Le shi et l'uta

Depuis l'antiquité, il existe deux veines distinctes dans la poésie japonaise : le shi 1 et l'uta 2 .

Le terme shi signifiait au Japon prémoderne la poésie écrite en chinois (les classiques chinois ainsi que les textes écrits par les Japonais) 3 .

Au tournant du XIXe au XXe siècle, le genre décline au Japon et le même mot va être utilisé pour traduire le terme anglais « poetry ». Un genre nouveau, sous l'influence de la poésie occidentale, voit le jour dans les années 1880 ; il se fait appeler « shi au style nouveau » (shintai-shi). Un premier recueil collectif du genre paraît en 1882 4 . Plus de la moitié des textes sont des traductions de poèmes en anglais. Les auteurs / traducteurs étaient de jeunes intellectuels de l'élite, pas nécessairement des littéraires. Le « shi au style nouveau » s'écrivait encore en japonais ancien 5 , et le rythme de base 7-5 mores 6 qu'il hérite de la tradition de l'uta (voir infra) sera pendant longtemps tangible, mais après de nombreuses tentatives, cette production n'aboutit pas à un véritable genre versifié 7 .

Le genre mûrit en tâtonnant tant sur le plan thématique que sur le plan formel, et au fur et à mesure que les poètes se le réapproprient, la précision « au style nouveau » se fait oublier et donne naissance au genre communément appelé shi. Après un âge d'or des textes en ancien 1 Ce mot signifie en chinois la poésie. 2 La première signification de ce mot en japonais est la chanson. 3 Le Japon antique étant sous l'influence culturelle de Chine, d'où il a importé l'écriture, la rédaction en chinois est une pratique qui remonte aussi loin que l'histoire de l'écriture. 4 Shintai-shi shô (Sélection de shintai-shi), Tôkyô, 1882. 5 Nous employons l'expression «le japonais ancien » pour le « bungo (langage dit écrit) » la langue écrite avec des mots variables aux conjugaisons anciennes. Il s'oppose au « japonais moderne », le « kôgo (le japonais dit parlé) ». Il s'agit du japonais écrit standard mis en place avec la modernisation, sur le modèle de la langue parlée de l'époque. 6 La more est l'unité minimale des sons japonais, l'équivalent de la syllabe. Les mores se différencient des syllabes en ce qu'elles peuvent ne pas comporter de voyelle et qu'elles sont supposées avoir toutes la même durée. Cette deuxième particularité a une incidence importante sur la versification japonaise, car elle permet d'entendre au sens musical l'identité de deux séquences au même nombre de mores. 7 Le rythme 7-5 sera plutôt évité après la seconde guerre mondiale.

 Ce mot, qui signifie littéralement « mots légers », est mis en circulation par Masaoka Shiki (voir infra) dans les années 1880-1890. 11 Ce nom du genre est venu du nom de son fondateur Karai Senryû (1718-1790). 12 On pourrait avancer l'hypothèse d'un tournant qui pourrait coïncider avec le passage au langage moderne, mais la raréfaction de pièces très longues ne signifie pas nécessairement l'apparition de pièces très courtes.

2 japonais rythmés principalement en mesure 7-5, émergera une poésie en japonais moderne, sans versification dans les années 1910/20. Le terme uta désignait traditionnellement la poésie développée en dehors de l'influence directe de la culture chinoise. En position de suffixe, son pictogramme se prononce /ka/, et on le retrouve dans plusieurs mots désignant les genres relevant de l'uta. Tel est le cas du mot waka 8 , en opposition avec la poésie en chinois. En découlent trois formes versifiées pratiquées actuellement. Le tanka 9 (versifié en unités de 5-7-5-7-7 mores), le haïku [START_REF] De | Cheveux emmêlés[END_REF] (ayant le mètre 5-7-5 et l'obligation de l'usage d'un mot évoquant une saison, répertorié tel quel) et enfin le senryû 11 (ayant la même contrainte métrique que le haïku, mais sans mot de saison ni esthétique de retrait de soi).

Cette situation de coexistence (la poésie non versifiée appartenant au shi et la poésie versifiée de la tradition de l'uta) rend assez singulier l'état de la poésie brève. Dans cet article, nous tenterons de mettre en évidence les différentes facettes de la modernité en jeu dans les deux branches.

La poésie brève non versifiée

À première vue, la composition des textes très brefs ne constitue pas une option très fréquente chez les poètes du shi. Lorsqu'on a créé la « poésie au style nouveau », beaucoup ont cru à la naissance d'un genre épique, historiquement quasi absent de la poésie japonaise, et il n'était pas rare qu'un texte dépasse des centaines de lignes. Il semble donc que les premiers shi comptaient plus de textes longs qu'aujourd'hui 12 . Toutefois, on peut se souvenir de certains textes de la première moitié du XXe siècle, par exemple de ces pièces de 3 Yamamura Bochô (1884-1924) Ces poèmes d'apparence simple ont toutefois des structures sonores bien élaborées, qui permettent de sonder leurs proximité et distance par rapport à la tradition de l'uta. « Prière » est composé d'unités de 5 et de 7 mores (5-7/5-7/5/5-7). Dans « Main », ces unités 13 Nous transcrivons les noms japonais dans l'ordre du patronyme suivi du nom de personne, sauf dans les informations bibliographiques données dans les notes. « traditionnelles » se retirent derrière les lignes en neuf mores, même si le poème est introduit par une séquence rythmique ressemblant à la mesure 7-5 16 . Les lignes en neuf mores (les lignes 3,5 et 6) comportent deux types de composantes poétiques sonores de nature différente : la répétition qui participe à la structuration du texte par le retour des mêmes éléments 17 , et des chaînes sonores qui font ressortir des motifs rythmiques 18 . Il s'agit d'autant de techniques sonores que la poésie de la tradition du shi a héritées de l'uta dans les années 1880-1890, consciente de l'inefficacité en japonais de rimer comme ses modèles occidentaux.

Yamamura appartient à la première génération des poètes qui ont commencé leur carrière sans avoir recours de façon systématique au rythme 7-5. Il est l'un des pionniers qui inaugurent la modernité, car c'est lui qui, pour la première fois, a pris pleinement conscience de la potentialité de la page et des dispositifs graphiques. Les poèmes cités sont contenus dans le recueil Sei purisumisuto 19 Sokuryô-sen 21 , le premier recueil de l'auteur, publié en 1930 et qui contient ce texte bref, 16 En réalité, il s'agit du rythme 7/6, mais le son /n/ placé à la fin de la ligne 2 peut être ignoré. 17 Trois fois la séquence [nom en quatre mores + particule « no » + nom + particule « no » + nom], deux fois le mot kanashimi (tristesse) et trois fois te (main). 18 Les consonnes /n/ et /te/ semblent « resserrer » le tissu sonore. Les voyelles /a/ et /i/ sont placées au début de deux lignes, avec la consonne /sh/ entre eux (/a/ + /a/ + /shi/ + /i/). Dans la troisième ligne, elles sont placées dans l'ordre inverse (/i/ + /shi/ + /a/). Ce chiasme fait ressortir très fortement la dernière voyelle ouverte /a/. 19 Le titre signifie en français Saint Prismiste. 20 « Yuki », Tarô wo nemurase, Tarô no yane ni yuki furitsumu. / Jirô wo nemurase, Jirô no yane ni yuki furitsumu. (« 雪 », 太郎を眠らせ、太郎の屋根に雪ふりつむ。 / 次郎を眠らせ、次郎の屋根に雪ふりつむ。), Shinkichi Itô, Gendai-shi no kanshô (Lectures de la poésie contemporaine), tome II, Shinchô-sha, Tôkyô, 1974, p. 139. Tarô et Jirô sont des noms de personne masculine communs.

avait également l'ambition d'explorer les différents potentiels de la poésie libre. L'accent y est mis plutôt sur la poésie en prose, mais il porte en exergue un poème de deux lignes, rythmé exactement comme un tanka 22 . Quant à la prosodie de « Neige », elle se mesure, de loin, en rapport avec le rythme du haïku, car, comme un haïku, ces deux lignes comportent chacune trois unités rythmiques (8-7-6). La répétition quasi identique des deux lignes peut refléter le souci d'intégrer la « poétique du retour du même » à l'échelle des vers, fondamentale à toute métrique occidentale.

Nishiwaki Junzaburô (1894-1982) 21 Bateau de sondage en français. 22 Haru no misaki / tabi no owari no / kamome-dori // uki tsutsu / tôku nari ni keru kamo, (春の岬旅のをはりの 鴎どり浮きつつ遠くなりにけるかも, Un promontoire du printemps / de la fin du voyage / des mouettes / tout en flottant / s'en vont au loin). 23 Le titre en alphabet latin dans l'original signifie « Ambarvales ». Il s'agit d'une fête religieuse de la Rome antique en l'honneur de Cérès, déesse de l'agriculture. 24 « Tenki » / (Kutsugaesareta hôseki) no yôna asa / Nanbito ka toguchi nite dareka to sasayaku / Sore wa kami no seitan no hi (« 天気 », (覆された宝石)のような朝 / 何人か戸口にて誰かとささやく/ それは神の生誕の 日), Lectures de la poésie contemporaine, tome 2, op.cit., p. 381. 25 « …a youthful wight / Smiling beneath a coral diadem, / Out-sparkling sudden like an upturn'd gem / Appear'd … », « Endymion III », The poetical Works of John Keats, réimpression des éditions originales avec les notes de Francis T. Palgrave, Macmillan and Co, London, 1884, p.127. est placé en tête d'une série de trois pièces de longueur variadeble. Ces jeux de discontinuité seront poussés plus loin dans son deuxième recueil Tabibito kaerazu 26 , composé de 168 fragments numérotés. Il semble que la complexification du jeu de la discontinuité devienne une des modalités plus importantes des poèmes courts de cette tradition.

Dans les trois cas, il s'agit de pièces insérées dans les textes de longueur variée -la brièveté devrait donc être évaluée comme élément de la variation à l'échelle du recueil -, composés par des auteurs soucieux de recherches formelles, pendant la jeunesse de la poésie moderne.

On ne saurait oublier, aux antipodes de cette filiation, la présence de poèmes brefs écrits par des auteurs qui se tiennent à l'écart des aléas des mouvements littéraires, donnant à l'écriture en quelque sorte un statut de cahier de bord du quotidien et de divers cheminements spirituels.

Sans apparaître dans les anthologies sommaires, leur production, jamais très abondante, s'étale jusqu'à nos jours 27 . Citons deux pièces écrites dans un langage simplissime, mais qui continuent à interpeller, du recueil Aki no hitomi (1925) 28 de Yagi Jûkichi (1898-1927). Le

La poésie brève versifiée

Nous centrerons ici nos propos sur les travaux de Masaoka Shiki, le fondateur du haïku et du tanka ; nous nous attarderons plus sur le haïku, qui semble condenser les particularités des formes brèves versifiées. Nos observations porteront sur la période autour du tournant du XXe siècle, qui précède d'une décennie celle qui a vu la production de remarquables poèmes non versifiés brefs de la première catégorie, résultant d'une nette intention de renouvellement formel. Nous écartons le senryû de nos réflexions : ce genre, historiquement distinct du haïku, commence, pour sa part la meilleure, à perdre toute frontière nette avec celui-ci. Le genre est populaire car c'est souvent à travers cette forme que la satire s'exprime. Il sert par ailleurs à promouvoir tels ou tels produits, pratiques, ou groupes sociaux, mais on hésitera à y 30 « Kyôchikutô », Ôzora no moto ni shinuru / Hatsunatsu no kokoro aa tada hitori / Kyôchikutô no kurenai ga / Hatsunatsu no kokoro ni shimiteyuku (« 夾竹桃 »,おほぞらのもとに 死ぬる//はつ夏の こころの ああ ただひとり//きょうちくとうの くれなゐが//はつなつのこころに しみてゆく), ibid., p.57. 31 Haruki Katsuhara, « 'Shiika no kindai' o meguru ni, san no kôsatsu : 'Shintaishi-shô', Nakamura Shûkô, Kindai Imayô (Deux ou trois réflexions sur la 'modernité de la poésie' : Extraits de shintaishi, Nakamura Shûkô, Imayô moderne) » dans Bungaku (Littérature), Iwanami Shoten, Tôkyô, 2008, p. 45. reconnaître une qualité littéraire, de la même manière que pour les phrases publicitaires.

Le haïku et le tanka n'appartiennent pas à la modernité de la même manière que le shi. Ces genres ont résisté plus que partout ailleurs au passage au japonais moderne 32 . De plus, ils ne connaissent pas l'usage en tant qu'éléments poétiques, de la page et des lignes, tel que nous l'avons observé pour la poésie non versifiée 33 . Ils sont modernes (et sans doute postmodernes), essentiellement par leur esthétique du shasei 34 , concept forgé par Masaoka Shiki (1867-1902). Il s'agit de la « description de l'objet tel qu'il est » 35 , mais il ne faudrait pas l'associer à la description du type roman réaliste. Donald Keen décrit avec vivacité la rencontre de Shiki vers 1894 avec le peintre du style occidental Nakamura Fusetsu (1866-1943), événement qui lui a été un déclic pour comprendre la direction à prendre pour redonner vie au haïku, devenu à ses yeux par trop vulgaire, ludique et intellectuel : Selon le critique Kajiki Gô, il faut attendre fin 1898 pour que Shiki affirme la part subjective de ce croquis, quand il emploie dans un essai le terme « kanjôteki shasei » 37 . Prenant pour exemple la description d'un poisson, celui-ci écrit qu'il ne faut pas se contenter de dessiner 32 Dans les années 1970 pour le tanka. Le haïku n'a pas encore franchi ce cap. Il ne le franchira sans doute jamais. 33 À la différence des genres qui portent les mêmes noms en Occident, on écrit ces textes en une seule ligne. Les unités métriques des poèmes brefs versifiés sont suffisamment perceptibles sans les faire doubler par les lignes. 34 Croquis sur le vif. Sur la notion du shasei, voir Makiko Andro-Ueda, « Le shasei dans l'esthétique de la poésie versifiée moderne (haiku et tanka) » Japon pluriel 8 : La modernité japonaise en perspective, Philippe Picquier, 2011, p. 135-155. 35 « Jojibun (Récit des faits) », 1900. Shiki zenshû (OEuvres complètes de Shiki), tome 14, Kôdan-sha, Tôkyô, 1976, p. 241-249. 36 Donald Keen, Masaoka Shiki, traduit d'anglais par Yukio Kakuchi, Shincho-Sha, Tôkyô, 2012, p. 118-119. Keen s'est appuyé sur le témoignage de Shiki dans « E (La peinture) », OEuvres complètes de Shiki,op.cit.,tome 12, Littéralement, croquis émotionnel. Gô Kajiki, Shasei no bungaku : Shiki, Sachio, Takashi (Littérature du croquis sur le vif : Shiki, Sachio, Takashi), Tanka-shinbun-sha, Tôkyô, 2001, p. 52-54. un grand point pour signifier qu'il y a des écailles (comme la peinture japonaise le faisait à une époque), mais ce n'est pas non plus utile de les dessiner toutes. Il faut simplement le dessiner tel que l'homme le perçoit et, en le faisant, viser à la fusion de la subjectivité et de l'objectivité 38 .

Malgré sa courte vie de trente-quatre ans, entravée pendant de longues années par une tuberculose pulmonaire, incurable à l'époque, Shiki était un homme de lettres très prolifique.

En plus d'être le « père » du haiku et du tanka modernes, il est l'un des auteurs plus importants dans le domaine de la poésie en chinois de l'ère Meiji 39 . Shiki est également connu pour ses essais rédigés dans son lit de malade, où le regard du shasei prend une étonnante acuité. Certains parlent de son influence sur l'écriture de son ami romancier Natsume Sôseki (1867-1916) 40 . Il a également laissé des nouvelles, une pièce du théâtre nô, des dessins, une classification de 120 000 haïkus de tous les temps dont il a pu prendre connaissance, un répertoire de rimes, etc. Sa base d'activités a été le quotidien Nippon 41 , où il de Shiki, tome 14, op.cit., p. 214-230. 39 1868de Shiki, tome 14, op.cit., p. 214-230. 39 -1912, l, l'ère impériale sous laquelle le Japon procède à la réouverture de ses frontières et la modernisation du Japon. 40 Gô Kajiki,op.cit.,p.52. 41 Le Japon, fondé en 1889 par le nationaliste Kuga Katsunan (1857-1907). 42 Poème léger en chaîne (5/7/5+7/7+5/7/5+7/7…), composé vers par vers par plusieurs auteurs, pratiqué depuis le XVe siècle. devant la neige. L'esthétique du shasei, inspirée par l'art pictural, peut être ainsi appliquée à des représentations non visuelles.

Feuilletant les haïkus de Shiki, on éprouve parfois l'impression de s'égarer dans le Japon de 1900, et de se promener tel un ethnologue ou un réalisateur de documentaires. Le shasei est sur le fond un appareil pour faire parler le monde. La modernité du haïku réside précisément dans ce besoin de l'extériorité. Mais c'est également pour cette place accordée aux faits, impliquant un manque de révérence devant l'intériorité souveraine de l'homme moderne, que le genre s'oppose à l'ensemble de la poésie moderne de la tradition du shi, ainsi qu'au tanka moderne (voir infra), expressions des rêves (ou des cauchemars) de la subjectivité moderne.

Ayant pressenti que son approche du réel -le volet novateur de l'esthétique du haïku -peut être appliquée également pour réformer le tanka (appelé jusqu'alors waka), genre directement lié à la tradition de la noblesse et de l'uta, il rédige en 1898 une série d'articles intitulée « Uta yomi ni atauru sho 46 », proposant une poésie de l'esquisse de ce qui est immédiat, libre de son arsenal rhétorique séculaire.

À la différence de ce qui s'était passé avec le haïku, où le shasei avait été partout adopté et devenait un dogme presque exclusif, la revendication de Shiki rencontra une opposition violente, essentiellement de la part de poètes romantiques qui voyaient dans le contact avec la poésie occidentale l'opportunité de libérer, ou de redécouvrir, l'expression de la passion (amoureuse). Le shasei et le romantisme vont ainsi en se stimulant jusqu'au milieu du XXe siècle. Précisons toutefois que le tanka romantique ne produira pas de dogme esthétique de l'envergure du shasei, tandis que celui-ci est approfondi par les successeurs de Shiki, notamment par Saïtô Mokichi (1882-1953), et règne sur la production du tanka jusqu'au moment où il se fait frontalement contester dans les années 1950 par l'avant-gardiste Tsukamoto Kunio (1920-2005).

Citons ici un tanka bien connu de Shiki et, en comparaison, un tanka fameux de Yosano 46 « Écrits à l'adresse des auteurs de l'uta », en français.

  de feuilles qui tombent Tendant la main du chagrin Secouant l'arbre La main du ciel en chagrin La main d'un parent d'automne Une spiritualité singulière se dégage de « Main », sans que l'on comprenne le rapport entre « la main du ciel » et la main qui « secoue l'arbre », ni le statut de la « main d'un parent » posée sous la lumière d'automne. Dans ce poème, la linéarité de la phrase ne porte pas seule le sens. Les lignes résonnent entre elles, grâce aux chaînes paradigmatiques constituées tantôt par des champs lexicaux (arbres, minéraux…), tantôt par la représentation organisée autour d'un axe vertical. Cet usage poétique de la page et des lignes fait partie des particularités de la modernité en poésie.

  de 1915, qui réunit des textes de styles variés, de piécettes comme celles-ci jusqu'à un long poème en prose. Différents effets visuels sont recherchés pour créer des signifiances extra-syntaxiques, mais le livre rencontrera un mur d'incompréhension pendant un demi-siècle. Parmi les pièces courtes ayant bercé notre enfance, certains penseraient au distique célèbre de Miyoshi Tatsuji (1900-1964) : « Neige » Elle fait dormir Tarô, la neige tombe s'accumule sur son toit Elle fait dormir Jirô, la neige tombe s'accumule sur son toit 20

  situe son écriture plus loin de la rythmique de l'uta. Voici le premier poème de son premier recueil Ambarvalia 23 , daté de 1933 : « Le temps qu'il fait » Matin comme des (bijoux renversés) Une personne au seuil de la porte chuchote avec quelqu'un C'est le jour de la naissance d'un dieu 24 Cette image éclatante d'un matin sur un fond de silence mis en relief par le chuchotement de deux personnages sans visage, donne une forte impression d'unité, comme s'il s'agissait d'une icône païenne, mais en réalité, le texte est traversé par une problématique complexe d'intertextualité et de discontinuité. Avec l'emploi des parenthèses autour du segment « bijoux renversés », le poème insiste sur le statut citationnel de celui-ci (que ses premiers lecteurs n'ont pas vu), tirée d'« Endymion » de John Keats 25 . Les deux lignes qui suivent sont inspirées d'une illustration d'une édition du XIXe siècle des Contes de Canterbury. Le poème

26«

  premier poème repose sur une symétrie simple entre deux parties, chacune composée d'une seule phrase. Dans le deuxième, les phrases et les lignes sont mises en tension, de sorte que le segment « ah tout(es) seul(es) » est suspendu syntaxiquement entre le narrateur et les fleurs, qualifiant les deux à la fois. Se crée une image d'osmose entre un jeune homme en perdition et des fleurs au parfum suffocant. Le voyageur ne revient pas en français. 1947. 27 Par exemple, nous pensons spontanément à Sugiyama Heiichi (1914-2012). 28 Prunelles d'automne en français. 29 « Kumo », Kumo no aru hi / Kumo wa kanashii // Kumo no nai hi / Sora wa sabishii (« 雲 », くもの ない日 /くもはかなしい//くものある日/そらはさびしい), Aki no Hitomi, Fuji-insatsu, Tôkyô, 1925, p. 61. 7 Le daphné » Sous le grand ciel meurt Le coeur du début d'été ah tout(es) seul(les) L'écarlate des fleurs de daphné Imprègne le coeur du début d'été 30 Nous avons observé une relative rareté des poèmes brefs non versifiés et une bipolarisation entre les textes composés avec l'ambition de trouver un nouvel horizon collectif, et les pièces « personnelles », au langage simple. Pour la transition à l'examen des poèmes versifiés, remarquons un fait relativement marginal : Katsuhara Haruki note une production importante de poèmes composés d'un seul quatrain, versifié en 7-5, à partir de 1888 durant une dizaine d'années 31 . Pour expliquer cette disparition après dix ans, le critique avance l'hypothèse d'une concurrence avec le haïku et le tanka.

  Fusetsu a insisté sur le caractère réaliste et la vraisemblance de la peinture occidentale. À la différence de la peinture japonaise, stylisée en général, la peinture occidentale était fidèle à l'objet. Par ailleurs, la peinture japonaise avait tendance à traiter seulement les paysages déjà peints et reconnus comme beaux par tous, tandis que la peinture occidentale prenait comme objets tous les paysages, même ceux qui n'ont apparemment pas de charme. Shiki se défend : « Mais le Mont Fuji est une belle montagne ». Fusetsu répond : « Elle est vulgaire. » Shiki dit : « Le pin, n'est-il pas un bel arbre ? » « Il est vulgaire. » répond Fusetsu 36 .

  s'était fait engager fin 1892 comme journaliste culturel. Il y publie pêle-mêle poèmes, textes critiques et essais théoriques. Dès 1893, Shiki propose d'isoler du haïkaï-renga 42 le premier vers mesuré en 5/7/5 et, par la suite, pour compléter le vide créé par la suppression du reste du poème, d'inscrire le fragment dans un réseau de signifiants organisés selon la logique du shasei, ce qui constitue une attitude nouvelle. D'autre part, comme le veut la tradition, dans l'intertextualité avec les textes canoniques, à l'usage d'un mot associé à une saison. Voici deux célèbres haïkus de Shiki, qui illustrent bien cette approche du réel. Nous empruntons la présentation en trois lignes du haiku occidental : 38 « Bungaku Bijjutsu hyôron (Critique littéraire et artistique) », décembre 1898. OEuvres complètes

  frontières51 . Tout en préservant le shasei comme doctrine de base, le haïku contemporain semble osciller entre des pièces très légères, frôlant le non-sens, et des textes qui trouvent leur authenticité à ne pas innover. Du côté du tanka également, on constate une bipolarisation entre la légèreté et l'anachronisme. La poésie versifiée manque plutôt de souffle au Japon, au moment même où, dans les pays occidentaux, est reconnue sa spécificité au sein de la famille de la poésie 52 .En essayant d'approcher la poésie brève en tenant compte de la relation complexe entre la poésie sans versification, implantée avec la modernisation, et la poésie traditionnelle versifiée à la recherche de la modernité, nous avons été conduite à nous intéresser à la période des trois décennies partant du tournant du XXè siècle. Le shi s'attache entre autres à forger des outils prosodiques adaptés jouent sur la répétition non rimique et l'intertextualité, le haïku, et partiellement le tanka, cherche le moyen de faire surgir le réel comme intertexte. A partir de ce premier constat, il serait intéressant d'établir un corpus des textes des deux branches traitant de sujets proches, dans l'espoir d'aboutir à une étude de cas sur les effets de la versification sur la temporalité, l'intertextualité et la représentation de la poésie brève japonaise.BibliographieANDRO-UEDA, Makiko, « Le shasei dans l'esthétique de la poésie versifiée moderne (haïku et tanka) », Japon pluriel 8 : La modernité japonaise en perspective, Philippe Picquier, Arles, 2011, p. 135-155. ANDRO-UEDA, Makiko, « Le sujet lyrique du haïku moderne », Modernités 36, Soi disant. Poésie et empêchement, Presses universitaires de Bordeaux, Bordeaux, 2014, p.47-66.

  14 « Inori », Tsuribari zo sora yori taretsu / Maboroshi no kogane no uo-ra / Samishisa ni / Samishisa ni / Sono hari wo nomi. (« いのり », つりばりぞそらよりたれつ /まぼろしのこがねのうをら/さみしさに/さみしさに/そのはり をのみ。), Sei purisumisto, Ningyo shi-sha, Tôkyô, 1915, p.91-92. La traduction des textes cités est faite par nous sauf indication contraire.

15

« Te », Miki wa shirogane / Chiru ha no kin / Kanashimi no te wo nobe / Ki wo yusuru / Kanashimi no ten no te / Nikushin no aki no te., (« 手 », みきはしろがね /ちる葉のきん /かなしみの手をのべ /木を揺る /一本の天 の手 /にくしんの秋の手。), ibid., p.51-52.

Voir par exemple Ogawa Keishû, Gendai haiku no kaizu : Shôwa sanjûnen sedai haijin-tachi no yukue (La carte marine du haïku contemporain : ce que sont devenus les poètes de haïku de la génération des années trente de l'ère impériale Shôwa),Kadokawa gakugei-shuppan, 2008.

On voit dans ces productions exploitées des potentiels parfois oubliés par le haïku japonais. Par exemple, les habitants de Vilnius (Lituanie), ont publié un recueil en l'honneur de la ville, en mettant à profit, non les mots de saison mais les toponymes, -une possibilité évoquée par Shiki -ou encore, le haiku macédonien est lié à la tradition du « dit memorable », loin de l'éphémère, cher au haïku japonais (Frosa Pejoska-Bouchereau, « De la poésie orale au haïku. La poésie macédonienne, une tradition séculaire », communication dans le cadre du séminaire doctoral de l'Inalco, « Les poésies du monde : vers une approche formelle », 2014.).


domain: phys spi

text:
C Tannous

D Weber

Magnetic properties of crystals

Keywords: numbers: 51.60.+a, 74.25.Ha, 75.00.00, 75.60.Ej, 75.75.+a Keywords: Magnetic properties. Magnetic materials. Hysteresis. Magnetic properties of nanostructures Contents I. Introduction II. Taxonomy of Magnetism energy Ean D. Magneto-crystalline anisotropy energy Emc E. Magneto-elastic anisotropy energy Eme F. Exchange energy

Soiid state magnetism is introduced to Physics and Electrical engineering students from a practical point of view enabling them to write energies, fields and forces for any magnetic system while avoiding pitfalls and traps originating from different approaches or systems of units. Prospects and applications of magnetism are also described.

I. INTRODUCTION

Partially filled atomic shells are responsible for magnetism in 79 elements out of 103 of the periodic table.

In solids, magnetism is scarcer than its elemental atomic counterpart, since only transition and rare earth metals (20 elements: 8 Ferromagnetic and 12 Antiferromagnetic) possess magnetic properties albeit with moments smaller than their atomic counterpart. This is due to the fact while atoms are distant from each other in a gas, in a solid crystal, ions are separated by small distances (few Å). In addition, there are environmental issues around the ions creating a crystal field impacting magnetic properties as well as a surrounding Fermi sea or electron gas that interact with the ions. Moreover a small inter-ion distance favors quantum effects (orbital overlap, exchange and spin effects) that also interfere with magnetic properties.

Traditionally solid-state magnetism has been confined to information storage (analog and digital tapes, hard disks...), electro-mechanical (levitation, motors...) and power systems (transformers, eddy currents...), compasses, Electromagnetic Interference (EMI) filters, sensors and actuators... whereas electronics has been used for computing, signal processing and communication.

In microelectronic technology a practical measure of miniaturization is the minimum feature F m (also called process node). It is the scale used by microelectronics foundries to control all sizes within chips and circuits such as length, width and depth of various properties (gate, gate-oxide, contacts, current transport channels...) related to some device (transistor) fabrication or in making contacts between different devices (metal wire width and thickness). Microelectronic circuits do not use inductance but telecommunication devices and circuits pertaining to RFIC (Radio-Frequency Integrated Circuits), Smartphones e-tablets... and RFID (Radio-Frequency Identification) need inductance whose miniaturization took some time to be solved due to a strong decrease of inductance quality factor with scaling [START_REF] Lee | The design of CMOS radio-frequency integrated circuits[END_REF].

As technology advanced over the years and F m became smaller and smaller (as in Table 1) getting closer to the nm and atomic dimensions triggering quantum effects (wave-function, exchange, spin, related effects...), making the number of transistors progress into billions while pushing the Joule effect to explode, solid-state magnetism based electronics became more and more attractive. In magnonics, for instance, microwave devices are extended to the nm scale.

In fact, magnetism is enhanced by smaller F m (magnetization is larger in clusters, nanometric grains, thin films than in bulk solids), promises zero Joule effect (in Spintronics, exchange effects allow to transfer information without charge transfer), absence of characteristic length (since cutting a magnet produces another magnet, regardless of the number of cuts), non-volatile memory like MRAM or Magnetic RAM. MRAM blurs the boundary between mass and memory storage, works without computer booting since the operating system with all applications are permanently stored once for all, Quantum computing and processing... The origins of Magnetism are linked to the many flavors of Electromagnetism (EM) depending on the way a magnetic field is generated. The Bohr-Van Leuwen theorem adds another brick required in solid-state magnetism by appealing to Quantum Mechanics (QM) essential elements such as exchange, , spin and other QM hallmarks in order to tackle the very existence of a magnetic moment, the origin of magnetic anisotropy (sensitivity of magnetization orientation with respect to intrinsic axes within the material)... There are several types of magnetic field generation:

1. From classical current (Coulomb-Oersted type): Faraday and Ampère laws as well as Biot-Savart. A current loop carrying a current I creates [START_REF] Landau | Electrodynamics of Continuous Media[END_REF], at a distance R, a potential vector A and induction B given by:

A = µ 0 I 4π dl R , B = curlA = µ 0 I 4π dl × R R 3 (1)
2. From Electricity and Special Relativity (Lorenz and Einstein type): Given a Galilean frame S with only an electric field E, in another Galilean frame S moving at constant velocity v with respect to S, we get from Lorentz transformation an induction B || = B || = 0 and B ⊥ = -γ(v × E) ⊥ /c 2 in S . We have γ = 1 √ 1-v 2 /c 2 and B || , B ⊥ are the inductions parallel and perpendicular respectively to velocity v.

3. From QM and matter (Bohr-Van-Leuwen type):

• Moments originate from atomic L, S, J, molecular Λ, Σ... and nuclear I quantum numbers and are related to Bohr or nuclear magnetons given by µ B = e /2m e and µ N = e /2m p respectively. m e and m p are the electron and proton masses whereas e is the electron-proton charge. Hence µ N ≈ µ B /2000 since m p = 1836m e .

• In a given solid, these moments are influenced by the local environment (crystal field) causing quenching of orbital angular momentum L implying its quantum average L = 0 in transition metals [START_REF] Kittel | Introduction to Solid State Physics[END_REF].

• Magnetic moments produce a field in 1/r 3 where r is observation distance like electric dipoles

• In the homogeneous case we have the macrospin representation which is equivalent to a coarse graining approximation consisting of averaging an ensemble of individual moments into a single average moment per unit volume M (macrospin magnetization).

• In the inhomogeneous case (M (r) is space dependent) we have domains that produce a field from Maxwell pole density div(M ).

4.

From QM notions such as Exchange, Spin, Fine and Hyperfine interactions...(Heisenberg, Pauli, Dirac... type):

• Magnetic exchange arise from wave-function overlap and electrostatic interactions

• Magnetic anisotropy arises from spin-orbit coupling

• Magnetic Resonance Imaging arises from nuclear spins and is sensitive to hyperfine interactions Basic solid magnetic materials are Ni, Co and Fe with their properties in table 2. M s is the saturation magnetization corresponding to total alignment of all moments in the solid:

M s = max |M |, that is M s = N A n B µ B /v where N A is Avogadro number, n B the
number of magnetons per mole, µ B = e /2m e is Bohr magneton, the magnetic moment of a single electron and v is unit volume. T c is Curie temperature signaling moment alignment loss.

Material Ms(emu/cm 3 ) at 300K Ms(emu/cm 3 ) at 0K nB at 0K molar mass (g) density (g/cm Basic ferromagnetic Ni, Co and Fe solids with their saturation magnetization (in emu/cm 3 since magnetization is moment per unit volume), effective number of Bohr magnetons n B per mole, molar properties and Curie temperature (adapted from Kittel [START_REF] Kittel | Introduction to Solid State Physics[END_REF]). Fe has the largest value M s = 1707 emu/cm 3 among transition metals.

Note that Kittel [START_REF] Kittel | Introduction to Solid State Physics[END_REF] uses gauss instead of emu/cm 3 units.

There are also different flavors of EM Theory depending on the various treatment of some inherent aspect to it such as Current, Force, Retardation Effects...as illustrated in the case of Weber versus Maxwell EM [4; 5]

(see Table in Appendix D).

A number of issues are also to be addressed by any complete EM theory such as EM duality, Symmetries, Emerging issues related to topological materials, QM effects in Cavity QED, Spintronics, conventional and non-conventional Superconductivity... A simple way for understanding the difference between electricity and magnetism on the Maxwell level is that electricity is dominated by polar vectors such as E, P , D... whereas magnetism contains axial vectors [START_REF] Landau | Electrodynamics of Continuous Media[END_REF] such as H, M , B ... with different behavior with respect to plane-mirror symmetry since an axial vector is in fact a rank-2 tensor element. For instance let A = (A x , A y , A z ), be a 3D polar vector, its mirror image with respect to the x -y plane is A = (A x , A y , -A z ). Taking a typical axial vector A as the cross-product of two polar vectors B × C and if we perform the mirror image with respect to the x -y plane of B, C and take their cross product to get

A = (-A x , -A y , A z ).
A number of issues related to EM theory, its meaning, developments and special features proper to Electricity and Magnetism separately should be addressed such as:

• Maxwell: Lorentz transformation invariance symmetry with asymmetry between electricity and magnetic sources i.e. divE = ρ/ whereas divB = 0 meaning that electricity is based on simple charges whereas magnetism is based on dipoles excluding the monopole whose existence is required for electric charge quantization by the Standard Model of Particle Physics as shown by Dirac [4; 5]. Photon mass is considered as zero.

• Proca: Photon mass is not zero and there is perfect symmetry between electricity and magnetism.

• Moon-Spencer: special treatment of retardation effects in EM

• EM concepts for emerging novel spintronic and topological materials:

1. Topological charge, current, symmetries and objects (Skyrmions)...

II. TAXONOMY OF MAGNETISM

The view of Magnetism evolved through time from compasses and toys to motors, transformers... and now is targeted for spintronic applications, quantum computing, processing and telecommunication.

Maxwell viewed magnetism as an internal orbital moment (by comparison with the gyroscope effect) whereas Pierre Weiss viewed it as an internal current following Ampère and Faraday. Weiss estimated from Iron magnetic moment an equivalent current (see Appendix 9) of 100,000 Amperes (10 5 Amperes) a fact that remained unexplained until Heisenberg [START_REF] Landau | Electrodynamics of Continuous Media[END_REF] developed QM wave-function overlap and exchange notions.

With the rise of quantum computing and communication needs and the emergence of topological materials, spintronics as well as advances in QED circuits... a number of novel properties are required to make useful devices demanded by these developments.

There are at least seven classes [START_REF] Coey | Magnetism and magnetic materials[END_REF] of magnetic materials characterized by the sign of their exchange constant J, the behavior of their magnetization with field or susceptibility χ = ∂M ∂H with temperature...: Paramagnets have a (χ > 0) susceptibility whereas diamagnets have (χ < 0). Both have zero magnetization at zero field whereas Ferromagnets have J > 0 (encouraging moment alignment) M = 0 when H= 0 with an hysteretic behavior versus field. Antiferromagnets have J < 0 (favoring opposing moments), M = 0 with moments canceling each other. Ferrimagnets have canted (making a small angle) opposing moments resulting in a small moment yielding weak ferromagnetism.

Helimagnets have a helicoidal order of their magnetization... In fig. 1, behavior of the magnetization for a para, ferro, anti-ferro and dia-magnet versus field are displayed.

A. Magnetic symmetry groups

Magnetic space symmetry comprise 1651 groups combining the 230 crystallographic space groups with time reversal symmetry as shown below.

According to Birss [START_REF] Birss | Symmetry and Magnetism[END_REF] we have the following classification following a color crystallographic group notation: W (White), B/W (Black or White) and Grey (B&W i.e. Black and White) . B uses Time Reversal Symmetry whereas W does not. Moreover color crystallographic groups have both symmetry and antisymmetry operators.

• Type I groups for W type materials (ferro or non magnetic)

• Type II groups for Grey (B&W) type materials (paramagnetic)

• Type III groups for B or W (B/W) type materials (antiferromagnetic, ferrimagnetic...)

Performing the individual sums:

• One-color (W) groups or simple Bravais: 230

• Grey (B&W) groups: 230

• Two-color (B/W) groups: 1191 Note that 230 is the number of ordinary space groups and is the same for Grey groups.

Thus the magnetic Space Groups breakdown [START_REF] Birss | Symmetry and Magnetism[END_REF] is: 230: (one-color) + 1191 (two-color) = 1421. Adding: 230 (Grey), we get: 1651. Total sum corresponds to what is known as Shubnikov space groups.

There are 1651 groups in the complete bicolor space group family, 230 of which are the regular space groups with Black symmetry operators, another 230 have simultaneously Black and White (grey) symmetry operators, and the remaining 1191 bicolor space groups have mixed Black and White operators.

Of the 1191, 674 groups contain no anti-translations [START_REF] Birss | Symmetry and Magnetism[END_REF] while 517 have anti-translations (bicolor Bravais lattices).

When all translations are removed, the bicolor space groups produce 122 bicolor point groups, 32 regular point groups, 32 grey point groups, and 58 with mixed black and white symmetry operators. Table 3: Point Symmetry Groups and Space Groups for different types of magnetic solids classified according to a color scheme with B using Time-Reversal while W does not and the color combinations correspond to W as for ferro or non-magnetic, Grey (B&W) to paramagnetic and finally (B/W) groups to antiferromagnetic, ferrimagnetic....

B. Rare-Earth and Transition metals

Atomic ground states represented as 2S+1 L J are obtained with Hund rules:

• max S value

• max L value compatible with max S and when L=0,1,2,3,4,5,6... we have respectively: S, P, D, F, G, H, I...

• J = |L -S| for less than half-filled shell, J = L + S for more than half-filled shell and finally: J = S, L = 0 for exactly half-filled shell.

• L = S = 0 for completely filled shell.

Rare-Earth elements have the external atomic configuration 4f n and the orbital quantum number for an f shell is = 3 yielding 2 (2 +1)= 14 elements (see table 4). Transition metal elements have the external atomic configuration 3d n and the orbital quantum number for a d shell is = 2 yielding 2 (2 +1)= 10 elements (see table 5).

Maximum S value is reached in each case when the shells are half filled (five 3d electrons or seven 4f electrons). In most cases, the energy separation between the ground-state multiplet level and the other levels of the same multiplet are large compared to thermal energy k B T .

For example Ce 3+ has 4f 1 configuration, thus S=1/2, L=3 since the orbital quantum number for an f shell is 3. J = |L -S|=5/2 thus a ground state given by 2 F 5/2 .

Another example is Mn 2+ giving a configuration 3d 5 which is a half-filled d shell. Thus max S = 5/2 and 2S +1 = 6. Since we have 5 electrons and the orbital quantum number for a d shell is 2, the configuration

↑ | ↑ | ↑ | ↑ | ↑ mapping to orbital values 2|1|0| -1| -2 yields L = 0 since L=2+1+0-1-2. Finally we get J = S = 5/2 and 6 S 5/2 .
Landé factor is given by:

g J = 3 2 + S(S+1)-L(L+1) 2J(J+1) originating from g J = g L L•J J 2 + g S S•J
J 2 with the orbital Landé factor g L = 1 and the spin Landé factor g S = 2.

Note that when J = 0 then g J = 3 2 . In the Er 3+ case, we have the configuration 4f 11

C. Amorphous magnets and Spin-Glasses

A magnetic ion is sensitive to its local atomic/molecular environment through a set of ligands representing the crystal field (CF). In an amorphous material [START_REF] Hurd | [END_REF], the CF is function of position meaning that local environment varies. Each magnetic ion will have a preferred alignment for its moment along a local easy axis determined by the local CF called single-site anisotropy characterized by D representing the CF strength. Selected ionic properties of the rare-earth 4f n elements. Experimental values agree with g J J(J + 1). Adapted from Kittel [START_REF] Kittel | Introduction to Solid State Physics[END_REF]. Selected ionic properties of transition metals 3d n elements. Experimental values agree with g S S(S + 1) or 2 S(S + 1). Adapted from Kittel [START_REF] Kittel | Introduction to Solid State Physics[END_REF].

Ion 4f n n Ground state L S J gJ gJ J(J + 1) Exp. La 3+ 0 1 S0 0 0 0 0 0 0 Ce 3+ 1 2 F 5/2 1/
Ion 3d n n Ground state L S J gJ J(J + 1) gS S(S + 1) Exp. K + , V 5+ 0 1 S0 0 0 0 0 0 - Sc 2+ , Ti 3+ , V 4+ 1 2 D 3/2 2 1/2 3/2 1.55 1.78 1.73 Ti 2+ , V 3+ 2 3 F2 3 1 2 1.63 2.83 2.83 V 2+ , Cr 3+ , Mn 4+ 3 4 F 3/2 3 3/2 3
Inside an amorphous solid, D is fixed and positive but the easy axes, are randomly distributed in direction. The first term is usually insignificant compared with the second for d-shell ions in an amorphous metal but the reverse usually applies to Rare-Earth f-shell ions.

In a Spin-Glass, we have random local exchange interactions [START_REF] Skomski | Simple models of magnetism[END_REF] J ij , producing random orientations of neighboring moments, resulting in zero average magnetization M = 0 with M 2 = 0.

As in ordinary glasses [START_REF] Elliott | Physics of amorphous materials[END_REF], it is necessary to distinguish between ensemble and time averages. Ensemble averages of glasses are liquid-like, meaning that atomic-scale snapshots of glasses and liquids look equally disordered, but time averages are solid-like, because atomic positions are frozen.

Spin-glass behavior [START_REF] Skomski | Simple models of magnetism[END_REF] is observed in a variety of disordered magnetic materials and involves both competing exchange and disorder. Examples of spin glasses are iron-series transition-metal atoms in noble-metal hosts, such as Fe x Au 1-x and Mn x Cu 1-x , metallic glasses, such as amorphous [START_REF] Skomski | Simple models of magnetism[END_REF] Fe-Sn, and chemically disordered oxides and sulfides such as Eu 1-x Sr x S.

A crystalline alloy comprising magnetic ions of a given species incorporated into a nonmagnetic host can freeze [START_REF] Hurd | [END_REF] into a spin-glass state when cooled below a critical temperature signaled by a cusp in the low-field a.c. susceptibility.

III. MAGNETIC ANISOTROPY

In a magnetic crystal, magnetization M orientation is sensitive to a set of special axes in the crystal.

Magnetic ions are sensitive to their surroundings through the CF. Moreover, in a magnetic ion, spin-orbit coupling provides additional anisotropy because spin and orbit degrees of freedom of unpaired electrons [START_REF] Skomski | Simple models of magnetism[END_REF] are coupled. Generally, orbit is also coupled to the CF via Coulomb interaction between unfilled orbital charge distribution and the electric field due to neighboring ligands. Thus coupling exists between an ion moment and the CF.

Experimentally, anisotropy is detected with the hysteresis curve change as we rotate the magnetic sample with respect to an applied field (section V explains mathematically this effect). This agrees with Maxwell point of view that describes magnetization as an internal orbital angular momentum L. It agrees also with QM since magnetization arises in fact from several angular momenta: L, S, J.... Physically this can be understood by considering an anisotropy energy E mc (M ) that depends on magnetization M orientation with respect to a set of special axes in the crystal.

E mc (M ) is minimized when M is along these special axes. These axes are called easy (or anisotropy) axes and their counterparts are called hard.

M H M s M r H c H k

Fig. 1: Hysteresis loop obtained for an arbitrary angle φ, between a magnetic field and the easy (anisotropy) axis. Quantities such as coercive field H c and remanent magnetization M r depend on φ. The hysteresis loop width increases as φ decreases from 90 • (field along hard axis) to 0 • (field along easy axis). When φ=90 • (field along hard axis), the hysteresis loop collapses into a line (blue line) that breaks at the anisotropy field ±H K values. Magnetization versus field of a ferromagnet with basic parameters such as M s , M r .H c , H K . The linear curve (blue line) is the case of a paramagnet as well as an antiferromagnet whereas the pink dashed line is for a diamagnet. Note the two distinct loops of the ferromagnet and the linear curve with a small negative slope of a diamagnet.

Magneto-crystalline anisotropy energy E mc originating from spin-orbit interaction. E mc depends on the underlying crystal symmetry such as cubic, hexagonal, orthorhombic... as well as an intrinsic property pertaining to local environment such as uniaxial, biaxial, triaxial... In the general 3D case, we call α 1 , α 2 , α 3 the direction cosines of M with respect to a set of orthonormal axes as displayed in fig. 2.

Using energy arguments based on Kramers theorem dealing with time reversal symmetry t → -t leading to M → -M one may infer that the anisotropy energy E mc must be such that E mc (M ) = E mc (-M ) meaning that E mc (M ) must contain even products of the magnetization components. Using Einstein summation convention, the anisotropy energy writes:

E mc = K ij α i α j + K ijkl α i α j α k α l ... (2)
where K ij is a second order tensor, K ijkl is a fourth order tensor and so on... In the following we write the energy in the different symmetry cases for the second and fourth order.

α i = cos θ i with i = 1, 2, 3 normalized by α 2 1 + α 2 2 + α 2 3 = 1.
A. Second order

Uniaxial

Uniaxial means that M tends to align along a single axis. Considering the z axis as that special axis (anisotropy axis) and regardless of the angles θ 1 , θ 2 , the important one in this case is θ 3 . Thus we have:

E mc = K 1 sin 2 θ 3 = K 1 (1 -α 2 3 ) = K 1 (α 2 1 + α 2 2 ) α 1 α 2 α 3    K 1 0 0 0 K 1 0 0 0 0       α 1 α 2 α 3    (3)
with α 3 = cos θ 3 the direction cosine of M . It is clearly obvious that energy minimum E mc = 0 is obtained when θ 3 = 0.

Biaxial

Superposition of uniaxial energies with respect to two different axes (say x, y) gives:

E mc = K 1 sin 2 θ 1 + K 2 sin 2 θ 2 = sin θ 1 sin θ 2 sin θ 3    K 1 0 0 0 K 2 0 0 0 0       sin θ 1 sin θ 2 sin θ 3    (4)
Minimum energy is obtained when θ 1 = 0 or θ 2 = 0.

Triaxial

Superposition of uniaxial energies with respect to three different axes (say Cartesian x, y, z) gives:

E mc = K 1 sin 2 θ 1 + K 2 sin 2 θ 2 + K 3 sin 2 θ 3 = sin θ 1 sin θ 2 sin θ 3    K 1 0 0 0 K 2 0 0 0 K 3       sin θ 1 sin θ 2 sin θ 3    (5)
The energy minimum is obtained when θ 1 = 0, θ 2 = 0 or θ 3 = 0.

α i = cos θ i with i = 1, 2 normalized with α 2 1 + α 2 2 = 1.
B. Fourth order

Uniaxial

As in the second order case, we may write E mc to fourth order when the anisotropy axis is along z as:

E mc = K 1 sin 4 θ 3 (6)
Energy minimum E mc = 0 is obtained when θ 3 = 0 as in the second order case.

Biaxial

Let us move from the uniaxial case to a thin film case by considering the x-y plane. What matters in the evaluation of E mc is the energy involved in the orientation of M with respect to a set of two axes (say x, y axes).

The fourth order energy writes, taking α 3 = 0 in eq. 9):

E mc = K 1 α 2 1 α 2 2 (7)
Obviously E mc = 0 when α 1 = 0 or α 2 = 0. E mc expression can be transformed with the new normalization condition α 2 1 + α 2 2 = 1 into:

E mc = K 1 4 sin 2 2θ 1 (8)

Triaxial or Cubic

In the cubic anisotropy case, one has to account for the 3D orientation of M orientation with respect to all 3D crystal axes as in fig. 2.

Let us call the sum s = (α

2 1 α 2 2 + α 2 1 α 2 3 + α 2 2 α 2 3
) and the product terms p = α 2 1 α 2 2 α 2 3 . The anisotropy energy expanded to sixth order as:

E mc = K 1 (α 2 1 α 2 2 + α 2 1 α 2 3 + α 2 2 α 2 3 ) + K 2 α 2 1 α 2 2 α 2 3 = K 1 s + K 2 p (9)
This can be derived by starting with the normalization condition: α 2 1 + α 2 2 + α 2 3 = 1 and raising it successively to second and third powers.

To eigth order one may write a product of two fourth orders as s 2 and this observation leads to any arbitrary order by simply combining the s, p terms to obtain the desired order.

It is possible to obtain an expression for E mc to the 20th order [START_REF] Berling | [END_REF] as:

E mc = K 0 + K 1 s + K 2 p + K 3 s 2 + K 4 sp + K 5 (s 3 + p 2 ) + K 6 s 2 p + K 7 (s 4 + sp 2 ) +K 8 (s 3 p + p 3 ) + K 9 (s 5 + s 2 p 2 ) + K 10 (sp 3 + s 4 p) + ... (10)
C. Anisotropy field

In the general non-uniform magnetization case, the functional derivative H A = -δEmc δM yields the field from the corresponding energy.

In the uniform case, the functional derivative becomes a simple partial derivative

H A = - ∂E mc ∂M (11)
This can be checked when one assumes the energy E mc has a Zeeman form such that E mc = -M • H f which evidently yields:

H f = -∂Emc ∂M .
For example, let us evaluate the anisotropy field in the simple uniaxial case where 2 with K u and M along the anisotropy axis defined by the unit vector u = M /M s and M s is the saturation magnetization.

E mc = K ij α i α j = K ij (M i /M s )(M j /M s ). In the uniaxial case K ij = K u δ ij resulting in E mc = K u (M/M s )
Adapting H A = -∂Emc ∂M to this case yields

H A = - ∂E mc ∂M /M s = -M s ∂E mc ∂M (12)
resulting in

H A = -(2K u /M s )u.

IV. TYPES OF MAGNETIC ENERGY IN SOLIDS

Micromagnetic theory [START_REF] Coey | Magnetism and magnetic materials[END_REF] appeared at the beginning of the 1930s and explains magnetization reversal process and hysteresis in ferromagnets. Today, it helps understand the stability of various magnetic nanostructures required for spintronics as well as their magnetization reversal dynamics.

Energies characterizing a ferromagnetic system, namely: Zeeman energy, magnetostatic energy, exchange energy and magneto-crystalline anisotropy energy should be considered. Summing these energies to obtain total energy to be minimized in order to determine equilibrium. The transition from one equilibrium state to another can be effected by several types of magnetization reversal. In fact, there are two basic reversal processes: uniform (Stoner-Wohlfarth) and non-uniform (curling).

The evolution of a magnetic system towards a global minimum or local energy is obtained by the Landau-Lifshitz-Gilbert equation. The latter describes the dynamics of magnetization M as a temporal evolution from one state to another or the oscillations or rotational precession of M around equilibrium... On macroscopic scale, ferromagnetism is often dominated by the presence of magnetic domains separated by domain walls. This is the non-uniform case.

Magnetization reversal occurs, in most cases, by nucleation, motion of walls or their disappearance. Minimization of the total energy is defined by the expression:

E tot = E Z + E D + E an + E X ( 13
)
where E Z , E D , E an and E X are Zeeman, magnetic anisotropy (magneto-crystalline, shape and magneto-elastic), demagnetization (dipolar) and finally exchange energies respectively.

A. Zeeman Energy EZ

Zeeman energy expresses interaction of an external applied magnetic field H with magnetization. It is given by:

E Z = - V dvM • H (14)
It tends to align the magnetic moments along the field H direction.

B. Dipolar and magnetostatic energies ED Note: In this section, Einstein summation convention is used whenever an index is repeated.

We will treat on the same equal footing dipolar and magnetostatic energies since both have same origin.

Considering each moment as a dipole, for any pair of magnetic moments m 1 and m 2 separated by r = |r|, the dipolar energy is expressed by:

E D = 1 r 3 m 1 .m 2 - 3 r 2 (m 1 .r)(m 2 .r) (15)
This energy behaves as 1 r 3 where r is the distance between moments. Consequently, dipolar interaction which is much weaker at short distance than exchange energy between neighboring moments becomes dominant at long distance giving rise to demagnetizing (or magnetostatic) effects.

The demagnetizing field H D by analogy with electrostatics, is due to volume and surface densities ρ = -div M and σ = M • n, respectively with n(|n| = 1) the normal outward to the surface of the magnetic body.

The expression of the demagnetizing field is thus:

H D (r) = V ρ(r ) (r -r ) |r -r | 3 dv + S σ(r ) (r -r ) |r -r | 3 ds (16)
This field is non-local and depends on the magnetization distribution in the material.

Note that, in the general non-uniform case, the functional derivative H D = -δE D δM yields the field from the corresponding energy. In the uniform case, the functional derivative becomes a simple partial derivative

H D = -∂E D ∂M .
In particular, H D can be expressed with the uniform magnetization M having a modulus M s , by

H D = -V N D M dv where N D = 4π N ij since the demagnetization energy is: E D = 2π N ij M i M j with 0 ≤ N ij ≤ 1 H D i-th component is -∂E D ∂Mi = -4π N ij M j .
N ij is a rank-2 tensor with 3 × 3 demagnetization coefficients with trace equal to 4π. This is the case of an ellipsoid of revolution.

When the magnetization is oriented according to one of the principal directions of the ellipsoid, the tensor is diagonal

N ij =    N xx 0 0 0 N yy 0 0 0 N zz    (17)
and the demagnetizing field is opposite to magnetization

[H D ] i = -4πN ij M j with i N ii = 1, where 0 ≤ N ii ≤ 1, ∀i = 1, 2, 3 (18)
If the reference frame axes Ox, Oy, Oz are taken along the ellipsoid axes, tensor N D is diagonal and verifies the relationship

N xx + N yy + N zz = 1.
The demagnetization energy is expressed in Zeeman form between magnetization and the demagnetization field H D , yielding:

E D = - 1 2 V M . H D dv (19)
Factor 1 2 is required to have compatibility between N ii = 1, and the demagnetizing energy,

E D = 2π N ij M i M j . In general : E = -M . H and H = -∂E ∂M . For H D component i, we have [H D ] i = -4π N ij M j .
Then we would get incompatibility between E D = 4π N ij . M i M j and E D = 2π N ij M i M j , hence the need to insert 1 2 in the energy expression to make dipolar and demagnetization approaches agree.

C. Magnetic anisotropy energy Ean

As described above, this energy respects crystal symmetries and due to the fact the magnetic moments, in a ferromagnet, orient themselves spontaneously along directions of easy magnetization.

In a magnetic material, different types of anisotropy energies may occur. There are at least the magneto-crystalline anisotropy E mc and the magneto-elastic anisotropy E me described below.

D. Magneto-crystalline anisotropy energy Emc

Magneto-crystalline anisotropy energy depends on crystal symmetry of the material (cubic, hexagonal, orthorhombic, tetragonal...). In the case of a cubic lattice (such as cases of BCC Fe and FCC Ni, for example), the magnetocrystalline anisotropy energy is expressed as a function of the cosine directors α i of M with respect to the crystal axes:

E mc = K 1 (α 2 1 α 2 2 + α 2 2 α 2 3 + α 2 3 α 2 1 ) + K 2 α 2 1 α 2 2 α 2 3 + ... (20)
With K 1 and K 2 the anisotropy coefficients of the material.

In the case of hexagonal symmetry (Cobalt HCP is an example), the density of magneto-crystalline anisotropy energy is:

E mc = K 1 sin 2 θ + K 2 sin 4 θ + K 3 sin 6 θ (21)
with θ the angle between magnetization and the easy axis. For a magnet having an anisotropy uniaxial described by a single coefficient K 1 , the energy of magneto-crystalline anisotropy is of the form:

E mc = K 1 sin 2 θ (22)
When elastic deformations of a magnetic crystal occur, we ought to introduce the notion of magneto-elastic anisotropy.

E. Magneto-elastic anisotropy energy Eme

When the crystal is deformed, local atomic environments are modified. An additional energy term appears. Magnetoelastic anisotropy relates crystal elastic deformations to magnetism.

The influence of deformations on magnetization is called magneto-elastic energy expressed with an interaction term linked to spin-orbit coupling. The associated physical phenomenon is magnetostriction [START_REF] Coey | Magnetism and magnetic materials[END_REF].

Linear magnetostriction coefficient λ is defined as the relative variation of the length l of the sample, when the material is brought from a demagnetized state to saturation. One observes elongation with λ > 0 (Iron case) or contraction with λ < 0 (Nickel case).

For a cubic crystal symmetry, the magneto-elastic energy is written [START_REF] Coey | Magnetism and magnetic materials[END_REF] as:

E me = - 3 2 V dvλσ cos 2 ψ ( 23
)
where σ is the uniaxial stress and ψ the angle between stress direction and magnetization.

F. Exchange energy EX Note: In this section, Einstein summation convention is used whenever an index is repeated.

Exchange energy has quantum origin and electrostatic character (denoted as QE). It is responsible for magnetic order and acts between nearest neighboring magnetic moments.

There are many types of exchange [START_REF] Coey | Magnetism and magnetic materials[END_REF]:

• Direct exchange: QE Heisenberg type between two moments -2J ij M i • M j located at two different sites i, j. This is localized exchange.

• Bi-Quadratic exchange: QE Quadratic Heisenberg type between two moments

-J ij [M i ] 2 • [M j ] 2 located at two different sites i, j.
• Indirect exchange: QE between two localized moments interacting through another medium such as an electron gas or Fermi sea.

• Double exchange: QE between one localized moment and another belonging to an extended medium such as an electron gas. For instance in a transition metal we have s -d exchange where s electrons belong to an extended medium (wideband) and d electrons belonging to a narrowband orbital.

• Super-exchange: Exchange over a large distance between two ions exploiting intermediate ions. For instance, in transition metal oxides that are insulators, 2p orbitals of oxygen ions act as bridges to provide Heisenberg exchange interaction between distant 3d ions.

• Exchange bias: Many body effect resulting into a horizontal shift of the hysteresis loop and originating from coupling between a ferro and an antiferromagnetic materials

• Anisotropic exchange or Dzyaloshinskii-Moriya interaction (DMI) given by D ij M i ×M j and found in non-centrosymmetric or chiral crystals with broken inversion symmetry r → -r. DMI is responsible for canting in some antiferromagnetic materials inducing a slight inclination of the moments rendering the material ferrimagnetic or weakly ferromagnetic.

The Heisenberg exchange energy density for a sample of volume

L 3 is [2]: 1 M 2 s A ij ∂M l ∂x i ∂M l ∂x j ( 24
)
Its unit is erg/cm 3 while for the exchange constant A ij is (erg/cm).

In the case of isotropic exchange constant: A ij = A X δij, we have:

E X = A X V dv (∇m x ) 2 + (∇m y ) 2 + (∇m z ) 2 (25)
where m x , m y , m z are the components of the reduced magnetization m = M /M s , M s being the saturation magnetization. A X is the exchange constant and E X the exchange energy density.

In this expression, the gradient operator presence means that exchange energy is short ranged. This interaction is isotropic and does not favor any orientation of the magnetization with respect to the crystal axes.

It is useful to consider an exchange H X field which corresponds to the uniform case of the m x , m y , m z magnetization components. When the components vary in space H X = -δE X δM , is the functional derivative of E X with respect to the magnetization. It is given by the expression:

H X = 2A X M s V dv (∆m x i + ∆m y j + ∆m z k) (26)
with the Laplacian ∆ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 and i, j, k the unit vectors of an orthonormal reference frame. The demagnetization energy density is: 2πN ij M i M j In the case of an infinite cylinder (N xx = N yy = 1 2 , N zz = 0). Comparing the two densities, we find:

1 M 2 s A X M 2 s 2 X = 2π N ij M 2 s (27)
We infer that exchange length is X =

A X 2π Nij M 2 s and N ij = 1 2 .
This is the distance within which exchange interactions are larger than the demagnetization field. When L >> X dipolar energy is predominant and the sample is subdivided into domains.

In fact, the presence of magnetic domains in a ferromagnetic material leads to a decrease in magnetostatic energy.

The creation of domains is accompanied by the occurrence of walls and zones of magnetization rotation at the boundary between domains. The creation of walls increases exchange energy limiting the number of domains. Moreover the competition between these energies, implies that a uniform ferromagnet cannot minimize the absolute energy required for an equilibrium state.

As a consequence, competition between E mc characterized by anisotropy constant K and exchange energy characterized by length X yield at equality:

1 M 2 s A X M 2 s 2 X = K M 2 s M 2 s ( 28
)
providing Bloch wall width w B separating domains, when X = w B = A X K . .

V. MAGNETIZATION REVERSAL

When the radius (or sample size) becomes larger than a critical radius R c , a multi-domain magnetic structure is obtained, while for R < R c , a single-domain structure occurs.

For a single-domain, there are several types (modes) of magnetization reversal, function of sample size, intensity and direction of the applied field:

1. Uniform reversal: Switching (fast) or rotation (slow) 2. Non-uniform reversal [START_REF] Chikazumi | Physics of Ferromagnetism[END_REF]: Curling, buckling, bulging...

Reversal by nucleation and wall motion.

In the case of a single-domain, there are two significant reversal modes: homogeneous rotation of the magnetic moments (Stoner-Wohlfarth) and inhomogeneous rotation of moments.

Note that the inhomogeneity in the magnetization manifests itself when the exchange and anisotropy energy densities (either in shape or magneto-crystalline) become comparable. Thus, we have a critical length (radius) signaling transition from the homogeneous to the inhomogeneous reversal mode.

A. Homogeneous reversal (Stoner-Wohlfarth)

In 1947, Stoner and Wohlfarth developed a model allowing to describe the behavior of magnetization subjected to an applied field in a single-domain. Magnetization modulus is constant but its direction changes with field.

Note that exchange energy remains constant during magnetization reversal. We suppose a particle represented by an ellipsoid of revolution, whose large axis is along the axis of easy magnetization. The anisotropy is uniaxial and the energies of magneto-crystalline anisotropy and shape anisotropy both have an expression in sin 2 θ, or

E an = K sin 2 θ ( 29
)
where K is the anisotropy constant of the magneto-crystalline and shape energies, that can make an energy barrier to reversal. Note that θ the angle between the moment and the easy axis. θ is the only degree of freedom.

In presence of an external field H making an angle φ with the easy axis, the magnetic energy is written:

E = K sin 2 θ -M s H cos(θ -φ) (30)
The stable configurations are obtained by minimizing the total energy at θ * with conditions given by:

∂E ∂θ θ=θ * = 0 and ∂ 2 E ∂θ 2 θ=θ * ≥ 0 (31)
Normalising the magnetization by its saturation value m = M/M s and the field h = HH K with H K = 2K/M s the anisotropy field, yields:

[sin(θ) cos(θ) + h sin(θ -φ)] θ=θ * = 0 and [cos(2θ) + h cos(θ -φ)] θ=θ * ≥ 0 (32)
Let us solve analytically the above for the two cases: φ = 0 (field along easy axis) and π/2 (field along hard axis) with the longitudinal magnetization i.e. the projection of M along H, m = M • H/(HM s ) = cos(θ -φ).

1. φ = 0: eqs. 32 give the solution:

θ * = cos -1 (-h) when h ≤ 1 otherwise θ * = 0, π. When h ≥ 1 we have also θ * = 0, π. This yields a square hysteresis loop in fig. 4 meaning fast reversal by switching at h = ±1 when ∂E ∂θ θ=θ * = 0 and ∂ 2 E ∂θ 2 θ=θ * = 0 simultaneously.

2. φ = π/2: eqs. 32 give:

θ * = sin -1 (h) when h ≤ 1 otherwise θ * = π/2 yielding the main diagonal (m = h) hysteresis loop in fig. 4. When h ≥ 1 the solution is cos θ * = 0, thus θ * = ±π/2. This is slow reversal by rotation from

θ * = -π/2 to θ * = π/2.
In fig. 4, we display hysteresis curves [START_REF] Press | Numerical Recipes in C: The Art of Scientific Computing[END_REF] for several angles φ: 90 • , 45 • and 0 • to show the progression from rotation to switching.

Note: For arbitrary angle φ it is possible to find the analytic solution in a tedious way as shown in the Appendix C. Thus we determine the value of the magnetization reversal field as:

H C = 2K M s (sin 2 3 θ + cos 2 3 θ) - 3 2 (33)
At equilibrium, the magnetization is oriented according to an angle θ 0 resulting in minimal energy.

At zero applied field, the magnetization has two equilibrium positions θ = 0 • and θ = 180 • separated by an energy barrier (anisotropy due) of height E 0 .

Let us take as initial condition the situation H = 0 and θ = 0 • . Under application of a field H with (ϕ > 90 • ), the equilibrium position of M evolves and is given by the solution of the nonlinear relation: ∂E ∂θ = 0 . Moreover, the difference ∆E between the first local minimum of energy, corresponding to the equilibrium position of the magnetization and the local maximum, decreases with the intensity of the field applied (decrease in barrier height). The evolution of equilibrium position of M is reversible with respect to applying the applied field until it has a value H C for which the energy barrier vanishes. The magnetization is then in a position of unstable equilibrium.

The cancellation of the energy barrier corresponds to the appearance of an inflection point for E(θ) given by the condition

∂ 2 E ∂θ 2 = 0.
Thus hysteresis is linked to the existence of two energy minima whose positions vary with the angle between magnetization and the applied field.

The coercive field is the critical field required to move from one minimum to another when energy first derivative is zero. Note the breaks in the line at h = H/H K = ±1 that allows to determine experimentally H K .

In order to simplify the description of magnetization reversal as a function of a field H, Stoner and Wohlfarth have developed a graphical method to find the moment direction of a magnetized particle. It consists of drawing, in an arbitrary plane (OX, OZ), an astroid of which sets the value of the coercive field H C depending on the field applied H along the direction perpendicular to the axis of anisotropy. In this plane, OX and OZ are unit orthogonal axes.

To determine the equation of the astroid, it suffices to apply the condition of minimization of the energy by putting the values of the field H projected according to both X and Z axes.

So, we get using H Z = H cos ϕ, H X = H sin ϕ :

h Z = H Z M s 2K = H Z H K , h X = H X M s 2K = H X H K (34)
where h = H H K the reduced field, H K = 2K Ms is the reduced anisotropy field and h Z and h X are the reduced values of fields H Z and H X obtained which by the projections of the external magnetic field H on the Z and X axes of the plane (OXZ) respectively. This yields:

h Z = -cos 3 θ, h X = sin 3 θ (35)
Transforming the above with the trigonometric identity cos 2 θ + sin 2 θ = 1 into a parametric curve, we get the Stoner Wohlfarth astroid:

h 2 3 X + h 2 3 Z = 1 (36)
It should be noted that a sudden magnetic moment reversal occurs whenever the point (H X , H Z ) crosses the astroid boundary (see fig. 5).

B. Inhomogeneous reversal (Curling)

In homogeneous reversal, the corresponding field can be expressed as: Outside, it is not possible

.

H n = - 2K 1 M s + 4 π(N zz -N xx ) M s (37)
In this mode, the magnetization rotates simultaneously everywhere in the ellipsoid. This mode is known as a homogeneous rotation mode or Stoner-Wohlfarth. A. Aharoni [START_REF] Aharoni | Introduction to the theory of ferromagnetism[END_REF] proposed another reversal mode he called curling that exists in single-domain magnetic particles of size larger than a critical value.

C. Critical radius Rc for an ellipsoid of revolution

For an ellipsoid of revolution, it is possible to find a critical radius R c (along the minor axis a = b = c) for the appearance of a single-domain state. Taking into account the average exchange energy over the ellipsoid volume and the magnetostatic energy of the structure, we get:

R c = 3A X 2πN c M 2 s ln 4R c a 1 -1 (38)
where N c is the demagnetizing factor along the c-axis of the ellipsoid, A X is the exchange coefficient and a 1 the typical nearest neighbor distance.

D. Critical radius Rc for an infinite cylinder

In the case of an infinite cylinder of radius R, the nucleation field is:

H n = - 2K 1 M s - 2A X q 2 R 2 M s (39)
where A X is the exchange constant, M s the saturation magnetization and q = 1.8412 is the first positive zero of the Bessel function J 1 (x) derivative dJ1(x) dx . The comparison of nucleation fields corresponding to (homogeneous) Stoner-Wohlfarth and (inhomogeneous) curling, gives us the critical radius separating the two reversal modes. If there are no other modes, the magnetization reversal in an infinite cylinder of radius R must begin with coherent rotation if R < R c , and by curling, if R > R c , where:

R c = q M s A X π (40)

VI. APPLICATIONS OF MAGNETISM

A. Recording

Magnetic materials are required in the development of high-density information storage for recording heads and recording media. The required properties for recording heads are high M s , low coercivity H c , low magnetostriction coefficient λ and large permeability µ(f ) over a broad frequency f range (typically several hundred MHz to several GHz). In order to speed the recording process (during read/write/erase with an applied magnetic field H), the media should be able to switch and not undergo other slow processes like rotation... This means H should be along the media easy axis. Recording heads and sensors in general require soft materials with narrow hysteresis width. In sharp contrast, the required properties of recording media are small M s and high coercivity H c in order to manipulate data easily and retain it for a long time. Media and permanent magnets require hard materials with broad hysteresis width.

Magnetic materials are described in table 6 along with their composition and hardness. Soft materials can be Amorphous or Polycrystalline (disorder decreases anisotropy) or nanocrystalline. Mu-Metal is excellent for shielding against magnetic fields.

B. Levitation

Levitation is achieved by opposing magnetic moments originating from permanent magnets, current loops or pole images. Lifting force is given by

F = - ∂E ∂r ( 41
)
where E is the total energy, U the potential energy and r the spatial position. Assuming the total energy E = T +U where T is the kinetic energy function of the velocities and not positions we get ∂E ∂r = ∂U ∂r and: magnetic field source and a small metallic object into which eddy currents are created in a way such that the respective moments oppose each other.

S N

S N +q +g +g -q

Grounded metal Superconductor We describe briefly the Levitron item. A Levitron [START_REF] Bassani | -1888) and Passive Magnetic Levitation[END_REF] consists of two magnetic objects, a top and a base ring with anti-parallel moment and induction. If the top of mass m rotates at an appropriate angular speed ω, the angular momentum creates a dynamic gyroscopic effect preventing the top from flipping and falling, making it remain in equilibrium above the base-ring at some height as displayed in fig. 8. Assume the top moment µ is always oriented in the -z direction and the repulsive magnetic induction B from the base magnet is in the +z direction.

The potential energy is U = -µ • B -m 0 g 0 z = µB z -m 0 g 0 z where g 0 is gravity constant and m 0 the top mass. There are two conditions for stable levitation. The potential energy must be minimized for levitation. Thus the lifting force -µ ∂Bz ∂z must balance the gravity term m 0 g 0 . Energy is a minimum when we have its second spatial derivative with respect to local coordinates ≥ 0 or µ ∂ 2 Bz ∂r 2 > 0 with r the top center of mass position. However, ∇ 2 B z = 0 at any point in space, thus the energy minimum condition cannot be satisfied in all directions. We get an unstable saddle point [START_REF] Bassani | -1888) and Passive Magnetic Levitation[END_REF] instead of a minimum. This is due to the fact B(r, z) in the region of interest is such that divB = 0 (absence of magnetic charge) and curlB = 0 (from base ring geometry).

The gyroscopic action must prevent the top from flipping and act to always align the top precession axis to B(r, z) direction.

When these conditions are met, U depends only on B(r, z) amplitude and gravity. While each component of the induction must satisfy Laplace equation (i.e. ∆B z = 0), the induction magnitude does not. This allows the curvature of the potential energy to have a minimum (and not a saddle point) at some height.

In order to proceed with the mathematical [START_REF] Bassani | -1888) and Passive Magnetic Levitation[END_REF] description, we make two assumptions:

1. The top is a magnetic dipole whose center is also its center of mass 2. The top is spinning fast with the angular momentum along the top rotation axis which is also the magnetic moment axis. That is, the angular momentum is Iω(µ/µ).

I is the top moment of inertia around the spin axis, ω is the angular frequency, and µ = |µ| is constant. ±ω signals respectively when µ is parallel or antiparallel to B(r, z). Orientation of the angular momentum impacts the stability of the top, precession wise. The equations [START_REF] Bassani | -1888) and Passive Magnetic Levitation[END_REF] that describe the motion of the top (ignoring air resistance and other losses, for simplicity) are the torque:

dµ dt = µ Iω µ × B
and Newton equation of motion for the top center of mass position r:

m 0 d 2 r dt 2 = ∇(µ • B) -m 0 g 0 ẑ
The precession frequency is found to be inversely proportional [START_REF] Bassani | -1888) and Passive Magnetic Levitation[END_REF] to ω adding to the difficulty of achieving Levitron stability.

C. Shielding

We can either shield H or B. H is shielded with a soft layer whereas a superconductor shields against B since it is the perfect diamagnet. This is a big difference with dielectrics where we can shield against E with a metallic Faraday cage but we still cannot shield against the displacement field D.

D. Ferromagnetic resonance

When a homogeneous magnetic field H acts on the moment magnetic M , it exerts a torque Γ equal to Γ = M ×H. At equilibrium (Γ = 0), however when (Γ = 0) and in absence of damping, M time evolution is given by:

dM dt = -γ 0 (M × H) (43)
where the gyromagnetic factor γ 0 = g J e 2 me = 2.2.10 5 mA -1 .s -1 with e the electron absolute charge, m e its mass and g J the Landé factor which is close to 2 for most ferromagnetic materials. This equation describes magnetization M precession around field H, with both magnetization amplitude |M | = M s and angle with H constant over time.

In the LLG equation we use instead of the field H the effective field H f defined from the energy H f = -∂E ∂M . This field can also be seen as the sum of the applied field H and internal fields associated with anisotropy energies (magneto-crystalline and magneto-elastic) H an and the demagnetizing field (or dipolar interaction field) H D .

Thus we have:

H f = H + H an + H D (44)
Introducing a damping term with dimensionless coefficient α to account for various phenomena and losses occurring during magnetization evolution, we get:

dM dt = -γ 0 (M × H f ) + α M s M × dM dt (45)
Damping slows down magnetization M precession pushing M gradually to point along field direction eventually reaching equilibrium. This equation is known as Gilbert equation.

Actually Gilbert equation is identical to Landau and Lifshitz [START_REF] Landau | Electrodynamics of Continuous Media[END_REF], once we account properly for damping [START_REF] Coey | Magnetism and magnetic materials[END_REF].

(1 + α 2 ) dM dt = -γ 0 (M × H f ) - αγ 0 M s M × (M × H f ) (46)
Henceforth the Landau-Lifshitz-Gilbert (LLG) equation is:

dM dt = -γ(M × H f ) - αγ M s M × (M × H f ) (47)
with γ = γ0 1+α 2

Magnetization motion described by the LLG equation retains the modulus of the saturation value M s after insertion of the relaxation term.

Compared to the total energy minimization method, the LLG equation takes into account the evolution magnetization and its temporal dynamics. Moreover, it offers a method suitable for the study of reversal processes and magnetization dynamics.

Ferromagnetic resonance (FMR) can be neatly described with the LLG model. It allows us to measure several parameters of magnetic systems such as M s , the anisotropy field and its dispersion, the Landé factor g J and the spectral linewidth arising from dissipative effects.

Applying a static magnetic field H separates energy levels of an ensemble of moments. This energy separation ∆E = E f -E i = g J µ B H results in a resonant absorption by the moments in presence of a small high frequency field h rf (ω) when the resonance condition ω r = g J µ B H is met. E f, i are the energies in the final and initial states and ω r is the resonance angular frequency.

In a spherical coordinate system, the magnetization vector M (resp. the field H) is identified by the polar θ and azimuth angles ϕ (resp. θ H , ϕ H ).

The M, H cartesian components when expressed with spherical coordinates, are:

M =    M x = M s sin θ cos ϕ M y = M s sin θ sin ϕ M z = M s cos θ H =    H x = H sin θ H cos ϕ H H y = H sin θ H sin ϕ H H z = H cos θ H (48)
The effective anisotropy (strength K f ) and Zeeman energies are written: E = K f sin 2 θ -M s H [sin θ cos ϕ sin θ H cos ϕ H + sin θ sin ϕ sin θ H sin ϕ H + cos θ cos θ H ] (49)

At resonance, the Larmor frequency associated with the effective field strength is equal to the microwave frequency of the exciting wave.

In order to evaluate the resonance frequency ω r , according to the Smit-Beljers formula [START_REF] Coey | Magnetism and magnetic materials[END_REF] from the total energy E, we write:

ω r γ 2 = (1 + α 2 ) sin 2 θ ∂ 2 E ∂θ 2 ∂ 2 E ∂ϕ 2 - ∂ 2 E ∂θ ∂ϕ 2 (50)
The evaluation of second order E partial derivatives yields the resonance frequency:

ω r γ 2 = [H f cos 2θ + H cos (θ H -θ)][H f cos 2 θ + H cos (θ H -θ)] (51)
This equation relates the effective anisotropy field to the applied static field H and resonance frequency.

The orientation of the magnetization is given by the angle θ 0 (ϕ = ϕ H is set) determined from energy minimization ( ∂E ∂θ ) θ0 = 0:

K f sin 2θ 0 = M s H sin (θ H -θ 0 ) (52)
Numerically the resonance field function of the angle θ H between H and the easy axis is fit in order to find the various physical parameters.

Permeability versus frequency is obtained from the linearized LLG equation and Fourier transforming it. -θ 1 -√ -θ 2 + √ -θ 3 .

(C6)

The square roots are selected according to:

-θ 1 -θ 2 -θ 3 = -q. (C7)

Add -a/4 to the roots of g(w) = 0 to get those of f (z) = 0 and select the roots such that ∂ 2 E ∂θ 2 ≥ 0 that is:

∂ 2 E ∂θ 2 = 1 -2 sin 2 θ + A cos θ + B sin θ (C8)
In fact, it is preferable to indulge into a numerical approach [START_REF] Press | Numerical Recipes in C: The Art of Scientific Computing[END_REF] to obtain the minimum angular values with a less tedious way.

  then max S= 3/2 obtained by having 7 ↑ and 4 ↓ to accomodate 11 electrons in the 4f shell as ↑ | ↑ | ↑ | ↑↓ | ↑↓ | ↑↓ | ↑↓ |. This leaves us with 3 ↑ electrons when mapped to orbital values 3|2|1 yield with Hund rule L = 3 + 2 + 1 thus L = 6 and J = L + S = 6 + /2 = 15/2 for more than half-filled shell. The resulting ground state is 4 I 15/2 .

Fig. 2 :

 2 Fig.2: 3D magnetization making angles θ 1 , θ 2 , θ 3 with orthonormal axes. The direction cosines are:α i = cos θ i with i = 1, 2, 3 normalized by α 2 1 + α 2 2 + α 2 3 = 1.

Fig. 3 :

 3 Fig.3: 2D magnetization making angles θ 1 , θ 2 with x, y axes. The direction cosines are: α i = cos θ i with i = 1, 2 normalized with α 2 1 + α 2 2 = 1.

ig. 4 :

 4 Stoner Wohlfarth hysteresis curve for several values of angle φ: 0 • , 45 • and 90 • between field and easy axis. m || = M • H/(HM s ) is the magnetization along the field h = H/H K . When φ= 0 • (field along easy axis) the curve is broadest (fast switching). The lower horizontal line corresponds to θ * = π and the upper horizontal line corresponds to θ * = 0. When φ= 90 • (field along the hard axis) (slow rotation from θ * = -π/2 at lower horizontal line to θ * = π/2 at the upper horizontal line), the curve collapses to a line like the paramagnet or antiferromagnet.

Fig. 5 :

 5 Fig.5: Stoner Wohlfarth astroid. Inside the domain bounded by the curve magnetization reversal is possible. Outside, it is not possible

Fig. 6 :

 6 Fig.6: Radius versus demagnetization coefficient N c for Iron, Nickel and Cobalt.

Fig. 7 :

 7 Fig.7: (left) Charge and current loop images with respect to a grounded metal sheet. (right) Pole and magnetimages with respect to a plane superconductor. Note that S|N is equivalent to a -g, +g dipole.

Fig. 8 :

 8 Fig.8: General configuration for gyroscopically stabilized magnetic levitation. The moments of the top and base ring are opposite to each other. The base ring creates the induction B(r, z) and the top carries a moment µ opposing B(r, z) while being subjected to the gravitational force m 0 g 0 .

Table 1 :

 1 Process node evolution with decrease of minimum feature F m and projection toward future (Intel and TSMC projections for 2024).

	10 µm 1971 6 µm	1974 3 µm 1977 1.5 µm 1981
	1 µm 1984 800 nm	1987 600 nm 1990 350 nm 1993
	250 nm 1996 180 nm	1999 130 nm 2001 90 nm 2003
	65 nm 2005 45 nm	2007 32 nm 2009 22 nm 2012
	14 nm 2014 10 nm	2016 7 nm 2018 5 nm 2020
	3 nm 2022 2 nm ≈ 2024

Table 2 :

 2

	3 ) Curie Tc (K)

Table 4 :

 4

			2 3 5/2 6/7	2.54	2.4
	Pr 3+ 2	3 H4	1 5 4 4/5	3.58	3.5
	Nd 3+ 3	4 I 9/2	3/2 6 9/2 8/11	3.62	3.5
	Pm 3+ 4	5 14	2 6 4 3/5	2.68	-
	Sm 3+ 5	5 H 5/2	5/2 5 5/2 2/7	0.84	1.5
	Eu 3+ 6	7 F0	3 3 0	0	0	3.4
	Gd 3+ 7	8 S 7/2	7/2 0 7/2 2	7.94	8.0
	Tb 3+ 8	7 F6	3 3 6 3/2	9.72	9.5
	Dy 3+ 9	6 H 15/2	5/2 5 15/2 4/3	10.63	10.6
	Ho 3+ 10	5 I8	2 6 8 5/4	10.60	10.4
	Er 3+ 11	4 I 15/2	3/2 6 15/2 6/5	9.59	9.5
	Tm 3+ 12	3 H6	1 5 6 7/6	7.57	7.3
	Yb 3+ 13	2 F 7/2	1/2 3 7/2 8/7	4.54	4.5

Table 5 :

 5

	/2	0.70	3.87	3.82

Properties of hard magnetic materials that might be used for Perpendicular Magnetic Recording. All these materials are uniaxial and capable of sustaining over 10 year storage time. Physical parameters such as, anisotropy field: H k = 2K u /M s , domain wall width: w B = π(A/K u ) 1/2 , exchange coupling constant A = 10 -6 erg/cm, are displayed. Adapted from Weller [15].

• Current loop and grounded metallic plate: The image of the current loop through the grounded metal opposes it hence the lifting. This is used in Maglev trains that hover over the railway.

• Superconducting lifting: A pole g above a superconductor creating an image g since the superconductor, being a perfect diamagnet does not allow a non-zero induction inside it such that B n = 0 (normal to superconductor surface) whereas the tangential component B t = 0. Thus the pole+superconductor system is the magnetic analog of an electric charge q above a grounded metal screening an electric field E such that E n = 0 (normal to metal surface) whereas the tangential E t = 0.

• Levitron: Dynamically stabilized magnetic levitation

• Eddy currents in a metal in presence of a current loop: Applying a time-dependent magnetic field to a metal creates eddy currents that can be used for lifting. A typical example is a current loop or a time-dependent Note: This section is written in SI units.

The Lagrangian [START_REF] Landau | Electrodynamics of Continuous Media[END_REF] of a particle with charge e, mass m and velocity v in presence of an electric potential Φ(r, t) with associated potential energy V e = eΦ(r, t) and a magnetic vector potential A(r, t) with associated potential energy

2 is the kinetic energy and V = V e + V m the total potential energy:

The Hamiltonian is obtained from the Lagrangian via the Legendre transformation (r, v) to (r, p) where p = ∂L ∂ ṙ = ∂L ∂v :

Evaluating p = ∂L ∂v = mv + eA, the velocity is obtained as v = (p-eA) m . The Hamiltonian becomes:

Another way to obtain the Hamiltonian simply is the use of Peierls substitution transforming p into p -eA such that the kinetic energy p 2 2m becomes (p-eA) 2

2m

.

Note that the Hamiltonian is given by H = T + V e and not H = T + V e + V m since the kinetic energy embodies the magnetic contribution through Peierls substitution.

Using Hamilton equations ṗ = -∂H ∂r , ṙ = ∂H ∂p , we obtain using Einstein summation rule:

) -e ∂Φ ∂r i

ṙi

Using independence of generalized coordinates ∂pj ∂pi = δ ij , we get the equation of motion after taking the time derivative of the velocity components ṙi = 1 m (p i -eA i ):

Using Hamilton equations and dAi dt = ∂Ai ∂t + ∂Ai ∂ri ṙj , we get:

The Lorentz force is E = -∂A ∂t -gradΦ and B = curlA.

Note that the equation of motion is not obtained from the Newtonian formula F = dp dt but from F = m dv dt = mr since p = mv.

Bohr-Van Leuwen

Note: This section is written in SI units. Starting with a gas of N atoms/molecules carrying each a charge e, we write the Hamiltonian as

When a magnetic field is applied to this ensemble, we use the Peierls substitution to get:

Placing this ensemble of particles with a thermal reservoir at temperature T allows us to exploit the Canonical ensemble thermal averages with the partition function:

The free energy is given by: F = -ln Z/β and the thermal average magnetization is

with V the volume.

Making the change of variables p = p -eA and integrating over p makes A disappear from Z and thus M = 0 since Z does not depend neither on A nor on B.

Dirac monopole

Note: This section is written in SI units. Given an electric charge Q placed in a field B created by a pole of strength g, its wave function ψ is modified by the Peierls substitution with a phase factor given by: exp(-iQ(A • r)/ ). Performing a path around the electric charge alters the phase with: ∆ = Q A • dr. Using Stokes theorem, we have:

The induction created by g at a distance r is B = µ0g 4π 1 r 2 and the flux Φ over the sphere of radius r is Φ = B × 4πr 2 , thus Φ = µ 0 g.

The phase picked up by the wave function after performing a complete loop around the pole should be 2πn where n is an integer:

Thus we get the charge quantization condition: Q = 2π gµ0 n implying that the elementary electric charge is e = 2π gµ0 .

It suffices to have a single magnetic pole in the Universe in order to have electric charge quantization. The search for the elusive monopole is still ongoing.

APPENDIX B: Multipole expansions in Electrostatics and Magnetostatics

In Electrostatics a multipole is defined from:

where ρ(r) is the charge density. When we have factors x i the multipole tensor Q ijkl... has 2 components. In traditional Magnetostatics there is no magnetic charge (pole) thus we should start with a dipole equivalent which is the current loop moment m= 1 2 dV (r × j) such that m= I S and integration performed over the loop volume V = 2πRs where R, s are the loop radius and section respectively (cf. fig. 9). We get 1 2 dV (r × j) = 1 2 R(e r × je θ )2πRs = ISz since V = 2πRs, S = πR 2 , e r × e θ = z and I = js (cf. fig. 9). Thus the magnetic counterpart of Q ijk... is the M ijk... tensor defined as:

Integration performed over the volume of the current loop (see fig. 9) with m i playing the role of the electrostatics ρ(r) x i term.

APPENDIX C: General solution of the Stoner-Wohlfarth problem for arbitrary angle φ

Let us write the normalized energy as:

The minimum condition requires ∂E ∂θ = 0 as given by The moment given by:

We transform the trigonometric form into a complex algebraic equation with Euler substitution:

) where z = exp(iθ) to obtain:

The solution of this complex quartic equation is given by the NIST [START_REF] Olver | NIST Handbook of Mathematical Functions[END_REF] Handbook of Mathematical Functions as follows:

Setting z = w -1, reduces f (z) = z 4 + az 3 + bz 2 + cz + d to g(w) = w 4 + pw 2 + qw + r, with: p = (-3a 2 + 8b)/8, q = (a 3 -4ab + 8c)/8, r = (-3a 4 + 16a 2 b -64ac + 256d)/256.

The discriminant [START_REF] Olver | NIST Handbook of Mathematical Functions[END_REF] of g(w) is:

The roots α 1 , α 2 , α 3 , α 4 of g(w) = 0 are related to the roots θ 1 , θ 2 , θ 3 of the resolvent cubic equation:

APPENDIX D: Weber Electrodynamics Wilhelm E. Weber introduced a generalised electrodynamic force that was first published in 1846, a few years before Maxwell's first works on EM. Weber's action-at-a-distance theory is not known today and often dismissed on the basis that it is inappropriate. Indeed there are historic reasons that led to the dismissal of Weber's theory and the success of Maxwell-Lorentz field theory, however there are several misconceptions about Weber work.

Weber EM is only one example of departure from orthodox Maxwell-Lorentz theory and some emerging ideas might lead to a revival of some Weber ideas as well as other workers dismissed maybe unfairly by History of Science. The appropriate system for Magnetism is CGS whereas SI is appropriate for Electricity.

The unit of current is 1 ampere= 1 Coulomb/s in SI but in CGS it is the Abampere or 10 ampere coming from Abampere=10 Coulomb/s. We explain below the origins and practical aspects of the SI versus the CGS systems highlighting their origins and applications.

Some of the CGS units expressed with their fundamental constituents, Abampere, cm, gram (g), and second (s) are: 1 emu= Abampere.cm that comes from the equivalence between surface S loop carrying I current yielding moment µ = IS and ordinary dipole definition M = gl with length l, thus unit of g is [g] = IL. Note: Dipole length l does not exist in magnetism since we do not have a characteristic length. This originates from the fact of cutting a magnet S|N in half will result in another magnet regardless of the number of cuts despite the fact magnetic matter is lost during cutting S|N. This brings a tremendous benefit to the magnetic industry since scaling of a purely device is not a major concern, in sharp contrast to the ferroelectric and nano-electronic industries.

Physical quantity

CGS SI Field of a pole H = g/r 2 (oersted) H = g/4πr 2 (ampere/m) Force between poles F = g1g2/r 2 (dyne) Unit conversions. CGS being a non-rationalized metrological system provides two definitions of magnetization in emu/cm 3 and gauss leading to confusion. For instance Iron M s =1707 emu/c 3 is sometimes expressed as 1707 Gauss whereas it should be 21,440 Gauss [START_REF] Cullity | Introduction to Magnetic Materials[END_REF]. e is the electric charge in SI whereas e = ec/10 ≈ 4.8 × 10 -10 esu (electrostatic units) is the electric charge in CGS.

Some of the SI units expressed with their fundamental constituents, ampere (A), meter (m), kilogram (kg), and second (s) are: newton (N) = kg m/s 2 , joule (J) = kg m 2 /s 2 , tesla (T) = kg/(s 2 A), weber (Wb) = kg m 2 /(s 2 A), henry (H) = kg m 2 /(s 2 A 2 ) Let us illustrate with some examples:

• µ 0 determination: In CGS force between poles is: F = g 1 g 2 /r 2 (dyne). Let us express the force in SI and convert it from CGS to SI. Thus in SI the force is: F = µ 0 g 1 g 2 /4πr 2 (newton). Transforming the CGS value of 1 dyne= (1 Abampere.cm) 2 /(1cm) 2 into SI, we get: 10 -5 newton= µ 0 /4π (0.1 A.m).(0.1 A.m)/(10 -2 .10 -2 m 2 ) yielding µ 0 = 4π.10 -7 Henry/m in SI. Note: From the metrological point of view, µ 0 must be determined from a current. In fact, µ 0 is determined from the force per unit length F/L between two wires separated by 1 m and traversed by opposing currents of 1 ampere. The attractive force should be 2 dynes/cm between the wires. Using F/L = µ 0 I 1 I 1 /2πd with • Moment conversion: unit of µ is emu. In CGS we use µ = I.S with I= 1 Abampere and S= 1 cm 2 . Converting to SI: 1 emu is equivalent to a 1 Abampere= 10 amperes current loop having a surface 1 cm 2 =10 -4 m 2 yielding 1 emu= 10 -3 ampere.m 2 .

• Magnetization conversion: M is moment/volume, thus unit of M in CGS is emu/cm 3 . Converting to SI: 1 emu= 10 -3 ampere.m 2 and 1 cm 3 =10 -6 m 2 , yield 1 emu/cm 3 = 10 -4 weber/m 2 since 1 Abampere=10 ampere. If we use 4πM in gauss, we have 4π× 10 -4 weber/m 2 .

• Field conversion: In CGS H = g/r 2 in Oersted. The conversion amounts to start from an SI expression, and convert every element from CGS to SI thereby finding the equivalence between the SI and CGS values. For instance g = 1 Abampere.cm, r = 1 cm give 1 Oersted. In SI the field is H = g/4πr 2 . Simply plug-in these values to get the SI field. This leads with g = 0.1, r = 10 -2 m to H = g/4πr 2 or H = 10 3 /4π A/m. Thus 1 oersted =10 3 /4π ≈ 80 ampere/m.

• Induction conversion: In CGS the energy is W = M.H whereas in SI W = M.B with an additional subtlety that µ 0 = 1 in CGS thus Oersted is equivalent to gauss. Since W = M.H is in ergs and W = M.B in Joules, we get from 1 erg= 10 -7 Joules and M =10 -3 ampere.m 2 , the conversion: 1 gauss= 10 -7 /10 -3 = 10 -4 Tesla.

• Permeability conversion: In CGS let µ= 1 gauss/oersted and convert to SI, using 1 gauss =10 -4 weber/m 2 and 1 oersted= 10 3 /4π ampere/m. We obtain: 4π × 10 -7 henry/m which is about 1.257 ×10 -6 henry/m. 2). Multiplying by the density of Fe (7.86 g/cm 3 according to Table 2), we get finally: M s = 1707 emu/cm 3 . This corresponds to 1.7 ×10 6 amperes/m since 1 emu is equivalent to 10 -3 ampere.m 2 (see Table 10). Weiss estimation of 10 5 amperes originates from a confusion of magnetization units since M s should be expressed in emu/cm 3 but often expressed in Gauss with another potential 4π factor mistake (see Table 10).