner | Dipolar Halbach magnet stacks made from identically shaped permanent magnets for magnetic resonance[END_REF]. This concept has been widely used for building prototypes because they are easy to assembly and their region of interest is easy to access. Despite the lowest homogeneity of Halbach type, it still better in comparison to ex situ magnets, but poor compared to magnet volume. For measurement of the relaxation time T 2 and T 1 or the spectrum, the fraction of inhomogeneity 1 and T 2 represent the relaxation of the nuclear spin magnetization, respectively parallel

Figure 4 - 1 :

 41 Figure 4-1: The position of magnets and their directions of magnetization.

F 4

 4 Figure 4-2:

-

  a saturation magnetization of 1.37 T -a coercively of H c = 1000 kA/m -The diameter and the length of the magnet are respectively 8 mm and 50 mm -The magnetization is oriented along the diameter -The maximal operating temperature is 120 °C -And the temperature coefficient is 0.11 %.°C -1 .



  -3-b represents the distribution of magnetic field in the XOY plane.Each shades of color represent a variation of the inhomogeneity is larger than 450 ppm (RADIA) while the correspondent value determined by ANSYS is 380 ppm. For an inhomogeneity lower than 100 ppm, the expected volume for experiment is 3 x 3 x 3 mm 3 .The variation of magnetic field B x on the XOZ plane is shown on the Figure4-4. The homogeneity of magnetic field is 300ppm in the region of 5 x 6.4mm as determined by RADIA, while ANSYS gives a result of 200ppm. The Figure4-4 shows the similarity of magnetic field distribution obtained by ANSYS and RADIA. However, the homogeneity calculated by ANSYS is always smaller than those of RADIA because of the method of calculation. It can be explained that the RADIA determines the homogeneity on the different region of the plane while ANSYS calculates the mean value of magnetic field on the whole plane.

Figure 4 - 3 :

 43 Figure 4-3: The magnetic field Bx distribution in the XOY plane (at Z=0). (a) the variation of magnetic field B x versus Ox axis. (b) the inhomogenity of the magnetic field Bx versus X and Y.

Figure 4 - 4 :

 44 Figure 4-4: Magnetic Field Bx distribution in the region of 5 x 6.4 mm in xOz plane. (a) the variation of magnetic field B x simulated by ANSYS. (b) the variation of magnetic field B x simulated by RADIA.

Figure 4 - 5 :

 45 Figure 4-5: The variation of magnetic field in Z direction at different gaps.

Figure 4 - 6 :Figure 4 - 7 :

 4647 Figure 4-6: Volume of homogeneous region is a function of the gap esp.

Figure 4 - 8 :

 48 Figure 4-8: Direction of magnetization of the shim magnets.There are three variables that need to be optimized: esp, r1 and dH. The optimization objective is to determine the values for esp, r1 and dH that maximized the volume for a given inhomogeneity, in our case 100 ppm. The flow chart shown in Figure4-9 describes the optimization process implemented with the software Mathematica for the algorithm of the optimization and RADIA for the calculus of the magnetic field. To avoid the superposition of the main magnets and the shim magnets, we set the range of r1 from 15 to 23 mm and esp ranging from 0.1 to 0.6 mm. The optimal value for esp considered here is different from the value considered before because the shim magnets modified the magnetic field.Each step of increase of r1 is 1 mm while correspondent value of esp is 0.1 mm. Each possible values of r1 are placed in a matrix. For all these values, the magnetic field is determined and then the three coordinates x,y,z for an homogeneity lower than 100 ppm. For each value of r1, a value of the volume is obtained. A selection of the value of r1 that correspond to the highest value of the volume is saved. The same process is repeated with esp and dH. After a variation of one parameter, the variation is refined around the best value previously obtained. It's very important to choose good initial condition and begin the variation of one parameter with plausible value for the others parameters. This method was preferred than the use of optimization function of Mathematica like FindMaximum.The optimization objective performed by RADIA is to determine the values for esp, r1 and dH that maximized the volume for a given inhomogeneity. For ANSYS, on the other hand, the

Figure 4 - 9 :

 49 Figure 4-9: Optimum flow chart of our configuration with 24 main magnets and 8 shim magnets performed by RADIA.

Figure 4 - 10 :

 410 Figure 4-10: Optimum flow chart of our configuration with 24 main magnets and 8 shim magnets performed by ANSYS.

Figure 4 - 11 .Figure 4 - 11 :Figure 4 - 13 :

 411411413  It shows that the inhomogeneity of the magnetic field calculated in the region of 7 x 8 mm is 90 ppm after shim while the value before shim is 370 ppm.The inhomogeneity of magnetic field calculated by RADIA is in good agreement with ANSYS. However, in the larger region, the results of ANSYS are always smaller than those obtained by RADIA because of the method of calculation. It can be explained that the results of RADIA is the highest value at the border of the region while that of ANSYS are the mean value for the overall region.

  -17-c where the magnetic strength is equal to 0.138 T remains constant for the distance of 10 mm. The shapes of the curves plotted on the Figure 6-16 are similar to those obtained by simulation on the Figure 4-5.

Figure 4 - 16 :Figure 4 - 17 :

 416417 Figure 4-16: Equipment is used to measure the magnetic field of the prototype.

Figure 4 - 18 :

 418 Figure 4-18: The magnetic field distribution in xOy plane obtained by measurement before shimming Figure 4-19 shows the magnetic field distribution measurement in xOz plane in the region of 7 x 20 mm 2 . In this region the homogeneity calculated by formula (4-1) is equal to 1426 ppm. This homogeneity is of 4415 ppm in the volume of 6 x 7 x 20 mm 3 .The magnetic field distribution is similar to the simulation as shown in the Figure 4-4.

Figure 4 - 19 : 4 . 5 . 2 Figure 4 -

 4194524 Figure 4-19: The measured magnetic field distribution in xOz plane before shimming 4.5.2 After shimming measurements Figure 4-20 shows the improvement of the magnetic field homogeneity in xOy plane in the same region 6 x 6.5 mm 2 . The homogeneity is respectively, 1399 ppm before shimming calculated by formula (4-1) and 817 ppm after shimming.

  Figure 4-4.

Figure 4 - 20 :

 420 Figure 4-20: Magnetic field homogeneity at z = 0 in xOy plane

Figure 4 - 21 :

 421 Figure 4-21: The magnetic field homogeneity in xOz plane

  Fig The resul ses to 2750 without mis to be simila

  Figure 4-23

Figure 4 - 24 :

 424 Figure 4-24: The magnetic field distribution in xOz plane for ideal case and with error of misalignment of the direction of the magnetic field

Figure 4 - 26 : 4 . 7 . 2 of simulation 4 . 7 . 2 . 1

 4264724721 Figure 4-26: The configuration is constituted by small magnets 4.7.2 Results of simulation

Figure 4 - 28 :

 428 Figure 4-28: The field distribution in the rectangle region of 5 x 6.4mm (XOZ plane)

Figure 4 - 29 :

 429 Figure 4-29: The change of magnetic field profile in Z direction at four distance of gap esp

  -30.

Figure 4 - 30 :Table 4 - 4 :

 43044 Figure 4-30: The configuration and its shim magnets is modeled by ANSYS. The process of optimization used in this case follows the flow chart as in the Figure 4-10. The optimal geometric parameters are presented in Table 4-4. Name of the parameters Dimensions esp 0.2 mm r1 20 mm dH 24.5 mm Table 4-4: The optimum positions of the shim magnets In the region of 7 x 8mm (XOY), the magnetic field homogeneity calculated by formula (4-1) is 100pmm. It slightly increases in comparison to 90ppm of the configuration of long magnets. In Z direction, the homogeneity significantly increases from 10mm (Figure 4-29-c) to 20mm (Figure 4-31-a). This increase is confirmed by the field map (Figure 4-31-b).

Figure 4 - 32 :

 432 Figure 4-32: Magnetic field distribution in the volume of 7 x 8 x 20 mm 3

  performed by Ansys (a) The magnetic field profile in Z direction after shimming (b) The magnetic field distribution in the region of 7 x 20mm (XOZ) after shimming

Figure 4 - 31 :

 431 Figure 4-31: The variation of magnetic field in XOZ plane

  -34.

Figure 4 - 34 :

 434 Figure 4-34: Magnetic force distribution in the magnets

  FigureMan

  Figure

Figure 5 -

 5 Figure 5ber of mesh d the magn alculated by

Figure 5 -

 5 Figure 5-5: 2-D simulation with Mandhalas is made from cube magnets (vector plot).

Figure 5 -

 5 Figure 5-6: 2-D simulation with Mandhalas is made from cylindrical magnets (vector plot).

Figure 5 -

 5 7 shows the magnetic field distribution in the region of interest of 50 mm diameter with the case of n = 8. Obviously, the magnetic field of the circle magnets (Figure5-7-a) is more homogeneous than that of square magnets (Figure5-7-b).

Figure 5 - 7 :

 57 Figure 5-7: The field distribution in the region of interest 50mm diameter with n=8.

Figure er 5

 5 Figure

Figure 5 - 9 :

 59 figuration w ngs placed in 0 mm. Thes ength of th nent magnet

Figure 5 -

 5 Figure 5-11-a shows the field map in the middle length of configuration with the size 40x50 mm in YOZ plane. The Figure 5-11-b shows the field distribution in the ROI of 40 mm diameter and 50 mm length in 3D.

Figure 5 - 10 :

 510 Figure 5-10: The variation of magnetic field at the position 150.5mm from the left or right of configuration.

( a )

 a Field distribution in the region of 40x50mm (YOZ) length. (b) The variation of magnetic field in the volume of 40mm diameter and 50mm. length

Figure 5 - 11 :

 511 Figure 5-11: Geometric parameter of Mandhalas in circle and square magnets.

  (a) Magnetic field profile in Oy axis (b) Field distribution in the region of 40 mm diameter

Figure 5 - 12 :

 512 Figure 5-12: The field distribution in YOZ plane (region of 40 x 301mm).

  Figure

4 :Figure 5 - 14 :

 4514 Figure 5-14: The flow chart describes the optimum process.

Figure 5 -

 5 Figure 5-16 and Figure 5-17 respectively. The homogeneity of magnetic field in the cylindrical volume 40 mm diameter and 50 mm length is 178 ppm while the correspondent value of the configuration without shim magnets is 638 ppm. It is approximate 3.5 times better than. Figure 5-18 shows the improvement of magnetic field homogeneity in the volume of 40 mm diameter, 90 mm length before shim and after shim.

  (a) Before shim (b) After shim

Figure 5 - 15 :

 515 Figure 5-15: The variation of magnetic field in the plane 40 mm diameter at the position 150.5 mm from the left or right of configuration (XOY).

Figure 5 - 16 :

 516 Figure5-16: The magnetic field profile in Z direction (150mm in the middle length of configuration).

  0

Figure 5 -Figure 5 -

 55 Figure 5-19-a. Each ring has 16 magnets which are covered by two aluminum frames as shown in Figure 5-19-b. These magnets can rotate in the frame holes to find the position that generate the best homogeneity obtained by simulation. When these magnets achieve the desired position, they

Figure 5 - 17 :

 517 Figure 5-17: The field distribution in the region of 40x50mm (YOZ plane) at the middle length of configuration.

Figure 5 - 18 :

 518 Figure 5-18: The field distribution in the volume 40 mm diameters and 90mm length.

  Figure e 5-5 shows this works.

  the field homogeneity with finite height magnet array, we calculate and simulate its field strength and homogeneity in 3D for fully analyses the properties of the configuration.

  0 field were compared to those obtained with a digital gaussmeter. The homogeneity in the magnet longitudinal axis and the field B 0 strength are similar.However, the homogeneity in transverse plane differs from simulation and measurement because of the quality of the magnets. In order to improve the homogeneity, we propose a new shim method.Index terms: Nuclear Magnetic Resonance (NMR); Low field; Portable Permanent Magnet; Halbach; Shim magnets; Homogeneity, Simulation, Finite Element Method (FEM).INTRODUCTION

Figure 6 - 1 :

 61 Figure 6-1: Geometric parameters of Halbach structure.

  magnetic field outside of a ring, two others rings are placed in alignment as shown in Figure 6-2-b. The geometric parameters are depicted in Figure 6-2 and Figure 6-7.

Figure 6 - 2 :

 62 Figure 6-2: Position of magnets and the direction of their magnetization.

Figure 6 -

 6 Figure 6-3-a represents the variation of the magnetic field B 0 along the Ox axis. The value of the magnetic field is almost constant for 1.8mm < x < 4.2mm. Figure 6-3-b represents the variation of the magnetic field B 0 in the plane xOy. Each shades of color represent a variation of 50 ppm of the inhomogeneity

Figure 6 - 3 :

 63 Figure 6-3: Magnetic Field Bx distribution at z = 0 (xOy plane).The variation of the magnetic field Bx on the xOz plane is shown in Figure6-4. The magnetic field homogeneity in a region of 5 x 6.4 mm² is about 300 ppm determined by RADIA, while ANSYS gives a result of 200 ppm.

Figure 6 - 4 :

 64 Figure 6-4: Magnetic Field Bx distribution in the region of 5 x 6.4 mm² in xOz plane.

  performed by Ansys (c) esp = 0.9 mm. (d) esp = 1.3 mm.

Figure 6 - 5 :

 65 Figure 6-5: The field profile for different values of the gap esp between the two rings.

Figure 6

 6 Figure 6-6: Volume (mm 3 ) for a variation of

Figure 6 - 7 :

 67 Figure 6-7: Halbach configuration with 24 magnets and 8 shim magnets, modeled with ANSYS and RADIA.

Figure 6 - 8 :

 68 Figure 6-8: Direction of magnetization of the 8 shim magnets.

Figure 6 - 9 :

 69 Figure 6-9: Flow chart of our configuration with 24 main magnets and 8 shim magnets.

Figure 6 - 10 :

 610 Figure 6-10: Magnetic field homogeneity in the xOy plane and z = 0.

  (a) Before shimming. (b) After shimming.

Figure 6 - 11 :

 611 Figure 6-11: Magnetic field homogeneity in the region 8 x 20 mm 2 along Oz axis

Figure 6 - 12 :

 612 Figure 6-12: Magnetic field profiles at the center of our Halbach arrangement of magnets.

  performed by Ansys (a) Before shimming. (b) After shimming.

Figure 6 - 13 : 3 .

 6133 Figure 6-13: Magnetic field distribution in the 3D sensitive volume: 7 x 8 x 20 mm 3 .

( a )

 a Shim magnets that can move along three degrees of freedom.(b) Halbach prototype with the slide-blocks used to move the shim magnets in radial direction.

Figure 6 - 14 :

 614 Figure 6-14: Picture of the prototype with shim magnets.

- 4 T

 4 in the range 0 -0.299 T. The micropositioner Signatone S-926 is used to control the probe movement in three directions as shown in the Figure 6-15. The resolution is 254 µm per knob revolution. Matlab software carries out the plotting of the measurable values.

Figure 6 - 15 :

 615 Figure 6-15: Measurement set-up of the magnetic field in the prototype.

  ). In this region, the homogeneity value is respectively 1399 ppm calculated by formula (1) and 380 ppm obtained by simulation. It means that the measurable homogeneity is approximately 3.5 times worse than that simulated (Figure6-3-b).

Figure 6 - 17 :

 617 Figure 6-17: The magnetic field distribution in xOy plane obtained by measurement before shimming.

Figure 6 -

 6 Figure 6-18 shows the magnetic field distribution measurement in xOz plane in the region of 7 x 20 mm². In this region the homogeneity calculated by formula (1) is equal to 1426 ppm. This homogeneity is of 4415 ppm in the volume of 6 x 7 x 20 mm 3 .

Figure 6 - 18 :

 618 Figure 6-18: The measured magnetic field distribution in xOz plane before shimming.

Figure 6 - 19 :

 619 Figure 6-19: Magnetic field homogeneity at z = 0 in xOy plane.

  while it is 1426 ppm without shim magnets. The magnetic field homogeneity with shim magnets shows in Figure6-20, achieved in the volume 6 x 7 x 20 mm 3 is 1335 ppm in comparison to 4415 ppm obtained without shim magnets. The magnetic field homogeneity is 3.3 times better. Figure6-20 shows the improvement of the homogeneity that is approximate 5.8 times worse than the simulated value. The measured and simulated field distribution in xOz plane are similar as shown in the Figure6-4.

Figure 6 - 20 :

 620 Figure 6-20: The magnetic field homogeneity in xOz plane after shimming.

Figure 6 -

 6 Figure 6mulation sho region of 6 or. As show

Figure 6

 6 Figure 6-21 dy the impa mogeneity. D four magnets

  distribution measurement of the prototype.

Figure 6 - 23 :

 623 Figure 6-23: The magnetic field distribution in xOz plane, for ideal case and with error of misalignment of the direction of the magnetic field.

  one encounters during the development of Insertion Devices for Synchrotron Light Sources. However, it can also be used in different branches of physics, where efficient solutions of 3D boundary problems of Magnetostatics are needed (Figure A-1).RADIA uses a boundary integral method to determine the 3D magnetic field created by an arbitrary set of coils, permanent magnets, and linear and nonlinear iron volumes. The method used in RADIA belongs to the category of boundary Integral Methods and differs strongly from the Finite Element Methods (FEM). Volume objects are created, material properties are applied to these objects. Each object can be subdivided into a number of smaller objects for which one tries to solve for the general problem in terms of the magnetization (Figure A-2).

Figure A- 1 :Figure A- 2 :

 12 Figure A-1: General methods for field sources can be solved by RADIA

  Figur

Introduction -The need of portable NMR devices -Portable NMR devices -The applications of NMR devices -The components of portable NMR devices -Motivation for this work Chapter 2: Portable NMR system design 2.1 Introduction

	The field of portable NMR devices has been rapidly developed in the past few years. In the
	early time, the NMR devices are large, expensive and complicated devices. They are usually used
	in laboratory and are not portable. They use superconducting magnet to generate the field B 0 . Such
	devices generate the homogeneous field, consume too much energy and need high cost of
	maintenance.
	The need of external applications of the laboratory motivates the development of portable
	NMR devices. There are varieties of portable NMR devices that have been done by previous
	researchers such as NMR-MOUSE, Mandhalas and well-logging etc... The shapes of the magnets
	depend on the end uses. For example, the Ex-situ devices are suitable for the samples with large
	size and placed outside the magnets while in-situ devices are suitable for the small samples and
	high resolution experiments due to high field strength and homogeneity are more easily achieved
	with such geometries. The portable NMR devices are used for many applications such as oil
	industry, medicine and materials….

Table 3

 3

	-1:
	Symbols	Description	SI	CGS
	H	Magnetic field strength	A/m	Oe
	B	Flux density	Tesla	Gauss
	M	Magnetization	A/m	emu/cm 3
		Flux	Weber	Maxwell

Table 3 -

 3

	SI		To		CGS
	H in A/m	* 4 /10 3 =	H in Oe
	B in Tesla	*	10 4 =	B in Gauss
	M in A/m	* 10 -3 =	M in emu/cm 3
	in Webers	*	10 8 =	in Maxwells
	BH max in J/m 3	* 40	=	BH max in GOe
	BH max in kJ/m 3	* 4 /10 2 =	BH max in MGOe

1: Fields and units in the SI and CGS systems

To convert between the SI and CGS systems, we use the Table

3-2

Table 3 -

 3

2: Conversation from SI to CGS units

Table 4 -

 4 The optimum positions of shim magnets found by two softwares RADIA and ANSYS is the same. Their values are presented in the Table 4-2. 2: Optimum positions of shim magnets.

	Initial value of esp
	Initial value of r1
	Increase dH from 1
	to 50mm
	Calculate the	Increase esp from 0.1 to 0.6mm
	homogeneity of
	magnetic field in the	Increase r1 from 15
	volume of 7x8x20mm	to 23mm
	correspond to each
	value of dH
	Record the
	values of
	homogeneity
	and variables
	Compare the
	homogeneity
	Select the best
	homogeneity with
	correspondent
	values of esp, r1 and
	dH

Table 4 -

 4 3 shows the value of B 1 and B 2 for 27 magnets and the misalignment angle α. B 1 and B 2 are respectively the magnetic field in the vicinity of the two faces of the magnet cylinder as shown on the Figure 4-22 Erreur ! Source du renvoi introuvable.. The average value of the magnetic field is around 27.3 mT. The misalignment anglerepresents the error of orientation of the radial direction of the magnetic field on each face. In ideal case, this angle value is zero but for some magnets this value can reach 17 degrees and the consequence, is an error of homogeneity of the magnetic field.

	Magnet	1	2	3	4	5	6	7	8	9	10
	B 1 (mT)	27.3	27.2	27.3	27.2	27.3	27.2	27.1	27.0	26.9	27.0
	B 2 (mT)	27.2	27.1	27.5	27.1	27.0	27.1	27.2	26.9	27.0	27.2
	(degree)	0	0	8	9	17	8	0	0	0	12
	Magnet	11	12	13	14	15	16	17	18	19	20
	B 1 (mT)	27.4	27.2	27.2	27.2	27.3	27.3	27.3	27.4	27.4	27.4
	B 2 (mT)	27.5	27.0	27.2	27.4	27.4	27.3	27.5	27.4	27.5	27.2
	(degree)	0	10	0	12	0	0	12	0	0	8
	Magnet	21	22	23	24	25	26	27	28	29	30
	B 1 (mT)	27.3	27.2	27.6	27.3	27.4	27.4	27.3
	B 2 (mT)	27.3	27.5	27.5	27.4	27.3	27.5	27.4
	(degree)	0	16	4	6	9	7	8

Table 4 -

 4

3: Magnetic field and misalignment angle measured on each tip of the 27 magnets.

Table 5 -

 5 both in magnetic field and homogeneity. 2: The summary results of 2D simulations and geometric calculations.

	n	Squares (mm)	Circles (mm)		B(T)	Homogeneity * (ppm)	Weight (kg)
		a	r	a	r	Squares Circles Squares Circles Squares Circles
	4	100 100 241.42 170.71 0.7014 1.1705	727	88745 6	90.24	412
	8	70.71 100 61.99 80.99 0.7475 0.6853 3517	164	90.24	54
	16 20.43 64.44 24.23 62.11 0.3011 0.3553	857	23	15.06 16.63
	32 8.3	55.9 10.86 55.43 0.1348 0.1795	110	7	5.07	6.68

CONSTRUCTION OF LIGHT WEIGHT PORTABLE NMR HALBACH MAGNET Hung

  Truong Cong, J. T. Rosillo, H. Dang Phuc, H.H. Pham, P. Poulichet, A. Fakri and L. Fakri-Bouchet. A Numerical method for design of implantable RF microcoil for MRI and MRS 4.7T. Dang Phuc 1 , Patrick Poulichet 2 , Tien Truong Cong 1 , Abdennasser Fakri 2 , Christophe Delabie 2 , Latifa Fakri-Bouchet 1

	The International Conference on Advanced Materials forPhotonics, Sensing and Energy
	Conversion Applications (AMPSECA 2012).
	5. T. Cong Truong, J.T. Rosillo, H. Phuc Dang, H.H. Pham, A. Fakri, L. Fakri-Bouchet. A
	Numerical methods for improving the performance of implantable NMR microcoil. 4ème réunion
	de Institut Thématique multi-Organismes (ITMO) Technologiespour la Santé, 8 et 9 octobre 2012
	Lille.
	Laboratoire CREATIS UMR CNRS 5220, INSERM U1044, INSA de Lyon, Universite de Lyon, Lyon 1
	3, rue Victor-Grignard 69616 Villeurbanne, France 1
	ESIEE, 2 BD Blaise Pascal 93162 Noisy Le Grand, France 2

DESIGN AND

Table 6 -1: Geometric parameters of our configuration.

 6

	Names of the parameter	Definitions	Dimensions
	r0	Radius of the ring	30 mm
	h	Height of the ring	50 mm
	ray	Radius of magnets	4 mm
	hS	Height of shim magnets	6 mm
	rays	Radius of shim magnets	2 mm

Table 6 -2: The geometric parameters of an optimal configuration.

 6

	Name of the parameter	Dimension
	esp	0.2 mm
	r1	20 mm
	dH	26 mm

Cette thèse est accessible à l'adresse : http://theses.insa-lyon.fr/publication/2015ISAL0007/these.pdf © [H. Dang Phuc], [2015], INSA de Lyon, tous droits réservés

Acknowledgements

gave me valuable suggestions for my works. Special thank goes to Mr. Christian Jardin, who introduced me to study in France and Mr. Abdennasser FAKRI for his help during I had studied in ESIEE, Paris. grandmother, who passed away during I had studied in France.

Chapter 3 Magnetic materials and simulation methods

is the permeability of iron.

-0 is the permeability of air -M 0 is permanent magnets.

-S 1 is boundary of Ω 1.

is conductivity.

The domain Ω 2 does not concern with magnetostatics. It is considered for electromagnetic problem. Thus, the readers can find the detail in reference [106]. Here, we discuss the domains Ω 0

and Ω 1 which refer to permanent magnets. When permanent magnets are considered, the constitutive relation becomes [106]:

Where is the remanent intrinsic magnetization vector. The equation 3-29 can be rewritten as:

(3-30)

Where:

is the reluctivity matrix.

is the reluctivity of free space.

The constitutive relations for the related electric field are:

Where:

-

is the electrical conductivity matrix.

-

is the permittivity matrix.

is the velocity vector.

(3-41)

Applied loads:

Where:

is the matrix of element shape functions for ;

(3-50) -is the vector element shape functions for .

is the source current density vector.

is the total current density vector.

id the volume of element.

is the coercive force vector.

-| | is the derivative of reluctivity with respect to the magnitude of magnetic flux squared (derived from input material property curve B versus H).

Magnetic vector potential results:

The magnetic flux density is defined as the curl of the magnetic vector potential as equation 3-51.

(3-51)

The Figure 6-16 shows the magnetic field profiles in Oz direction for four different gaps esp between the rings. The optimal gap is displayed in Figure 6 (II-a) applying any subdivision to the objects created;

(II-b) constructing an interaction matrix corresponding to the problem (I) and particular subdivision (II-a), and executing a relaxation procedure;

(II-c) computing any components (as field induction, field integrals along straight line, potentials or forces) of the magnetic field created by the "relaxed" objects; 3D magnetostatic problems solved by RADIA based on these equations:

In this approach, one applies some kind of segmentation to the field-producing objects (typically iron) but, contrary to the FEM approach, one does not need to mesh the vacuum.

Poisson equation for Scalar magnetic potential: Solution through volume and surface integrals: Magnetic field created by uniformly magnetized volume: In ANSYS, the values of H and B can be obtained by two solutions, which are the magnetic scalar potential and the magnetic vector potential.

The scalar potential method as implemented in SOLID5, SOLID96, SOLID98 for 3D magnetostatic field. Magnetosatic means that time varying effects are ignored. This reduces Maxwell's equations for magnetic field to:

.

(B-2)

The vector potential method as implemented in PLANE13, PLANE53 and SOLID97 for both 2D and 3D electromagnetic fields. Considering static and dynamic fields and neglecting displacement currents (quasi-stationary), the following subset of Maxwell's equations apply:

(B-3) (B-4)

.

(B-5)