Electromagnetic field analysis in coils used for high-field NMR experiments in HR-MAS probes.

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NMR experiments in HR-MAS probes.

Baudouin Dillmann

To cite this version:

Baudouin Dillmann. Electromagnetic field analysis in coils used for high-field NMR experiments in

HR-MAS probes.. Electromagnetism. Université Louis Pasteur - Strasbourg I, 2007. English.

�tel-00207457v3�

Universit´

e Louis Pasteur, Strasbourg

Institut de Chimie de Strasbourg

Ecole doctorale de Chimie-Physique

´

Analyse du champ ´

electromagn´

etique,

dans les bobines des sondes de

HR-MAS utilis´

ees pour la RMN `

a

haut-champ

TH`

ESE

pr´esent´ee et soutenue publiquement le 25 mai 2007

pour l’obtention du

Titre de Docteur Ing´

enieur de l’Universit´

e Louis Pasteur

(sp´

ecialit´

e Chimie-Physique)

par

Baudouin Dillmann

Composition du jury

Président :

Pr. Daniel Canet (Universit´e Henri Poincar´e)

Rapporteurs :

Pr. Herv´e Desvaux (Universit´e d’Evry Val d’Essonne)

Pr.Pierre Panissod (Institut de Physique Chimie des Materiaux de Strasbourg)

Examinateurs :

Dr. Martial Piotto (Directeur d’Applications, Bruker Wissembourg)

Dr.Frank Engelke (Directeur BEENPK, Bruker Elektronik Karlsruhe)

Présents :

Dr. Jacques Laeuffer

Pr. Bruno Kieffer

En onsidérant es longues années de formation Do torale, je ne peux qu'être re on-naissant envers tous eux qui, même indire tement ou par le jeu des ir onstan es, ont ontribué à l'heureuxaboutissementde e travail.

Je souhaite remer ier la so iété Bruker Biospin qui m'a permis d'utiliser mes outils de produ tiondans le adre de l'étude fondamentale présentée i i.Enparti ulier je veux rendrehommage auDr. Tonio Gianottipour son soutienet ses en ouragements.

Dr. Frank Engelke est pour moi le plus bel exemple d'intégration réussie entre une démar he s ientique de fondetune produ tivitéindustrielleremarquable. Il m'aprouvé que la re her he fondamentale est né essaire, même dans un environnement qui ne peut pas être "purement a adémique". Il m'a rappelé plusieurs fois et fort justement, que les réalisations te hniques ne peuvent se passer d'une onnaissan e ertaine par les auses. J'aieulajoiedetravaillerave luipresque10ans,lale turedeDouglasAdamsetleplaisir des ontroverses s ientiques, fontvite oublierle stress liéau travailde produ tion.

Mit seiner Erfahrung und seiner beeindru kenden Persönli hkeit ist Dr. Heinz Zeiger eine unauswei hli he Autorität für jeden der Probenköpfe entwi kelt. Und somit mö hte i hhiermitmeineAnerkennungundmeineFreunds haftausdrü ken.Dankseiau hBenno Knotts me hanis hem Wunderwerk, ohne dem eine stabile Rotation um den Magis hen Winkelni htmögli hgewesen wäre.Dadur hkonnten wirMilliardensyn hronisierte Dre-hungenerzeugen.

Comment puis-je oublier de remer ier le Pr. Bruno Kieer qui m'a formé sur ses spé tros de haute-résolution? Depuis mon premier papier à l'ESBS, il y à une dizaine d'années,mesprojetsont ertesévoluévers desproblèmesdiérents,maismonexpérien e dans le monde de la Biophysique reste inoubliable. J'ai une pensée pour le Dr. Yves Nominé,et leDr. Emeri Wasielewski,ainsi que pour les autresmembres de l'ADDAl.

Parailleursjedoismestoutespremières ourbesdenutation,auPr.DanielGrukerqui m'a fait travailler sur ses onsoles experimentalles. Ma première image d' IRM je l'aieu sur une onsole équipéd'un aimantà 3 Gauss!

Je ne peux oublier d'évoquer la mémoire du Pr. Ja ques Hartong, et du Pr. Jean-FrançoisLefèvre, quim'ont en ouragé dans lavoiedu do torat.

Mer i au Dr.Gerhard Altho,pour ses onseils avisés etses rele tures du manus rit, e travail lui doit la tradu tion du résumé à l'Allemand. Les mots me manquent pour lui dire ombien je lui suis redevable. Sans les en ouragements du Dr. Hans Förster je n'aurraipas poussé mes performan es j'usqu'au bout.

in orporesano!,ThorstenMarquardsen poursonprofessionnalismeetpourl'équipement qu'il amis ànotre disposition,enn pour Alexander Krahnauquel je souhaite beau oup de han epour sathèse.

Ein herzli hes Dankes hön für Doris Herzig, Norbert Forger, Heinz S heele, Gilbert Ball, Thoralf S hippmann, Remy Klumpp, Modest Urban, Jurgen Ganz, Hartmut Glau-ner, Rainer vonHeyden und alleMitarbeitervon Bruker Elektronik GmbH!

I would like to express my aknowledgements to Dr. Werner Maas, for his invitations at Bruker-Biospin In . headquarters in Billeri a. My very best to all my olleagues in the U.S, in parti ular Patri k Saul, Dr. Yit Aun Lim, Dr Mark Xiaozhong Zhang and spe ially to Dr. Wurong Zhang, who made my journeys pleasant and fruitful, I hope we willhavetheo asiontoex hangeagainontopi ssu hasfren hXIX

th

enturyliterature and probe ir uits.

Et mer i à Joaquin Salvador Lavado (Quino) pour ses trois plan hes de dessins, em-puntés dans ses ouvrages `Humanose na e' et`Potentes, Prepotentes, Impotentes'.

Introdu tion générale xiii

Part I

Hardware aspe ts of the High Resolution Magi Angle Spinning

Spe tros opy : oils and rotors 1

1

General prin iples of HR-MAS experiments

1.1 RF Solenoidal oils . . . 3

1.2 The sample ontainer: The 4 mmrotor . . . 6

1.3 Ma ros opi sus eptibility dieren es . . . 7

1.3.1 Sus eptibility ompensation . . . 8

1.3.2 Sus eptibility mat hing . . . 9

1.3.3 Shimminggradients . . . 9

1.4 Mi ros opi sus eptibility dieren es under MAS . . . 11

1.4.1 General onsiderations on the averaging on magneti sus epti-bilities dieren es under MAS . . . 11

1.4.2 Averaging of magneti sus eptibilities present at the sample-rotor interfa es under MAS . . . 13

1.5 Gradient oilte hnologiesfor MAS . . . 14

1.5.1 Spe i ity of the HR-MAS gradients . . . 14

1.5.2 Pulsed eld gradient inhomogeneities under MAS . . . 16

Part II

Fundamental on epts, derived from lassi al Ele trodynami s, to be employed in Probe design. Engineering approa h of

B

1

19

2

Maxwell equations

2.1 Maxwellequations and the des ription of ir uits . . . 21

2.1.1 Maxwell-Ampere ina parallel ir uit . . . 22

2.1.2 Maxwell-Faraday equation ina series ir uit . . . 24

2.1.3 EM eld distributionand ir uitsimulation . . . 25

2.1.4 Maxwell equationsfor the mi ros opi Field . . . 26

2.2 Continuity equations . . . 28

2.2.1 Generaldenition of the ontinuity equations . . . 28

2.2.2 Appli ationtothe ase of anNMR probeshielding . . . 31

2.3 Fieldmappingby perturbation methods: the Ball-shift . . . 33

2.3.1 Magneti lling fa tor . . . 36

2.4 Quality fa tor . . . 37

2.5 Ma ros opi Magnetization: lassi al des ription . . . 38

2.5.1 The Nutationexperiment . . . 40

2.6

B

1

eld as afun tion of the input power . . . 42

3 Des ription of waveguides used in NMR probes 3.1 Propagationof Ele tromagneti eld in TEM . . . 45

3.1.1 Maxwell equationfor mono hromati waves . . . 45

3.1.2 Isotropi Propagation . . . 46

3.1.3 Separation of axial and longitudinal omponents in

(X, Y, Z)

frame. . . 47

3.1.4 Derivation ofthe Helmholtzprobleminthe ase of guided pro-pagation . . . 50

3.2 The TEM wave . . . 51

3.2.1 Orthogonality of

E

~

t

and

B

~

t

. . . 52

3.2.2 Phase Velo ity . . . 52

3.3.1 Appli ationto the spiral . . . 53

3.3.2 Appli ationto the Coaxial line . . . 54

3.3.3 Appli ationto the pair of oupled lines . . . 55

3.4 Propagation inhollowwaveguide, and resonant avities . . . 55

3.5 The helix model. . . 57

4 Numeri almethods to solve Maxwell equations 4.1 Introdu tionto Finiteintegration te hnique . . . 63

4.2 Considering Materialproperties for numeri alsimulation . . . 67

4.3 Pra ti alaspe ts of CST-basedEM al ulations . . . 69

4.3.1 Role of the dierent type of ex itationports in CST . . . 69

4.3.2 Means toevaluate onvergen e . . . 72

4.3.3 Eigen mode al ulations . . . 73

4.4 Strengths and weaknesses ofFinite time domain methods, and in par-ti ular toFIT used by CST . . . 74

5 Equation of re ipro ity and mathemati al expression of the signal in NMR 5.1 Polarizationof the RF eld pulses. . . 75

5.2 Re ipro ity asderived from the Maxwell equations . . . 76

5.2.1 Sour es atthe emission stage and atthe re eption. . . 76

5.2.2 First equation of the Voltagemeasured by the re eptor . . . 79

5.3 Generalized re ipro ity in materials . . . 80

5.4 Con lusions onele tromagneti re ipro ity . . . 85

Part III

Inuen e of the Axial and Transverse omponents of the ma-gneti eld on the NMR signal : Physi al des ription 87

6

6.1 Denitionof the NMR Hamiltonian. . . 90

6.1.1 Nu lear spin Operator . . . 90

6.1.2 S hrödinger'sequation . . . 91

6.2 Density of probability and quantum state evolution . . . 92

6.2.1 Expe tation value of anobservable . . . 92

6.2.2 Transitionprobability . . . 93

6.2.3 Equation of evolution of the density matrix . . . 94

7 The RF eld in a solenoid oil 7.1 Evolution of magnetizationunder the RF eld in MAS onditions . . . 102

7.2 Models for the RF eld-spin intera tion . . . 106

7.2.1 Model0 : Perfe tly homogeneousRF eld : . . . 106

7.2.2 Model1:AxialRFeld onstantandRadialRFeldof onstant amplitude and normalto the ring (Fig. 7.1A) : . . . 107

7.2.3 Model2 : Cosine modulationof the amplitude of the axialand radial elds(Fig.7.1B) : . . . 108

7.3 Predi tion of the quantum states with the master equation . . . 109

8

B

1

eld simulation for solenoidal oils from rst prin iples 8.1 Results obtained fromanalyti al methods . . . 111

8.2 Results obtained using the numeri al dis retization of Maxwell equa-tions fromCST . . . 115

9 Field modulation ee ts indu ed by sample spinning : results and dis- ussion 9.1 Rotary E ho experiment . . . 123

9.2 MLEV16and DIPSI2 experiments . . . 126

9.3 Nutationexperiment . . . 133

9.3.1 Pra ti al aspe ts of the adjustment of experimental nutation urvesusing theexpression ofthe measuredsignal given by the re ipro ity . . . 133

Part IV

Origin of the Temperature in rease indu ed by HR-MAS on

samples 145

10

Thermal issues in samples investigated under MAS

10.1 Heat produ edby me hani al ee ts . . . 148

10.2 Heat produ edby the

E

1

athigh frequen ies . . . 151

10.2.1 Joule heating inaqueous samples . . . 152

10.2.2 The dieletri dispersionmodel . . . 153

10.3 Relative inuen e of the indu tive and diele tri losses . . . 160

10.4 Conservative and non- onservative

E

1

. . . 161

10.5 Experimentalmeasurmentof heating inbiologi alsamples . . . 162

10.5.1 NMR thermometers . . . 162

10.5.2 Using Bi ellestoprobe RF heating . . . 162

10.6 Drawba ks of RF heating . . . 165

10.7 Con lusions on

E

1

. . . 167

Part V Introdu tion of a novel Low E- eld oil to minimize Heating of biologi al samples in Solid-State multinu lear NMR experi-ments 169 11 Z- oilas an example of multinu lear low-E eld resonator 11.1 Introdu ingoptimal oils for ondu tive samples . . . 171

11.2 Z- oil resonator geometry . . . 173

11.3 RF eld mapping . . . 174

11.3.1 Ele tromagneti eld simulation . . . 174

11.4 Experimental, omparativestudy of the dierent oils . . . 179 11.4.1 Experimental determination of

B

1

eld proles for oils of

va-rious geometries. . . 179 11.5 Enhan ement and sensitivity measured at 800 MHz . . . 180 11.5.1 In rease of temperature inBi ellesamples . . . 181 11.5.2 Comparison of NMR nutation elds for regular solenoids and

Z- oils . . . 183 11.5.3 Comparison ofqualityfa tors forregular solenoidsand Z- oils . 185 11.6 Con lusionson Z- oil . . . 188

Con lusion 191

Glossaire 193

Larésonan emagnétiquenu léaire(RMN)estunete hniqued'analysepuissanted'une trèsgrandepolyvalen e.Peu de spe tros opiespeuventen eetseprévaloird'unegamme aussiri hed'appli ations.Del'imageriemédi aleàl'analysephysi o- himique,lematériel étudiépeutallerdesgazauxmétaux,enpassantparlespolymères,lespeptidesensolution ou dans leur milieu membranaire. Ils peuvent être aussi des amalgames omplexes tels que les tissus et ellules. Dans e as, le mélange de petites molé ules dans une matri e anisotrope exige une te hnique très parti ulière : la rotation à l'angle magique (MAS), an d'obtenir un spe tre de haute-résolution (HR-MAS). C'est grâ e à une appro he éle tromagnétique et quantique que peut s'établir une ompréhension approfondie de la nature du signal re ueilli. L'expérien e de RMN, en eet, né essite la mise en oeuvre de deux hampsmagnétiques:le hampstatique

B

o

quipeutêtre réépardesbobines supra- ondu tri es, etun hamp radio-fréquen e (RF) appelé

B

1

, de moindreintensité, quiest rééparlabobinedanslasondedeRMN(unexempled'untel ir uitestdonnépar[155℄). Le hamp

B

1

irradie le volume de l'é hantillonsoumis à

B

o

et permet les transitions de rééquilibrage des moments inétiques nu léaires, peuplant les états d'énergie quantique du spin observé. En général et idéalement, les bobines de RMN se doivent de satisfaire simultanément les onditions suivantes portant sur

B

1

:

(i) Produireun maximumde hampRF dansune orientationquiest perpendi ulaireau hamp statique(

B

o

).

(ii) Produire une distribution de hamp aussi homogène que possible dans des é han-tillonsqui peuvent être de naturetrès diérente.

(iii) Atténuerautantquefairesepeutles omposantesdu hampéle triqueindisso iable du hamp

B

1

àl'intérieurduvolumeo upéparl'é hantillonetainsiréduirelerisque d'é hauement provoqué par ladispersion dans lesmilieuxpolarisés.

C'est pour ela qu'après une brève introdu tion de notre appli ation spé ique de la RMN (partie I) nous aborderons ertains on epts issus de la théorie lassique de l'éle -tromagnétisme(partieII)quinous semblentné essaires ande mieux dé rirel'impa tde l'inhomogénéitédu hamp

B

1

surlesignalRMN(partieIII).Ensuitenous avons examiné

les eets d'é haufement qui s'opposent a la ré ipro ité et peuvent ompromettre des ex-perien es de RMN (partie IV). Finallement (partie V) nous introduirons un résonateur innovant quiameillore l'homogénéitédu hamp

B

1

etreduit le hamp éle trique

E

1

.

***

Notre intérêtpourla artographiedu hampRFdanslesbobinesdes sondesHR-MAS a ommen é en 2001, à partir de résultats surprenants obtenus après appli ation d'une séquen e dite TOCSY ouramment utilisée sur une série d'é hantillons étudiés par HR-MAS. En eet, une même séquen e donne un signal d'intensité amoindrie et de phase dé aléeà ertaines fréquen es de rotationmé anique bien pré isesettoujours en relation ave l'intensité du hamp magnétique

B

1

[48℄. Après une étude approfondie mêlant une onnaissan e de la distribution de hamp magnétique et une des ription de l'évolution de la Matri e Densité (équation de Liouville-von Neumann), il s'est avéré que l'origine du problème est l'interféren e entre l'évolution spatiale d'un noyau de spin

1/2

dans son mouvement de rotation à l'intérieur de la bobine, et la séquen e d'impulsions elle-même.Parailleurslemêmetyped'observation futreproduite en hangeantles onditions expérimentales, ommepar exemple :

(i) letypedeséquen eTOCSYd'abord,puisdetypeCOSY(

90

o

x

-délai-

90

o

−x

-a quisition), ensuite. Plus tard e type de phénomène fut aussi observé dans le as d'une expé-rien e de nutation,telle qu'elle peut être utilisée dans presquetous leslaboratoires pour évaluer l'amplitude etl'homogénéité du hamp RF;

(ii) des é hantillons de nature diérente furent utilisés, aussi bien solides que liquides. Ce même eet est reprodu tible à partir du moment ou les noyaux se dépla ent à lamême vitesse angulaire autourdu entre de la bobine;

(iii) enn diérentes variantes de bobines RF furent utilisées pour s'aran hir d'un artéfa t qui seraitpropre à un solénoïde parti ulier.

Ainsi une généralisation à toutes les bobines de géométrie héli oïdale fut possible. Il a été établi, qu'au ours de sa période de révolution àl'intérieur du rotor, l'é hantillonest amenédans des régionsde hamps

B

1

distin ts, du fait de ladistribution inhomogènede elui- i dans le plan transverse. En eet, par dénition geométrique, une héli e ne pos-sède pas d'axe de symétrieautourduquel le hamp seraitidentique. Cetteinhomogénéité `périodique' donne lieu à une modulation dans l'évolution de l'observable qui dé rit la valeur moyenne de l'aimantation, telle qu'elle est déte tée par RMN. Des al uls basés sur la résolution de l'équation maîtresse de l'évolution de l'opérateur densité ont permis de reproduire lesrésultats observés expérimentalement.

des riptionsimpliéedu hampradialappli able àtoute bobinede géométriehéli oïdale, dans un ir uit bien équilibré.Ces modèles de distribution du hamp

B

1

ont été validés par larésolution aussi bien analytique quenumériquedes équationsde Maxwell àla fois pour des bobines régulières, des solénoïdes à pas progressif et des bobines enroulées à partir d'une spirale. Il est important de signaler qu'au une expli ation satisfaisante du point de vue de l'éle tromagnétisme n'avait été proposée. Depuis, des eets semblables ontété rapportés par d'autres équipes.

EnRMN l'impa tde la omposante éle triquedu hampRF (

E

1

) sur lesé hantillons dispersifs,estparti ulièrementinteressanteàhaute-fréquen e.Ilyaaumoinsquatreeets observables qui sont ausés par la ondu tivité de l'é hantillon:

(i) elui- i représente l'introdu tiond'un élémentdispersifdans le s héma éle triquede lasonde.Ilenrésultedon unegrandeinuen esurlafréquen ederésonan e, elle- iétant fortementde aléevers lebas par l'introdu tiond'unrotor ontenantun tel é hantillon. Celui- i est l'élément le plus fortement ouplé au hamp éle trique de la bobine. La onséquen e de e premier eet est que, pour une bobine lassique, il onvient d'élargir vers le haut la gamme d'a ord pour toutes les voies d'une sonde(lavoie

1

Hen parti ulier), pour leur permettre de `rattraper' laperturbation produite par ette inuen e du hamp éle triquesur l'é hantillon;

(ii) laprésen ed'unmilieudispersifsdégradefortementlefa teurdequalité(Q)desvoies d'irradiationetpar onséquentdégradeaussilasensibilitédelamesure,quantiable en termesde rapportsignal-sur-bruit;

(iii) si des é hantillons dispersifs sont introduits dans un hamp RF, le prol de la distributionde elui- iserasensiblementmodié,et elaen ontribuantàaugmenter leseets produits par l'inhomogénéité

B

1

;

(iv) lespertesdontl'origineestlapolarisabilitédel'é hantillon,setraduisenttoujoursen dissipationénergétique par eetjoule,et don par un é hauement de l'é hantillon onduisant rapidement à sadénaturation.

L'établissementde et eetindésirableau hamp éle triquepour un é hantillonvisqueux apu être observé dire tementpar RMN dansune expérien e dé ritesur des Bi elles[36℄. Ce problème, qui a été pris en ompte dès les débuts de la RMN, a onnu ré emment un regain d'intérêt, en parti ulier dans les appli ations biologiques.L'étude des Biopsies à haut- hamp (dès 500MHz) est onfronté à des élévations de température qui ne sont pas en ore maîtrisées. La quatrième partie introduit don quelques-uns des mé anismes d'é hauement,que esoità ausedelavitessederotation,ave propagationdela haleur

par onve tion,ou par irradiationà ausedu hamp

E

1

.

Ainsi, les eorts faits pour trouver une bobine à faible inhomogénéité

B

1

et don au hamp

E

1

réduit se trouvent justiés par l'étude de l'impa t d'une élévation de la tem-pérature sur l'é hantillon. C'est pourquoi, dans ette dernière partie de notre étude, est introduitunnouveau typedebobinehéli oïdaleappeléeZ-Coil ([153℄).Ce résonateurfait partie de la famille des héli es. Il peut se substituer au solénoïde dans le même ir uit d'a ord. Pour éloignerle hamp éle trique du entre de l'é hantillon,l'usage de spirales qui ontiennent elui-làdans leplan radial,est tout àfait adapté.Au entre, un ylindre fendu a umule le hamp magnétique alors plus homogène et réduit l'é hauement des é hantillons ondu teurs; enn es propriétésremarquablesserontprésentées dansla in-quièmepartie,grâ eàuneétude omparativerealiséesur plusieursbobinesdansun même ir uit.

***

Cette thèse, qui a débutée par une problématique liée à l'inhomogeneité du hamp magnétique, se on lu sur l'invention d'un résonateur àfaible hampéle trique. En her- hant à omprendre le prol du hamp

B

1

nous en sommes venus à essayer de trouver un résonateurquidiminueraitl'inhomogeneitétransversale.L'intérêt d'unteleortàété, nalement, de pouvoir ainsi obtenir une rédu tion du hamp éle trique à l'intérieur du volume o upé par l'é hantillon. Mais ette démar he prouve aussi que, dans une onde éle tromagnétique,les omposanteséle triques etmagnétiquesne sontpas indépendantes lesunesdes autres.D'une partelles sontgérésparleséquationsdeMaxwell;d'autrepart, la géométrie de la bobine ou du résonateur, ainsi que le ir uit et le positionnement de l'é hantillon(sa lo alisationet son extension dans la bobine) sont autant de ` onditions auxlimites'dontilfautprêterattention. Dansladernière partiel'étudedétailléed'un ré-sonateur àfaible hamp éle triquemontreque, lefaitd'allervers une symétriedu hamp éle tromagnétique est une ondition né essaire pour réduire l'é hauement. En poussant e raisonnement jusqu'au bout, on pourrait dire qu'un résonateur dont l'axe prin ipal serait aussi axe de symétrie ( ommepar exemple un ylindre fermé), ne donnerait ertes plus de hamp éle trique, mais malheureusement il ne pourrait pas non plus fournir de omposante magnétique

B

1

. On omprend e ompromis qui est fondé sur la première équation de Maxwell (Maxwell-Ampère) et qui peut se formuler de la façon suivante : toute amélioration de l'homogénéité spatiale du hamp magnétique RF dans le volume de l'é hantillon implique une diminution de la quantité d'énergie éle trique a umulée dans elui- i.

Hardware aspe ts of the High

Resolution Magi Angle Spinning

General prin iples of HR-MAS

experiments

HR-MAS experiments are usually performed on standard liquid-state NMR spe tro-meters equipped with a dedi ated HR-MAS probe that allows tospin the sample at the magi angle.Likewise,the gradient oilgeneratesagradienteldstrength similartowhat isfoundona liquidstateprobe(ie.

50 G.cm

−1

).HR-MASprobesare not designed torun ross-polarization,nonwithstanding ertainCP-MAS probes an beemployed toperform HR-MAS-likeexperiments.Theshapeofthesampleandthesample ontaineritself(MAS rotor)are designedtominimizetheee t ofthedierentsour esof

B

o

eld inhomogenei-tiesinordertoobtainanintrinsi linewidthsimilartotheoneobtainedonstandardliquid high resolution probes. In the following se tions, some general onsiderations on erning the design of HR-MAS probes will be presented starting from the most important : the RF oil.

1.1 RF Solenoidal oils

Heli es and solenoids are synonyms for a passive stru ture ommonly employed in ele troni s alled"indu tan es" : two-port elementswith positiveimaginary omplex im-pedan e. Indu torsnd their pla e together with apa itors inany os illating ir uit for analog ele tri designs. These stru tures, exhibiting a parti ular symmetry, are used as delay lines in travelling wave tubes[130℄or aslters [145℄ for radio-frequen y ele troni s appli ations.Asamereexample,they arepresentinnowadaysmobilephonete hnologies, asantenna omponents.Asigni antnumberofpubli ationsexistforthe various models and elds of appli ations for solenoidal oil stru tures. For the early work the reader is referredtothe reviewof referen e [122℄,while someof themore re entresear h literature relevant for magneti resonan e an be found in referen e[49℄.

The use of indu tors is the most ommonway of generating magneti elds, as they allow maximum urrent density. Solenoidal oils have been used to generate the radio-frequen y eld

B

1

sin e the beginning of NMR in the late 1940's. It is reported in the experiments of Pur ell and Torrey (see the review of Soutifand Gabillard[5℄) that1 mi- rohenryindu torswereusedtoprodu eaRFeldat30MHzfor

1

Hresonan efrequen y. Nowadays onlyafewtens ofananohenry arerequiredtogenerate eldsat900MHz.The prin ipleofre ipro ity,formulatedfor NMRby Hoult[74℄, waspresented rst by the use of solenoidal oils. Itwas shown that, inprin iple,su h oilspossessaperforman e signi- antly higher than Helmholtz oils (saddle oils) in terms of a hievable signal-to-noise ratio, provided that the same ylindri al dimensions are applied. In the 1970's, the de-velopment of ross polarization, magi angle spinningand proton high-powerde oupling te hniques ledto the development of multi-tuned single- oilNMR probes for solid-state NMR. Double-resonan e MAS probes and probes suitable for

1

H CRAMPS experiments emerged, homebuiltaswellas ommer iallyavailable,and be ameastandard instrumen-tationinsolid-state NMRspe tros opy [27,132, 55,108, 107,102,9,59℄.It wasnot until theend ofthe 1980'swiththeinvention ofREDORte hniques[67,69℄,thatthe ne essity arosetohavea esstosingle- oiltriple-resonan eMASprobes-andhere,again,the ver-satilityof solenoidal oilsrepresented anobviousand e ient te hni al solutionforthese solid-state NMR MAS probes. An alternative idea to nd a te hni ally e ient solution to fulll the dierent requirements for ir uits and oils for widely diering frequen ies (e.g., for

1

H and

13

C) was pursued by Doty [42, 44℄. He departed from the strategy of a multiply tunedsingle oiland used instead a dedi ated resonator stru ture for the high-frequen y hannel and a solenoidal oil for the low frequen ies in his HR-MAS probe. Sin e the 1980's RF homogeneity of oils was given in reased attention. One approa h to in rease the RF homogeneity of the solenoidal oilwas to vary the pit h angle of the solenoidal windings or the width of the oil wire [76, 134, 116℄. Leifer [90℄ proposed an inverse strategy to optimizehomogeneity of the RF eld in 6and 10 turn oils with dia-meters larger than 10 mm operating at frequen ies of 85 and 21 MHz, respe tively. By usingChebyshev polynomialstodes ribethe eld,he uses aninverse te hniqueproposed by Turner [139℄ to derive the urrent distribution that generates the desired eld and, inturn, fromthe urrentdistribution the positioningof the nonequidistant windings an be derived. Sun and Ma iel [134℄ have proposed and built an RF oil for MAS probes where the plane of ea h oil turn is tilted su h that the amplitude of the transverse RF eld is in reased while the oil axis is oriented at the magi angle. They laimed, that for su h a oil the signal-to-noise ratio ould be improved by 17

%

ompared to a stan-dard solenoidal oiloriented along the MAS axis. Apart from the detailed oilgeometry,

ele tri albalan e of the oilis another important point to a hieve good homogeneity of the RF magneti eld inside the oil, high sensitivity of the probe ir uits as well as a minimum oupling of ele tri elds to the sample diele tri . Ele tri al balan e means, thatatagiven frequen y the twoportsof asolenoidal oilare onne ted toa ir uitthat provides equal omplex impedan e to ea h port su h that the ree tion fa tors on both portsare equal toea h otherandasymmetri standingwaveappears withthe os illating maximum RF magneti eld in the oil enter. In earlyspe trometer designs,where the transmitterand/or preamplierwere usually integrated mu h loser tothe os illator ir- uitthat made up the probe,symmetri operationof both wasmu hmore ommon(see, e.g., [104, 28, 21℄). In order to redu e losses in biologi al samples having relatively high ondu tivity ausedbythepresen e oftheRFele tri eldof oilsimplantedinbiologi al tissues,Murphy-Boes handKoretzky[109℄apply symmetrizationorbalan ing ofthe oil using additional apa itors in a single-resonan e ir uit. As a result they demonstrate a signi antlyin reasedQfa toroftheprobe ir uitandadrasti in reaseofsignal-to-noise ratioof

31

Pspe tra at97MHz.De orps etal.[31℄ haveshown thatwell balan ed ir uits anbea hieved aswellby indu tive ouplingof the NMR oiltothe RF ir uit.The key point for these two appli ations is that balan ing leads toa minimization of the ele tri RF elds inside the oil and therefore diele tri and ondu tive losses appearing in the NMR sample an be aleviated.Another ase where a balan ed ir uit is advantageous is when the NMR oil has to be mat hed to the transmission line onne ting the oil to a liquidnitrogen ooledpreamplier.Thisset-upwas studiedbyConradiandEdwards[22℄ intheirlow-temperatureprobe withthe goaltominimizeJohnsonnoise originatingfrom linelosses. Finally, asmentioned above, a balan ed oilpossesses awell dened lo ation wherethe ele tri aleldisminimumandthevoltagetogroundisequaltozero.This old point an be used to tap the oil,for example,with ashield whose purpose is to remove stray elds fromthe oil[32℄.Solenoidal oils have been used asRF oils inNMR probes dedi ated for smallsampleswith volumina inthe range of afew mi rolitersand for sub-mi rolitersamples as well [146, 105, 106℄. The main advantage of these mi ro-solenoidal oilswithsub-mmdiametersisthatthey providehighllingfa torsand thereforeallowa relativelyhigh signal-to-noise ratio per volume unit for smallNMR samples. A ommon hara teristi for solenoidal oils of sub-mm diameter for mi roliter samples as well as for solenoidal oils in the mm range of diameterapplied as RF oils in HR-MAS probes onsistsinthe fa t that theirpresen e leadsto aperturbation of the homogeneousstati eld,sin e(a)usuallythemagneti sus eptibilityofthe oilmaterialsisdierentfromthe magneti sus eptibilityof theirenvironmentand (b)these oilsdonotrepresent ylinders with homogeneous walls and are not aligned with their main axis along. Te hni al

solu-tionstominimizethe resultingeld distortionswillbedis ussedinthefollowingse tions, but before we willfo us onthe sample inside the rotor.

1.2 The sample ontainer : The 4 mm rotor

Therotorisanessential omponentofthehardwareusedtorunHR-MASexperiments. Duetothe strong entrifugalfor es thatexertthemselvesonthewallsof therotorathigh spinning speeds, the rotor has to be extremely resistant. For that reason the material of hoi e is usually ZrO

2

erami s. In order to optimizethe sensitivity of the probe, the rotor an be designed with an inner volume that mat hes the dete tion volume of the solenoidal oil. This experimental set-up allows to dete t all the material ontained in the rotor. The signal to noise ratio also (SNR) will depend on the amount of material involved. Good sample positionning in the rotor is an important pre-requisite for (1) olle tingthebest homogeneousmagneti eld, (2) avoidingair-bubblesthat would spoil theme hani alequilibrium

1

andnally(3)preventingpartofthesampletobedenatured through heating produ edby ele tri elds at the ends of the oil.

Fig.1.1 Des riptionof thedierent parts required topa k thesample in a 4 mmRotor usedforHR-MASexperiments.BottomSpa er (1),Upper Spa er (2),Grub s rew(3) and Rotor Cap (4) are engineered to redu e the perturbation of both

B

o

and

B

1

eld. The sample remains the most riti al part.At the time beeingHR-MAS rotors (both12

µ

l and 50

µ

l) do not have a bottom spa er but a massive ZrO

2

bottom. See Fig. 1.3.

Sin ethedete tionvolumeisunfortunately smallerthanthetotalvolumeofthe rotor, samplerestri tionis requiredde fa to[16℄. Toknow theposition ofthe samplesandtheir volumes,is a prerequisite for agoodHR-MAS experiment.

The Fig.1.2shows arotorfor4mmstators,used forsampleswithup to50

µ

lvolume. The left side denotes the bottom of the rotorand the right side representsthe top where the sampleistobeinserted, followedbyaTeonplugtorestri tthe samplevolume.The axial extend of the oil is refered from the bottom part at the left side. This has been measured inthe ase ofa5-turns oil,madeoutof atwire, atthe upperpartof the oil,

1

Piottoandal.[113℄have learlyshownthattheresolutionoftheHR-MAS spe trumisnotae ted bytheairbubblespresentinthesample

4

Fig.1.2 Rotor 4 mm positioned in a oil holder, used in this work.

opposed tothe oilleads.Thereisa markinred tosignalizethe middleof any 4mm oil fromthe bottomoftherotor.The 2mmheight,representedatthe rightend, orresponds tothe absoluteminimum volume that must be keept free inorder toallow the insertion of the ap(3 mm are required for safety in Fig. 1.3). The ilindri al rotor and the oil aroundhimare positionned asdes ribed by Fig. 1.5.

Two dierent types of inserts are often employed, we present here two : a 50

µ

l that ts the dete tion volume of the oil (Fig.1.2without the upper Teon plug) and a 12

µ

l spheri alsample that allows to positiona smaller quantity at the magneti enter of the oil(9.5 mmfromthe bottom).The insertsfor 15

µ

l were originalymadeout of teonas shown inFig.1.1,withthe disagreementof sa riingthe bottomplug ea h time.Thishas hanged sin e, the new 4 mm rotors are of dierent types as presented in Fig. 1.3, and thebottomislled up withZr0

2

with abetterme hani aland ele tromagneti behavior. Theinnersample an o upy adiameterofmaximum3mminthese ongurations,other rotors an be provided with thi ker wallsinorder toredu e the entrifugalfor e, insu h ase the sample isrestri ted in the radial dimension (2mm).

In the followingtwose tions we willsee that the magneti perturbation ae ting the ultimate resolution of a HR-MAS spe tra an be divided in two dierent ategories : perturbations external to the rotor system (ma ros opi ) and perturbations originating from the rotor system (mi ros opi ). The main dieren es between these two sour es of perturbationis that the former isnot modulatedby MAS while the later isaveragedout tozero by MAS.

1.3 Ma ros opi sus eptibility dieren es

The experimentalset-upof aMAS experiment onsists ofasample ina erami rotor pla ed inside a solenoid oil. The solenoid itself is embedded inside a erami stator. Pla edin the stati magneti eld

B

o

the oilassembly is unfortunately not transparent

Fig. 1.3  Rotor 4 mm used for HR-MAS experiments, and inserts employed for an ellipsoidal sample.

tothe magneti uxlines. The magneti eld dis ontinuities presentatthe surfa eof the dierent partsofthe MASset-upmayresultinasevere linebroadeningofthe spe tra.In order to alleviate this problem, the following strategy is employed : First, the magneti sus eptibility (

χ

m

dened in the next hapter by Eq. 2.10) jumps are redu ed during probe onstru tion by sus eptibility ompensation or by sus eptibility mat hing of the dierentelementsofthe probe.Se ondly,theresidualweakmagneti sus eptibilityjumps are shimmedout using dedi ated shim oils.

1.3.1 Sus eptibility ompensation

The materialin HR probes should not ex eed 3 ppm in sus eptibility when disposed lose to the samples [41℄. Sin e the oil wires are less that 0.5 mm far from the rotor wall,sus eptibility ompensationisrequired. Thiste hnique onsistsof adaptingthe oil sus eptibilitytothelo alenvironmentinorderto an elthedis ontinuityofthemagneti permeability. The most popular material used to produ e NMR oils is the opper- lad based wire

2

[42℄, eventually platted with a goldenalloy toin rease the ondu tivity and ompensate the weak sus eptibility of the opper. The quality of this te hnique an be

2

Copperhasamagneti sus eptibility

χm

of-9.6ppm,werefertothetablesgivenbyDotyfordierent ompounds su hasmetals,diele tri sandsolvents

evaluated by the half line width of the

13

C in adamantane samples whi h should not ex eed 5 Hz. Surfa e oating is intended to allow sus eptibility ompensation. Zelaya's experimentalarti le[156℄onRh-plated opper ylindersproves,experimentallyand theo-reti ally, that for a dened oating surfa e the additional magnetization introdu ed by the opper is totally ompensated adding a ertainlayer of rhodium.

1.3.2 Sus eptibility mat hing

A ording toele tromagneti eldtheory[146, 156℄,aperfe tlyuniformandinnitely longhollow ylinderarrangedperpendi ulartoastati magneti eldwillgiveaperfe tly uniform, albeit redu ed in magnitude, magneti eld inside the ylinder. Compensation ormat hing te hniques make itpossibletonegle tthe demagnetizationeld produ edin the oil due to sus eptibility jumps. The ase of MAS requires to onsider nite length for the rotors and Barbara [8℄ has proved that the modulation of the demagnetization elds,perturbing the

B

o

homogeneity,isaround 5times greateratthe rotors edges than at the enter of the rotor. Through graphi al representation of the magneti elds lines obtained by numeri ally solving the Lapla e equation of the magneti s alar potential [78℄, Ku hel etal. [85℄ shows that the best suited geometries are innitely long ylinders alignedparallel tothe magneti eld

B

o

. In the pra ti e of HRNMR mat hing plugs are used to enlarge the apparent longitudinal dimension of a sample (Shigemi and Doty are referen es amongothers for these produ ts).Kuboet al.[84℄ haveproved that the shape of the sample,has adire t inuen e onthe numberof sidebands in NMR :a prolate like sample within a length to diameter ratio of

12 : 1

will show almost no side bands, and same rotation frequen y an oblate like sample with geometri ratio equal to

0.8 : 1

will exhibitalmost 20 satellites.

1.3.3 Shimming gradients

Propershimmingwillredu e thedis ontinuitiesprodu edby thepresen eofanybody in the vi inity of the rotor. Shimminga sample spinning at the magi angle requires to reate gradients in the tilted frame along the MAS axis [50℄. The standard shim system found on most high resolution NMR instruments an be used to orre t the shims of a HR-MAS probe along the MAS axis to third order [129℄. For a HR-MAS probe with a statoralong the x axis, the zonalshims along the MAS axis

B

M AS

Z

1

, B

_Z

M AS

2

and

B

M AS

Z

3

are relatedto the laboratory frameshims by the following relations:

Fig.1.4

13

Cresolutionandlineshapeofa31.8mgsampleofadamantane.Spinningspeed at 5 kHz.

13

C RF eldamplitude during CP of 83 kHz, is followed by a

1

H de oupling at

∼

30 kHz during 800 ms. Theprobe used wasa 900 MHz H/C/N SB BL3.2with a Ma or stator. Linewidth at 50

%

: less than 2.0 Hz; at 10

%

: less than 8 Hz; at 5

%

, 13 Hz.

B

_Z

M AS

1

=

1

√

3

B

LAB

Z

1

−

s

2

3

B

LAB

X

B

_Z

M AS

2

= B

_(X

LAB

2

_−Y

2

₎

− 2

√

2B

_ZX

LAB

B

_Z

M AS

3

= −

2

3

√

3

B

LAB

Z

3

−

1

√

6

B

LAB

Z

2

_X

+

5

√

3

B

LAB

Z(X

2

_−Y

2

₎

−

5

3

√

6

B

LAB

X

3

(1.1)

The set of shims des ribed in Eq.1.1 is su ient to shim a well designed HR-MAS probe[113℄.

Here we present a good shimmingsetup that an be obtained from the

13

C signal of adamantane. This te hnique is in prin iple far better than using the lo k of deuterated water, due to the strong thermal gradients present during the me hani al rotation. The gure1.4,obtainedat900MHzshowstwodierent arbonpeaks,presentinadamantane, and the orrespondingsatellites.The s alar ouplingof twoadja entCarbons

13

C(

1%

in naturalabundan e) an beobserved after16 s ans(

J ∼ 30

Hz).Ba kgroundsignals from

theprobe ir uititselfareavoided 3

.AspresentedbyFig.1.4,spinningatthe magi angle isof primary importan efor the averaging of mi ros opi inhomogeneites.

1.4 Mi ros opi sus eptibility dieren es under MAS

1.4.1 General onsiderations on the averaging on magneti sus- eptibilities dieren es under MAS

As it was mentioned in the introdu tion, the averaging of magneti sus eptibility dieren es by MAS is an essential aspe t of HR-MAS spe tros opy. In this se tion, we evaluate the stati magneti eld seen by a spin in a medium onsisting of a magneti sus eptibilities distribution. If the dieren es in magneti sus eptibilitiesare reasonably smalland,ifthey areisotropi ,the additionalmagneti eld reated by avolumeelement of magnetization

M

~

j

of oordinates

r

j

= (r

j

, θ

j

, φ

j

)

at a point

r

i

= (r

i

, θ

i

, φ

i

)

an be treated as a dipolar intera tion. In the laboratory frame and for stati samples, this dipolareld isgiven by [94, 91℄ :

B

(r

i

, θ

i

, φ

i

) =

µ

o

4π

.

X

j

~

M

j

r

3

ij

(3 cos

2

_(θ

ij

) − 1)

2

(1.2)

Where

r

ij

isthe distan ebetween the points

r

i

and

r

j

,

θ

ij

istheanglebetween

B

o

and theve torjoining

r

i

to

r

j

,andthesummationexpresses thefa tthatpoint

r

i

experien es the sum of allthe dierent magneti dipoles

M

~

j

inits vi inity.

When the sample is rotatedat anangle

β

respe t tothe main magneti eld

B

o

ata speed

ω

r

,Eq.1.2 an be rewritten as in[4℄:

B

(r

i

, θ

i

, φ

i

; t) =

µ

o

4π

.

X

j

~

M

j

r

3

ij



₁

4

(3 cos

2

β − 1)(3 cos

2

β

_ij

d

_{− 1)}

+

3

4

sin 2β sin 2β

d

ij

cos(ω

r

t + φ

ij

)

+

3

4

sin

2

_{β sin}

2

_β

d

ij

cos(2ω

r

t + 2φ

ij

)

i

(1.3) Where

β

d

ij

is the angle between the ve tor joining

r

i

to

r

j

and the axis of rotation of thesampleand

φ

ij

isaphasefa tordes ribing theangularpositionof

r

i

.If

β

isset tothe

3

MAS probes based on transmission line (TL) ir uits, permit remote tunning and mat hing, and helptoeliminatespuriousprobesignals.TheadventageoftheTLdesign[155℄,isthatitremoveslumped elements, as hip- apa itors whi h may produ e su eptibility jumps lose to the stator, as dis ussed previously.

magi angle(

54.7

o

)inEq.1.3,theterm

3(cos

2

_{β −1)}

vanishesandonlytwotime-dependent terms modulated in

ω

r

and

2ω

r

remain.These two terms lead to spinning side bands at frequen ies

ω

r

and

2ω

r

with respe t to the main peak. Under magi angle rotation, the ontributionof

B

(r

i

, θ

i

, φ

i

; t)

,and onsequently ofvolumeelementsofdierentmagneti sus eptibilities,tothewidthofthe NMRresonan evanishesand onlyspinningsidebands remaininthespe trum. Thefa tthatinhomogeneousbulkmagneti sus eptibility anbe e iently removed by MAS was demonstrated both experimentally and theoreti ally by Dosko ilova[38℄,VanderHart[143℄,andGarroway[62℄inthe aseofliquidsandsolidswith random orientation.More re ently, Barbara [6℄ used arguments based onele tromagneti al ulationsto explainthis averagingpro ess of MAS.However, the magneti sus eptibi-lity

χ

m

isnotalwayspurelyisotropi [143,142,30℄andasubstantialamountofanisotropy an be present in the sample. The magneti sus eptibility

χ

m

is no longer a s alar and must be des ribed by atensor. Asanexample, onsider thewellknown anisotropyof the benzene ring.In the presen e of amagneti eld, the intensity of the ring urrentsof the benzene ring depends onthe orientationof the ringwith respe t to

B

o

.This impliesthat the intensity of the magneti dipole

M

of the aromati ring depends on its orientation with respe t to

B

o

. During the MAS averaging pro ess, the magnitude of

M

will vary and willinterfere with the quality of the averaging pro ess. This time-dependen e of

M

during sample rotation has important onsequen es, as it an be shown that MAS an onlyaverageout theisotropi (s alar)partofthe hemi alshiftdispersiontensor,butnot the anisotropi part [142℄. In the ase of asample likea peptide bound toa polystyrene-basedWangresinwith alarge amountofheterogeneity,the hemi alenvironmentof ea h peptide mole ule will be unique : The number, the distan e and the orientation of the neighboringaromati mole ules willbedierentforea h peptidemole ule. Aspreviously seen, the magneti dipole reated lo allyby all these phenyl rings annot be ompletely averagedout to zeroby MAS. The ombinationof these two ee ts willresult inabroad NMR line made of a superposition of dierent hemi al shifts (inhomogeneous broade-ning). These onsiderations are extremely important for HR-MAS when onsidering the nature of a Wang resin whi h is made essentially of highly anisotropi ross-linked aro-mati groups. The diusion pro esses an modify the traje tory of spins and therefore interfereinanin oherentway withtherotation[121℄.Theseadditional ompli ationswill not be onsidered here.

1.4.2 Averagingofmagneti sus eptibilities present atthe sample-rotor interfa es under MAS

Thepreviousse tionexplainshowMASe ientlyremovesthelinebroadeningdueto magneti sus eptibility gradients present inside heterogeneous systems. MAS plays also a very important role in averaging out magneti sus eptibility gradients present at the sample-rotorinterfa e. A typi alHR-MAS arrangement is su h that the wholesample is ontained within the dete tion volume of the solenoid oil. In some ases, the sample is onnedinsidea15

µ

lvolumeofa4mmrotor(Fig.1.1).Thisisaremarkablefeaturesin e thisexperimentalset-upallowsthewholesampletobedete tedandallowsforthehighest sensitivity. Under these experimental onditions, magneti sus eptibility jumps exist at the sample-rotor interfa e along the long axis of the rotor and at the top and bottom of the sample. Typi ally, this interfa es are between a solvent ontaining heterogeneous substan es and the rotor material (ZrO

2

and Teon for the insert). On a stati sample, thesemagneti sus eptibility jumps giverise toimportantmagneti eldgradients whi h are un orre table with the shim system of the spe trometer. MAS has the very unique property of being also able to average out these magneti sus eptibility to zero. This property anbeexplainedusingargumentssimilartothosedevelopedinse tion1.4.1.The onsequen e isthat the rotor materialwhi his the sour eof the magneti perturbations is rotating at the same time as the sample and its ee t on the sample vanishes. This veryimportantpropertyofMASexplainswhytheshimofHR-MASisalmostsampleand solvent independent provided that the rotor is always at the same position in the probe andthat the volumedete ted isalways the same.One set of shimsis su ientforall the solventsusedinHR-MAS.Itisimportanttonotethatperturbationsexternaltotherotor, and thereforenot rotatingat the magi angle, willnot be averagedout by MASand will have to be orre ted by the shim system. This is true, as mere example, for a apa itor pla edintheneighborhoodofthe sample.Itsdipolareldwillperturbthe magneti eld seenbyaspin pa kettravellingonathin ir leinaplanperpendi ulartothemainaxisof the rotor.These spins willsee aperiodi modulationof the main magneti eld thatwill result in a sharp line anked by spinning side bands. Another spin pa ket taken at the dierent position along the main axis of the rotor will resonate at dierent frequen ies. Theresultingfrequen ydistributionalongthemainaxisoftherotor anonlybe orre ted using a ombinationof shims that a t along the magi angle axis [129℄.

1.5 Gradient oil te hnologies for MAS

1.5.1 Spe i ity of the HR-MAS gradients

Pulsed eld gradients are widely used in modern high-resolution liquid-state NMR spe tros opy. The sele tion of oheren e order pathways with gradient pulses ina single s an allows to obtain faster results than with a phase y ling pro edure. Gradients are parti ularly useful for inverse experiments and solvent suppression [75, 141, 114, 115℄. With the advent of HR-MAS probes equipped with gradients, similar experiments may beenvisionedoninHR-MAS[96,18℄.Avarietyofdesignsforgradient oilswereproposed for MAS probes. The design of a gradient oil for MAS NMR spe tros opy diers in a number of points from traditional gradient oildesigns used in standard high-resolution NMR.First,theorientationofthesample(MASrotoraxisin linedat

54.7

o

withrespe tto themainstati eld)violatestheoverallsymmetrydi tatedby

B

o

.Se ondly,thesampleis rapidly spinning,su hthat agivenspin inthe sampletravelsalongama ros opi spatial pathway and therefore may rea h a quite distin t spatial regions after a gradient pulse. Mathemati ally,whilethe magneti eld itselfrepresents ave toreldwith intuitiveand relativelysimplepropertieswhenwe onsiderarotationofthe oordinatereferen eframe, the situationis more ompli atedwith gradientelds sin e they have tobedes ribed by se ond-rank tensors. Although we are eventually only interested in three omponents of su h a nine- omponent eld, namely

∂B

z

/∂x

,

∂B

z

/∂y

,and

∂B

z

/∂z

the omplete se ond-ranktensorhas tobetakenintoa ountif weareperforminga oordinatetransformation fromthelaboratoryframewiththe

z

axisparalleltothe stati eld

B

o

totheMASframe with the

0Z

axis in lined at the magi angle relative to

B

o

(Lab frameis represented in the gure 1.5).

This ompli ationwhenperformingrotationsofthereferen eframeree tsitselfquite expli itlywhendesigninggradient oilgeometries.Gradient oilsin onjun tionwithMAS wereproposedoriginallybyWind,etal.[149℄,andimplementedexperimentallybyCory,et al.[25℄,(using,atthattime,Golay oils enteredaroundthespinneraxis)andS hauss,et al.[120℄,(usinganti-Helmholtz oilsandalsoGolay oils enteredaroundthespinneraxis) for imagingofsolid samples spinningatthe magi angle.Forthese imagingexperiments, the urrentsthroughthegradient oilsneedtobemodulatedinordertogeneraterotating gradient elds syn hronouslyto the MAS rotor motion.Apart fromthe idea towind the gradient oils onsurfa es of ylinders with axesin lined atthe magi angle, Bowtell and Peters [13℄wenttopursuitthegoaltond agradient oil ongurationwithwireswound ona ylinder withanaxisparallelto

B

o

,but generatinga

z

gradientalongthe dire tion

Fig.1.5 Representation of the

(x, y, z)

laboratoryframe(gray) andof the

(X, Y, Z)

oil frame(bla k).Theradialandaxial omponentsofthemagneti eld

B

1

,areproje tedinto the oil frame.Only the upper part of the rotor isdisplayed for larity. As anillustration, for a given angle

θ

, the spin at a distan e

ρ

from the enter, experien es a eld

B

1

that admitsas proje tions in the oil frame the ve tors

B

ax

1

and

B

ra

1

.

the oil-wire paths on the ylinder along

z

, two te hniques were used : (i) the usage of a ombinationof ananti-Helmholtzpair with Golay oils, and (ii)the appli ation of the `target-eld approa h' as proposed by Turner [139℄.Experimental results were obtained and dis ussed, in luding imaging data and NMR spe tra. Magi angle gradients were appliedin high-resolution2D NMR inorder tosuppress long-range dipolar ouplings.

Based upon the transformation properties mentioned above of the se ond-rank gra-dient eld tensor, Cory et al. [96, 23℄, derived a gradient oil geometry that was (i) ompatible with existing MAS stator geometries, (ii) generating a

z

omponent of the stati magneti eld that in reased linearly along the MAS spinner axis, and (ii) where the

z

omponent of the gradient wasuniform inplanes perpendi ular to the spinneraxis

0Z

. Su h a gradient, when properly adjusted, should not lead to temporally modulated

NMR signal. One parti ular rotating spin will always sample the same magneti eld strength.

Fyfe and oworkers [60℄ as well as S hnell, et al. [53℄, proposed a design for their modied MAS system by winding an anti-Helmholtz oil pair on entri to the spinner axis.In that ase, however, the rotatingspin willexperien e a gradientmodulation.

Barbara, etal. [8,7℄ took adierentapproa h tond oil geometriesfor magi angle gradientsbyextendingthete hnique proposed byBowtelland Peters[13℄. Intheirdesign of magi angle gradient oils they relyon Turner's[139℄inverse target eld method.

To our knowledge, both types of gradient elds, either with magi angle gradients as proposed by Cory et al. [96, 23℄, and Barbara et al. [8, 7℄, or with on entri

anti-Helmholtzgradient oilsasproposedbyFyfe,etal.[60℄,andS hnell,etal.[53℄,havefound appli ationsinsolid-stateMASNMRinquiteavarietyofdierentNMRexperiments.For example, oheren e pathway sele tion was a hieved by means of pulsed gradients in

1

H MASdoublequantumexperimentsintheworkofSpiess,etal.[58℄ondipolarsolidsandin

27

Al 3QMASexperimentsonzeolitesbyFyfe [60℄.Similarexperiments wereperformedas wellfor gradientheteronu lear orrelationexperimentsby Maas, etal. [95℄,Be ause eld gradients provide spatial sele tivity, they an be applied to map the RF eld generated by theRF oilinMASprobes. Sin e

1

HFSLGde ouplinginsolidssensitivelydependson RF homogeneity,experimentswith pulsedeld gradients an bedesigned [16℄ thateither restri tthe sample volumeto ahomogeneousregion of the RF eld, oralternatively, an be used to map the RF eld distribution. Finally, similar to high-resolution NMR, in organi solids with residual solvents like in mi ro rystalline proteins, the need arises to suppress the solvent peak in

1

H MAS NMR experiments applying

B

o

pulsed gradients [53, 17℄.

1.5.2 Pulsed eld gradient inhomogeneities under MAS

The appli ation of eldgradientpulses toasample spinningatthe magi angleleads to a spe i set of onstraints regarding the spatial hara teristi s of the gradient pulse. Under MAS, aspin pa ketwilltravelalong a ir lein aplane perpendi ular tothe main rotor axis oriented at the magi angle. During their journey along the ir ular path, the spins should experien e a onstant eld gradient so that ea h of the spins in the ir le experien es the same gradient. Ideally the eld gradient should therefore be oriented exa tly along the axis of the magi angle. This ideal onguration means that a se ond refo using gradient pulse an be applied at any time during the rotation of the sample torefo us the entire magnetization.Ifthis ondition isnot satised,magnetizationlosses will be observed sin e the se ond gradient pulse will be not exa tly reverse the ee t of the rst gradientpulse. Signal losses were observed by Lippens etal.[148℄ in HSQC and diusion experiments whenusing non rotor-syn hronized gradient pulses.The remedy to this problem is to use pulse sequen es where gradient pulses are applied only when the rotor is lo ated at a given position. This solution requires that the delay separating two gradients pulses is set to an integer number of rotor periods. It is worth noting that a gradient oil generating a gradient exa tly at the magi angle may be less ae ted by these ee ts than a gradient oil onsisting of two anti-Helmhotz oils.

1.6 Con lusions to this introdu tion to HR-MAS

Here we briey des ribe the basis of HR-MAS instrument, starting from the sample holderinaninside-outexplanation:therotor,the oil,theMASgradients.Ourgoalisto pin-pointthemain hara teristi softhedierentprobesthat anbeused inanHR-MAS experiment. The versatility of HR-MAS makes it ompatible with MAS hardware origi-nallydesigned for high-powerappli ations, as itis the ase along this study. Nevetheless from this rst part it is lear that the oilremains the entral hallenge of an HR-MAS appli ation.In order to rea h a good resolution, the issues on erning sus eptibility and goodshimmingwillbeassumed fromnow. Inthis thesis,wefo us onthe taskof develop-ping new oil-stru tures, toimprovethe

B

1

homogeneity but alsoto redu e the thermal ee ts produ ed in visquous, ele tri aly dispersive samples, as those used in biomedi al appli ations.

Fundamental on epts, derived from

lassi al Ele trodynami s, to be

employed in Probe design. Engineering

approa h of

B

Maxwell equations

Theset ofMaxwellequationsappears fortherst timein1864,asasynthesisof F ara-day'slawsofindu tion,the lawsAmpere dis overedduringthersthalfofthenineteenth entury, as wellas Gauss laws on harges.The originalarti lewent so far inthe des rip-tion that it gave birth tothe rst uni ation of ele tri and magneti phenomena in the frameoftheso alled`Ele tromagneti Waves'[101℄.Theequationoflight,anthegeneral equations of wave propagation

4

, were already derived by J.C Maxwell in his founding paper.

2.1 Maxwell equations and the des ription of ir uits

We pretend here, only to introdu e their range of appli ations for the present work, and also to prove that all al ulations and models used for ele tri al al ulations, as for nodalanalysis, an benallyrefered toMaxwell's eldtheory. Inthis workwe willuse a ma ros opi des ription of these equations that allows us to onsider spatially lo alized sour esof urrent denoted by the surfa e density of urrent

J

s

and dened point- harges denoted by

q

. The benets of this lassi des ription are seen in a frame where ir uit theory has it's pla e. But in addition, we need to onsider the variation of magneti ux produ ed by the spin u tuations as a eld distribution. Indeed, the spins an not be purely des ribed as lo alized parti les. A eld's approa h should be employed at the mi ros opi level to hara terize the intera tions between the sour es of magnetization and the signal dete tion. This plays a role in the sequel to what we have to say later regardingre ipro ity (derivingthe equation 9.7), and diele tri dispersion inthe part IV (10.2).

4

After these prerequisites, Maxwell equations are rewritten as:

−→

rot ~

H = ~

J

s

+

∂ ~

D

∂t

−−→

rot ~

E =

∂ ~

B

∂t

div ~

D = q

Ω

div ~

B = 0

(2.1)

here

J

s

is dened in

A/m

2

,

q

Ω

the volumi density of harge in Coulomb.m

−3

,

E

~

is the ele tri eld ve tor (also alled

E

1

) in

V /m

,

D

~

is the ele tri ux density ve tor in

As/m

2

,

H

~

orresponds to the magneti eld expressed in

A/m

,

B

~

is the magneti ux expressed inGauss (CGS) or

V s/m

2

(SI).

To illustratethese notations, we an use the following ir uits models. We nd them usefull tointrodu e the sour esof EMeld. These what orrespond to lo alphenomenae andlaydowninsome(pre ise)partofthe ir uit.Thephysi al ir uitwilladmita ertain thi kness, so the ondu tors will dene a volume ( alled

Ω

1

). With this statements the two rst Maxwell equationsfrom2.1 an be respe tivelyillustrated with aparallel and a series ir uit.

2.1.1 Maxwell-Ampere in a parallel ir uit

The rst Maxwellequation, also alledMaxwell-Ampere equationdeals with urrents owing in parallel bran hes. As a appli ation of the Kir hho's node law the urrent in the main bran h is asum of the urrent owing inthe two other bran hes.

Fig. 2.1 Volume ir uit used to illustrate the rst equation of 2.1.

In addition,the surfa e is the ross-se tionof the wire,and a losed loop

l

,is dened tosurroundthis surfa e,asdes ribedbypi ture2.1. The urrentdensity

J

ows through

this surfa e

Σ

.

I

o

denotes the urrentboundtothe ir ulationalonga losedlooppath,ofamagneti

ex itationve tor

H

~

.Itisstated by Ampere's law. This losed loop, asit ir umvents the ele tri albran h, denes a Stokes surfa e (

Σ

in g. 2.1) insu h a way that the following surfa e integral an be formed :

I

o

=

I

Σ

−

→

_{H dl =}

Z

Σ

−

→

_{∇ ×}

−

→

_{H d~}

_Σ

(2.2) byappli ationoftheStokesintegraltransformation.Where

H

isa losedpathonasurfa e

Σ

.

Theothertwobran hes arryadispla ement urrent(

I

2

)andasour eof urrent (

I

1

). This sour eis produ ed by the ux of the surfa e density

J

s

. Sowe may write :

I

1

=

Z

Σ

−

→

_J

s

d~

Σ

(2.3)

Andatdispla ement urrent, produ edby the variationof hargesatthe surfa eof a ondu tor yieldsto:

I

2

=

Z

Σ

∂ ~

D

∂t

d~

Σ

(2.4)

I

2

has nothingtodowith movingpoint hargesbe auseasdened by J.C Maxwell

`Ele -tri al Displa ement onsists in the opposite ele tri ation of the sides of a mole ule, or parti les of a body whi h may or may not be a ompanied with transmission through thebody'[101℄.That'swhy weillustrate

I

2

asthe urrentina ondenser apa itor, alled displa ement urrent

Fig.2.2 Equivalent ir uit used to des ribe the Maxwell-Ampere equation.

The node equation is applied in the two last bran hes in su h a way that we have, afterremovingthe surfa e integral, and if we repla e the rotationaloperator (

rot

~

) by its

ve torialpendant(

∇×

~

) :

−

→

_{∇ ×}

−

→

_{H = −}

→

_J

s

+

∂−

→

D

∂t

(2.5)

Inthisequationweshouldpayattentiontothefa tthatbothlo alizedandeldterms are mixed.

−

→

_J

s

is a density of urrent, at one point of the ir uit. On the other hand

D

~

and

H

~

are elds i.e., spa ial distributions.

2.1.2 Maxwell-Faraday equation in a series ir uit

The derivation whi h an be done for the se ond equation, named Maxwell-Faraday equation, looks the same. We use the Kir hho equation for meshes, whi h means that the ve tor sum of voltagesequals zero.

Fig. 2.3 Volume ir uit used to des ribe the se ond equation.

In additionthe surfa eisarea dened bythe ir uit,andthe losedlooppath

l

,isthe ir uit path asdes ribed by pi ture2.3. If we assumethere is no eld inthe ondu tors,

V

o

denotes a voltage sour e .The onservative eld

E

~

stands perpendi ular between the

two surfa esofagap onsidered inthe bulk ir uit.Sin e the losedloop

l

en ompassthe ele tri al ir uit, aStokes surfa e is dended, and so the potential

V

o

an be written:

V

o

= −

I

Σ

~

E dl = −

Z

Σ

~

∇ × ~

E d~

Σ

(2.6)

Here again we onsider two ontributions tothis voltage

V

o

, indu ed by the variation of magneti ux. We assume two dierent sour es of magneti ux variation. The rst one is produ ed by the self indu tan e of the ir uit, alled the magneti ux density and denoted by

B

~

. The se ond one is somehow intera ting with the onsidered ir uit, by meansof mutual indu tan e (right hand of Fig.2.4). In the present ase this a tuator will be the distribution of spins present in the sample volume.It represents the sample magnetization

M

~

, obviouslynot deliveredwith the probe andatthe same time,strongly oupled to it. Both u tuations of the indu tan e a t over the same surfa e

(Σ)

. We

distinguish themin the followingequations :

V

1

=

∂

∂t

Z

Σ

~

B d~

Σ

V

2

=

∂

∂t

Z

Σ

~

M d~Σ

(2.7)

Fig. 2.4 Volume ir uit used to des ribe Maxwell-Faraday equation.

ThemeshesequationofKir hhoisappliedinthe ir uitdes ribed bygure2.4,sin e the surfa e

Σ

is the same for all wewrite the se ondMaxwell equation :

−

−

→

∇ × ~

E =

∂ ~

B

∂t

+

∂ ~

_M

∂t

(2.8)

in orporating the magnetization as ause of NMR signal, and as eld distribution, as it willbe lear whilederiving the signal equation later (inse tion 5.2of this part).

2.1.3 EM eld distribution and ir uit simulation

From the pre eding, we see howKir hholaws an givea ni e illustrationof the two rstMaxwellequations.Thisanalogyservesustorefer ir uitsimulationto ele tromagne-tism.Duringthisstudy alargenumberofprobe ir uitshavebeendesigned andanalyzed in the frequen y domain using a freeware program alled `Vipe '

5

. It onsists of a light C

++

odeusedto al ulatetheimpedan ematrixonalinearsystem.Nodalanalysiseasily treats apa itan es (

C

), indu tan es (

L

) as well as sour es of voltage and of urrent, as lumped elements. The geometry dependan e of

C

and

L

is the reason why nodal ana-lysis requires some prior knowledge, based on Maxwell equations. A good example, to

5

Vipe is developed under the terms of the GNU li ense,by J.Rossouw and E.Jansen, and an be obtainedatthefollowinghomepagehttp://vipe .sour eforge.net

showthedependen yof anodal-analysismodelon

E

1

elddistribution,isthe problemof stray apa itan estoground. These are of greatinuen e in the ir uit behaviorand an onlyberevealed when solving thefull 3D ele tromagneti distribution forthe onsidered stru ture.Nevertheless, assoonassour esof urrent,and voltage,aswellastheelements generating

B

~

elds (Indu tan es

L

)and

E

~

elds( apa itan es

C

)havebeenlo alized,an equivalent ir uit an beset todes ribethe impedan e, resonan e frequen y, and quality fa tors of a real ir uit.

Mostnetworkanalysisprogramswillformanadmittan ematrix(Y-matrix)internally and invert the matrix tond a solutionof the basi equation:

V = Y

−1

.I

The network oflumped elementshasnolength.Dephasingthrough propagationisnot onsidered there. The physi al phenomena are propagated immediately.This hypothesis may be true for a large numberof problems when the ex itation made by ele tri waves is larger than the distan es hara teristi s of the onsidered network. For example at 900MHzthewavelengthinfreespa eequals30 m.Inotherwords,assoonaswe onsider devi es longer than 3 m, we should onsider a transmission-line equivalent model in the ir uit, belowthese distan es stray indu tan es and apa itan es satisfythe network approa h.Onlyin ase of TEM mode the quasi-stati approa hbehind ir uitanalysisis valid. Resonan eappears whenastandingmodeisestablished. Thismode allowstosolve 3D problems in a plane as soon as the dire tion of propagation is known and supposed to be innite. The limitationstands in the fa t that most EM eld distribution present a dierent pattern than the TEM, and so they do not admit the quasi-stati approa h, this is the ase of most oils. Modeling a oilas anindu tan e in ir uit analysismay be possible in a rst approximation, but turns de ievingat high frequen ies, the usage of a heli al transmissionline modelisre ommended instead.

2.1.4 Maxwell equations for the mi ros opi Field

The equations given by Eq. 2.1 are suited for ma ros opi dete tion and, therefore, they over the engineering approa h of the present part. The auxiliary eld

D

~

and

B

~

are introdu edwhen notall hargesare in ludedassour es. Theseeldshowever ontain impli itelypartsofthesour es,namelythebound hargeand urrentdensities,intheform ofpolarizationeldsand urrents.Theseauxiliaryeldsin ludethematerialresponse(the ele tri polarizationandthe magnetization)totheappliedelds;theyare onne tedwith

~