Professor V N Seleznev

E Maksimova

To the memory of

Keywords: borate de fer-gallium, synthèse de cristaux, résonance magnétique électronique, résonance magnétique nucléaire, anisotropie magnétocristalline, magnétisme de surface iron-gallium borate, crystal synthesis, electron magnetic resonance, nuclear magnetic resonance, magnetocrystalline anisotropy, surface magnetism

Propriétés magnétiques et résonances magnétiques de monocristaux à base de borate de fer:

Études expérimentales et modélisation

La thèse porte sur la synthèse et l'étude des propriétés magnétiques de borates de fer-gallium, Fe x Ga 1-x BO 3 avec 0x1. Ces matériaux sont prometteurs pour les applications; en plus, grâce à la présence, en fonction de x, de différents types d'ordre magnétique, ils sont bien adaptés au traitement de nombreux problèmes du magnétisme des solides.

Le borate de fer, FeBO 3 est un antiferromagnétique possédant un plan de facile aimantation et un faible ferromagnétisme. Les caractéristiques du borate de fer sont radicalement modifiées par substitution isomorphe fer -gallium diamagnétique.

Nous avons mis au point une route de synthèse de monocristaux Fe x Ga 1-x BO 3 de haute qualité. Comme principales techniques expérimentales, nous avons choisi les résonances magnétiques électronique (RME) et nucléaire (RMN). Selon le contenu du fer, nous avons observé:(i) la résonance antiferromagnétique, (ii) la résonance de clusters magnétiques et (iii) la résonance paramagnétique électronique (RPE).

Les différents états magnétiques ont été identifiés et leurs caractéristiques -la température de Néel, le champ de Dzyaloshinskii-Moriya; les paramètres de l'hamiltonien de spin de Fe 3+ , etc.

-ont été déterminées. La coordinence et la symétrie de sites de 11 B et 71 Ga ont été précisées par RMN à rotation sous l'angle « magique » (MAS). Moyennant la simulation des spectres de RPE et de MAS RMN, à l'aide de codes mis au point ad hoc, les distributions de paramètres dues au désordre local ont été déterminées. L'analyse théorique, tenant compte de contributions du champ cristallin et de l'interaction dipôle-dipôle, permet d'expliquer l'anisotropie magnétocristalline de volume et de surface.

Sommaire

L'intérêt de la recherche du borate de fer, FeBO 3 est principalement lié à ses remarquables caractéristiques magnétiques, magnéto-acoustiques, optiques, magnéto-optiques, etc. Ce matériau est prometteur pour les applications dans des diverses branches des sciences expérimentales et de l'ingénierie. En particulier, éléments de mémoire magnétique à haute densité d'enregistrement, transducteurs magnéto-acoustiques et magnéto-optiques, instruments de mesure des champs magnétiques ultra-faibles, de la température et de la pression peuvent être fabriqués sur la base du borate de fer. FeBO 3 peut être appliqué dans les batteries lithium-ion afin d'augmenter leur capacité. Par ailleurs, le borate de fer peut être utilisé comme un monochromateur du rayonnement synchrotron pour la spectroscopie Mössbauer.

Le premier chapitre « Crystal and magnetic structure of iron borate » présente la structure cristalline et magnétique de FeBO 3 . Ce cristal a une structure de calcite rhomboédrique de groupe d'espace 6 3d D , possédant un axe C 3 orthogonal au plan de base. Du point de vue de la structure magnétique, en dessous de la température de Néel T N =348 K, FeBO 3 est antiferromagnétique à deux sous-réseaux, possédant un plan de facile aimantation qui est le plan de base ; cependant, les aimantations des sous-réseaux ne sont pas tout à fait antiparallèles, ce qui engendre un faible ferromagnétisme dû à l'interaction Dzyaloshinskii-Moriya.

Par ailleurs, les caractéristiques physiques du borate de fer sont radicalement modifiées par substitution isomorphe d'une partie du fer paramagnétique par le gallium diamagnétique. Nous avons mis au point la route de synthèse de monocristaux de borates de fer-gallium Fe x Ga 1-x BO 3 , par solution en fonte, décrite dans le deuxième chapitre « Synthesis of iron-gallium borate single crystals ». Des monocristaux de haute qualité ont été synthétisés dans toute la gamme des compositions 0 1 x   . À l'aide de l'analyse thermodifférentielle et de la technique de sonde, nous avons déterminé les rapports de composants dans la charge et les modes de température correspondants, optimaux pour la synthèse de monocristaux avec différents x. Après la synthèse, la composition des cristaux et les paramètres du réseau cristallin ont été déterminés par la spectrométrie de fluorescence et diffraction des rayons X (SFX et DRX, respectivement). La SFX a révélé une certaine distribution de concentrations d'ions Fe 3+ et Ga 3+ dans les cristaux extraits du même creuset. L'analyse par DRX a montré que dans les borates mixtes de fer-gallium la modification des paramètres du réseau cristallin suit de près la loi de Vegard.

Les cristaux Fe x Ga 1-x BO 3 présentent d'un grand intérêt, tant pour la physique du solide que pour la science des matériaux, dans la mesure où ils permettent :  de suivre la transformation des propriétés magnétiques sous la transition entre l'état magnétiquement ordonné et l'état paramagnétique ;

 d'interpréter les propriétés magnétiques des cristaux dilués, en particulier, l'anisotropie magnétocristalline, pour mieux comprendre la nature de ces propriétés dans le borate de fer pur. En effet, les différents mécanismes responsables des propriétés magnétiques de FeBO 3 possèdent différentes dépendances en température et en concentration en fer ;

 la solution en fonte permet de synthétiser les cristaux de haute qualité avec les propriétés magnétiques prédéterminées, adaptées aux applications pratiques.

Les chapitres suivants traitent les études expérimentales et théoriques de borates de fergallium. Pour autant que nous soyons intéressés par les propriétés magnétiques des cristaux, nous avons choisi comme principales techniques expérimentales les résonances magnétiques électronique (RME) et nucléaire (RMN).

Le troisième chapitre « Electron magnetic resonance of iron-gallium borate single crystals with 0.2≤x≤1 » présente les études de Fe x Ga 1-x BO 3 avec 0 1 x   par RME. Cette technique permet d'identifier les états magnétiques des cristaux à différentes teneurs en fer et à différentes températures. En fonction de x, les spectres de RME montrent un passage graduel de la résonance antiferromagnétique (RAFM) à x = 1 vers la résonance paramagnétique électronique (RPE) d'ions de fer dilués à 1 x  , en passant pour 0.34 0.85 x   par une coexistence de RAFM et de la résonance de clusters magnétiques (RCM), provenant, respectivement, des régions du cristal magnétiquement ordonnées et partiellement désordonnées ; pour 0.34 x  seule la RCM est présente.

Les températures de L'intensité de la raie de RMC, observée dans tous les cristaux avec 1 x  sauf à des valeurs de x très faibles, ne suit pas la loi de Curie en 1 T  , ce qui suggère que cette raie est due aux clusters magnétiques. L'écart d'énergie anisotrope, déterminé pour le cristal avec x = 1 à 77 K, a permis de calculer la constante effective de l'anisotropie hexagonale; ce dernier a été utilisé dans l'analyse de l'anisotropie magnétocristalline de base.

La RPE d'ions Fe 3+ isolés dans les cristaux avec 1 x  a été utilisée pour déterminer la symétrie des sites de fer et les paramètres de l'hamiltonien de spin dans les borates de fer-gallium. Ces données sont nécessaires pour obtenir une description complète de l'anisotropie magnétocristalline dans ces cristaux. Une facette essentielle des études de RPE est l'analyse numérique -modélisation par ordinateur -des spectres expérimentaux. Le quatrième chapitre « EPR of iron-gallium borate single crystals with low x » décrit la paramétrisation des spectres de RPE d'ions Fe 3+ à l'aide de codes informatique mis au point ad hoc. Dans un premier temps, nous avons développé un code basé sur l'hamiltonien de spin conventionnel, ce qui a permis de bonnes simulations des spectres expérimentaux, au moins, en ce qui concerne les champs de résonance. Cependant, nous avons constaté que dans cette approche deux jeux de paramètres différents sont possibles, dans la mesure où ils aboutissent à la même matrice de l'hamiltonien de spin. Cette dichotomie a pu être résolue en testant la compatibilité de ces deux jeux de paramètres avec les prédictions du modèle de superposition de Newman. A cet effet, nous avons utilisé l'hamiltonien de spin général pour les sites de symétrie trigonale. Les tests ont montré sans ambiguïté que seul l'un de ces jeux de paramètres est compatible avec le modèle de Newman.

À l'aide des simulations utilisant les deux hamiltoniens de spin -conventionnel et généralon obtient un bon ajustement des positions des raies de résonance ; en revanche, d'importantes divergences subsistent entre les amplitudes et largeurs des raies correspondantes dans les spectres expérimentaux et théoriques. Nous avons attribué ces divergences, à l'existence d'un certain degré de désordre local dans les cristaux. Dans le but de prendre en compte ce désordre dans l'hamiltonien de spin, nous avons exprimé celui-ci au moyen des opérateurs équivalents tesséraux à deux vecteurs, le spin effectif et le champ magnétisant. Cette approche nous a permis d'établir des relations entre d'une part, les distributions dues au désordre, des coordonnées d'oxygènes dans l'entourage de fer, et d'autre par les distributions des paramètres de l'hamiltonien de spin. Les simulations par ordinateur effectuées en utilisant un code basé sur ce modèle ont permis d'obtenir un très bon accord entre les spectres théoriques et expérimentaux en ce qui concerne à la fois les positions des raies et leurs amplitudes et largeurs. Ainsi, l'hypothèse de la présence du désordre local dans les cristaux fut confirmée.

Les études par RMN à rotation sous l'angle « magique » (MAS) des cristaux Fe x Ga 1-x BO 3 avec 0 0.02 x  

, sont présentées dans le cinquième chapitre « NMR studies of iron-gallium borates ». Le double objectif de ces études a été de préciser la coordination et la symétrie de sites de bore et de gallium ainsi que corroborer, de façon indépendante, la présence d'un certain degré de désordre local dans ces cristaux. Les simulations par ordinateur des spectres de MAS RMN de noyaux 11 B et 71 Ga confirment la triple coordinence et la symétrie intrinsèque 3 C pour le bore, et la sextuple coordinance et une symétrie plus faible que la symétrie cubique pour le gallium. Pour les deux noyaux, l'élargissement des spectres de MAS RMN avec l'augmentation de teneur en fer a été attribué aux distributions des paramètres quadripolaires et du déplacement chimique, dans le cas de 71 Ga, causés par le désordre local, comme attendu. Comme dans le cas de RPE, les spectres ont été simulés par ordinateur ; dans le cas de RMN nous avons utilisé deux différents modèles de la fonction de distribution des paramètres, celles de Czjzek et de Maurer. Une analyse comparative des ces modèles a été réalisée en vue de leur application pour tenir compte de l'élargissement des spectres de MAS RMN induite par le désordre local dans les cristaux. Le modèle de Czjzek est bien adapté au cas des solides fortement désordonnés ; cependant, il ne tient pas compte d'ordre local partiellement conservé dans les cristaux présentant un certain désordre. En effet, il ne prend pas en considération les valeurs moyennes non nulles des paramètres quadripolaires et il ne contient qu'un seul paramètre de largeur de distribution. Au contraire, le modèle de Maurer n'a pas ces limitations, il est donc mieux adapté aux simulations des spectres de RMN expérimentaux dans les cristaux à faible désordre local.

Les résultats des études de résonance magnétique ont permis de donner une nouvelle description théorique de l'anisotropie magnétocristalline dans le borate de fer et les borates mixtes de fer-gallium. Cette anisotropie comprend deux contributions : (i) du champ cristallin et (ii) de l'interaction dipôle-dipôle.

Dans la mesure où les ions Fe 3+ dans FeBO 3 se trouvent dans un fort champ d'échange, la première contribution peut être calculée dans la théorie des perturbations, en exprimant les constantes d'anisotropie par l'intermédiaire des paramètres de l'hamiltonien de spin déterminés par RPE. D'autre part, auparavant la contribution dipôle-dipôle a été considérée en négligeant la taille les dipôles (dipôles « ponctuels ») auquel cas elle contribuerait uniquement à l'anisotropie uniaxe (suivant l'axe C 3 ), mais pour des raisons de symétrie ne donnerait pas de contribution dans l'anisotropie de base. En effet, dans le borate de fer cette dernière est hexagonale, alors que l'énergie d'interaction de dipôles ponctuels dans le plan de base du borate de fer reste isotrope. Nous avons réexaminé ce problème en tenant compte des dimensions des dipôles (dipôles « étendus »). En supposant un rapport non négligeable entre la taille du dipôle et la distance dipôle-dipôle et en développant l'énergie d'interaction dipôle-dipôle en série de Taylor, les différents termes de ce développement permettent de rendre compte d'interactions de différentes symétries. Toutefois, pour explicitement calculer ces termes, on a besoin d'un modèle du dipôle étendu.

À cet effet, nous avons présenté une description théorique de trois modèles : (i) une sphère uniformément aimantée, (ii) une boucle de courant circulaire (un courant ampérien) et (iii) une paire de deux « charges magnétiques » fictifs. Nous avons démontré que les trois modèles donnent des résultats identiques à grandes distances. En effet, le premier terme du développement de Taylor de l'énergie dipôle-dipôle dans les modèles d'une boucle de courant circulaire et d'une paire de charges magnétiques coïncident avec l'expression de cette énergie pour les dipôles ponctuels (ou pour deux sphères uniformément aimantées non pénétrantes mutuellement).

Pour le calcul de la contribution dipôle-dipôle à l'anisotropie magnétocristalline du FeBO 3 , nous avons utilisé la sommation sur le réseau cristallin pour deux modèles de dipôles étendus, à savoir une paire de charges magnétiques et une boucle de courant circulaire. La comparaison avec les données expérimentales élimine le modèle d'une paire de charges magnétiques ; en revanche, le modèle d'une boucle de courant circulaire paraît tout à fait adapté dans la mesure où il permet de rendre compte de la symétrie hexagonale de base et, accessoirement, fournit deux estimations plus ou moins réalistes de la taille des dipôles associés à ion Fe 3+ . L'énergie d'interaction dipôle-dipôle pour les borates mixtes Fe x Ga 1-x BO 3 a été calculée par la même technique que pour FeBO 3 . Afin de modéliser le un réseau cristallin diamagnétiquement dilué, nous avons utilisé la technique de Monte Carlo. Ainsi nous avons obtenu les contributions dipôle-dipôle aux constantes d'anisotropie magnétocristalline uniaxe et de base pour des cristaux avec différents x. Les résultats de cette analyse sont décrites dans le sixième сhapitre « Magnetocrystalline anisotropy of iron borate and iron-gallium borates ».

Les propriétés surfaciques constituent un véritable défi compte tenu de leur importance primordiale pour la miniaturisation des éléments de micro-électronique. Auparavant, l'anisotropie magnétocristalline de surface dans le borate de fer volumique avait été décrite en ne tenant compte que de la contribution dipôle-dipôle. Dans le septième сhapitre « Surface magnetocrystalline anisotropy of iron borate single crystals » nous avons étendu la théorie du magnétisme de surface de borate de fer, en tenant compte de la contribution du champ cristallin.

Nous avons développé un modèle de distorsions structurales dans la couche surfacique pour la face   1014 de borate de fer, dans l'hypothèse que les positions de tous les ions dans cette couche soient décalées proportionnellement à leur distance d'un plan de référence, supposé rester immobile, parallèle à la face du cristal. Afin de calculer les paramètres de l'hamiltonien de spin pour les ions Fe 3+ dans la couche surfacique en tenant compte de la baisse de symétrie, nous avons utilisé l'hamiltonien de spin généralisé exprimé au moyen des opérateurs équivalents tesséraux à deux vecteurs (voir chapitre 4) et le modèle de superposition. Pour les ions fer dans la couche surfacique, nous avons utilisé les mêmes paramètres du modèle de superposition que dans le volume, v. chapitre 4, dans la mesure où ces paramètres devraient rester inchangés tant que les distorsions structurales restent faibles. La contribution du champ cristallin à l'énergie d'anisotropie surfacique a été calculée dans la théorie des perturbations. Les résultats de ce calcul montrent que la prise en compte des modifications du champ cristallin dues aux distorsions structurales produit une contribution significative à l'anisotropie surfacique. La comparaison avec la détermination expérimentale de l'anisotropie magnétocristalline surfacique du borate de fer permet de conclure que les distorsions (contractions) relatives dans la couche surfacique sont proches de 1% .

Les résultats de ma thèse ont été présentés dans un certain nombre de conférences internationales et donné lieu à plusieurs publications, voir la liste à la fin de la thèse.

Contents

Introduction

The research interest in iron borate FeBO 3 is mainly due to its remarkable magnetic, magneto-acoustical, optical, magneto-optical, resonance, etc. characteristics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. This material is a first-rate candidate for practical applications in various branches of experimental science and engineering. In particular, magnetic memory elements possessing high density of recording, magneto-acoustic and magneto-optical transducers, instruments for measuring ultra-weak magnetic fields, temperature and pressure can be created on the basis of iron borate. FeBO 3 can be applied in modern lithium-ion batteries in order to substantially increase their capacity [17][18][19]. Besides, iron borate can be used as an excellent monochromator for synchrotron Mössbauer spectroscopy [20,21].

Chapter 1, "Crystal and magnetic structure of iron borate" describes in detail crystal and magnetic structure of FeBO 3 . This crystal has a rhombohedral calcite structure of 6 3d D space group [22]. From the viewpoint of magnetic structure, iron borate is a two-sublattice easy-plane antiferromagnet with the Néel temperature T N = 348 K, showing a weak ferromagnetism caused by the Dzyaloshinskii-Moriya interaction [22].

have different concentration and temperature dependences; consequently, the series of Fe x Ga 1-x BO 3 crystals offers a possibility to get a better insight in their nature.

(iii) The solution in the melt technique allows synthesizing high quality crystals with predetermined magnetic properties, suitable for practical applications.

Thus, mixed iron-gallium borates are of a major importance both in solid state physics and in materials science.

The following chapters deal with experimental and theoretical studies of iron-gallium borates. As far as we are mainly interested in magnetic properties of the crystals, as key experimental techniques we have chosen magnetic resonances, namely Electron Magnetic Resonance (EMR) and Nuclear Magnetic Resonance (NMR).

Chapter 3, "Electron magnetic resonance of iron-gallium borate single crystals with 0.2≤x≤1" describes EMR studies of Fe x Ga 1-x BO 3 crystals with 0 1 x   . This technique allows identifying magnetic states occurring for different iron contents and at different temperatures. In particular, the Antiferromagnetic Resonance (AFMR) in crystals possessing magnetic ordering allows specifying the Néel temperature, the Dzyaloshinskii-Moriya field; studying the magnetocrystalline anisotropy of the samples. The Electron Paramagnetic Resonance of isolated Fe 3+ ions in crystals with 1 x  serves as a powerful technique for specifying the symmetry of iron sites and determining spin Hamiltonian parameters of isolated Fe 3+ ions. These data are necessary for a consistent description of the magnetocrystalline anisotropy in the crystals. An essential facet of the EPR spectroscopic studies is the numerical analysis -computer modellingof the experimental spectra. Chapter 4, "EPR of iron-gallium borate single crystals with low x" describes the full parametrization of the EPR spectra of diluted Fe 3+ ions by means of detailed computer simulations with the help of laboratory-developed codes, explicitly taking into account parameter distributions and thus allowing to estimate the degree of local disorder in the crystals. We have used different spin Hamiltonians in order to resolve the ambiguity in the choice of different sets of parameters.

The Magic Angle Spinning (MAS) NMR studies of Fe x Ga 1-x BO 3 crystals with 0 0.02 x   , described in Chapter 5, "NMR studies of iron-gallium borates", have been made in order to specify coordination and site symmetry of 11 B and 71 Ga nuclei. As in the case of EPR, we have followed the approach of computer simulations, applying different distribution functions of quadrupole parameters and chemical shift. A comparative analysis of the Czjzek's and Maurer's models of the joint distribution density of NMR quadrupole parameters has been carried out in view of their application to account for MAS NMR spectra broadening induced by local disorder in the crystals.

The results of the magnetic resonance studies have allowed to give a new theoretical description of magnetocrystalline anisotropy in iron borate and mixed iron-gallium borates. This anisotropy includes (i) crystal field and (ii) dipole-dipole contributions. The first contribution has been calculated in perturbation theory, expressing the anisotropy constants through the spin Hamiltonian parameters determined by EPR. The second contribution is usually considered for "point dipoles" having a negligible size. In this case only uniaxial anisotropy is accounted for, basal anisotropy being ruled out on the grounds of symmetry. Indeed, the basal anisotropy in iron borate has hexagonal symmetry, whereas the interaction energy of point dipoles in the basal plane is isotropic. We have reexamined the dipole-dipole contribution using the concept of "extended dipoles" having non-negligible size. Taking into account higher-order terms in the expansion of the dipole-dipole interaction energy in a Taylor series in the small parameter dipole size/interdipole distance opens the possibility of describing more sophisticated issues, in particular, the hexagonal magnetocrystalline anisotropy. However, in order to explicitly calculate these terms we need a physical model of the extended dipole. For this purpose we have put forward a theoretical description of three models of the extended dipoles: (i) a uniformly magnetized sphere, (ii) an Ampérian current and (iii) an assembly of two fictitious "magnetic charges". For calculating the dipole-dipole contribution to the magnetocrystalline anisotropy we have used the lattice-sum method; for the mixed borates, a diamagnetically diluted crystal lattice has been generated by Monte Carlo technique [25]. A comparison between the calculation and the experimental data provides a consistent description of the basal magnetocrystalline anisotropy; moreover, it allows estimating the size of the extended dipoles. The results of this analysis are described in Chapter 6, "Magnetocrystalline anisotropy of iron borate and irongallium borates".

Surface properties represent a true challenge, as far as these properties are important for miniaturization of elements of microelectronics. Previously, the surface magnetocrystalline anisotropy in bulk iron borate had been described taking into account only the dipole-dipole contribution [6]. In Chapter 7, "Surface magnetocrystalline anisotropy of iron borate single crystals", we extend the theory of surface magnetism, taking into account the crystal field contribution. With this aim in view, we have put forward a model of structural distortions in the near-surface layer, allowing a complete description of the surface magnetocrystalline anisotropy of iron borate.

The results of my thesis have been presented in a number of international conferences and given rise to several publications, see the list at the end of the thesis.

  1 2 h r r h r r 1 2 2 sin 3 1 2 cos . a a c a      (1.1)
The rhombohedral, see Figure 1.2, and hexagonal unit cells of FeBO 3 contain two and six formula units, respectively. In order to specify the atomic arrangement in iron borate, the so-called oxygen parameter h x should be determined. [1.3].

Atoms

Hexagonal setting Rhombohedral setting Fe 1 2 0, 0, 0; 0, 0,

1 1 1 2 2 2
0, 0, 0; , ,

B 3 1 4 4
0, 0, ; 0, 0,

3 3 3 1 1 1 4 4 4 4 4 4 , , ; , , O    1 1 1 h h h h 4 4 4
,0, ;0, , ; , ,

x x x x   * 1 1 1 1 1 1 r r r r r r 2 4 2 4 4 2 , , ; , , ; , , x x x x x x     * 1 r h 4
x x  

Magnetic structure

Neutron diffraction studies have shown that iron borate is an antiferromagnet with weak ferromagnetism [1.4]. Indeed, FeBO 3 crystals possess two magnetic sublattices. The magnetic Since the neutron diffraction measurements by Pernet et al. [1.4] have been carried out on a polycrystalline iron borate, they could not determine the orientation of a in the basal plane.

Various vectors used in describing the magnetic structure of FeBO 3 are defined in Table 1.4.

Vector Definition

Sublattice magnetizations

* 1,2 1,2 1 2 N  m M Ferromagnetic vector or spontaneous magnetization 1 2 1 2 N    M f M M Antiferromagnetic vector 1 2 1 2 N    L a M M Reduced ferromagnetic vector 1 2  M m M ** Reduced antiferromagnetic vector 1 2  L l M * N is the iron concentration ** M is the norm of the sublattice magnetization. For FeBO 3 at 0 K 0 520 G  M [1.3]. Note that 2 2 1 l m   (1.2)
Taking into account the value of the tilt angle, 55 The magnetic energy of FeBO 3 includes exchange energy E E and magnetocrystalline

   [1.5], cf.
anisotropy energy A E : E A   E E E . (1.4)
As far as for Fe 3+ ( 5 3d electron configuration) the orbital moment equals zero, E E in a good approximation is isotropic. Thus, A E includes only contributions of dipole-dipole and crystal field terms [1.6].  

2 2 E 1 1 2 2 12 1 2 1 2 E E E      E M M M M (1.5)
where

    2 2 2 2 2 E 1 12 E l m E m l            E M , (1.6)
and taking into account eq. (1.2), we get:

2 E 0 1 2 E Em   E (1.7) where   2 0 1 12 = E E E   M and 2 12 4 E E   M .
We expand E in a series of products of tesseral harmonics of the components of unit

vectors and l m   l m   : ( ) ( ). l m l m m m l m l l l l m m j j i i i j i j i i i j i j K l m Z Z   E   (1.8)
Here , , , l l m m i i i j are integers numbering the harmonics, 0

l i  , 0 m i  , 2 l m i i p     0, 1, 2, p  
is the rank of the corresponding expansion term; , ,

l m l m l m l m j i i i     , , , 1, ,
, and m m l l i j i j K are the corresponding constants.

As far as experiments have attested the existence of hexagonal anisotropy in the basal plane of FeBO 3 [1.7], the expansion of E should be extended up to the sixth rank in tesseral harmonics. Obviously, its expression can include only terms invariant under all transformations of the relevant point group. Besides, as in the case of FeBO 3 m l  , cf. eq. ( 1.3), for

1 p  we can neglect terms with 1 m i  .
Taking into account all above considerations, eq. (1.8) reduces to .

                                    0 0 0000 0 0 1 1 1 1 1 1 1 1 1111 111 1 2
K Z Z lm K Z Z K Z Z K l Z Z K m Z Z K l mZ Z K l Z Z K l mZ Z K l Z Z                   E                   (1.9)
Note that in order to possess the required symmetry, the relation 1111 1111

K K  should hold.
Explicit expressions of the tesseral harmonics featuring in this expression are [1.8]:

            0 1 0 1 0 1 1 1 0 2 3 3 2 2 3 3 2 3 3 3
Z Z Z Z Z Z Z Z Z                                                  (1.10)
where ,    

. Substituting these expressions in eq. (1.9), we get:

               
K K lm K l K m K l m K l K l m K l                                                         E (1.11)
This expression can also be written in terms of components of and l m :

               
K K l m l m K l l K m m K l l l m K l l l l K l l l l l m K l l l l l l                                E (1.12)
Simplifying the notation of the constants, up to a constant term we get:

                2 2 2 2 2 2 2 2 2 2 2 2 4 2 2 4 2 2 1 1 1 2 6 6 3 3 3 3 3 9 1 14 . x y y x z z x y x z x y y z x y z x z x x y y x y Em D l m l m a l l b m m c l l l m d l l l l f l l l l m e l l l l l l                   E (1.13)
Here the first term on the right-hand side describes isotropic exchange, cf. eq. (1.7); the second term accounts for antisymmetric exchange -the Dzyaloshinskii-Moriya interaction, responsible for weak ferromagnetism; the third and fourth terms represent uniaxial magnetocrystalline anisotropy; the remaining terms describe higher-order magnetocrystalline anisotropy. Relations between the constants featuring in eqs. (1.12) and (1.13) are straightforward.

Data available for certain constants in FeBO 3 are summarized in Table 1.5.

Table 1.5 Values of some constants for FeBO 3 at 0 K, taken or calculated from the corresponding references.

Constant

Value, The parameters and d e have not been experimentally determined. In Chapter 6, "Magnetocrystalline anisotropy of iron borate and iron-gallium borates", we provide a theoretical analysis of these quantities, resulting in the following order-of-magnitude estimates In equilibrium E is at a minimum. Minimizing E with respect to the components of m at fixed l , we get the equilibrium orientation of m with respect to l :

    2 2 2 1 1 2 3 3 3 9 1 , , 3
z y x x y x c f l D D l l l l l E b E b E b                         m . (1.14)
Obviously, z m is negligible in comparison with and

x y m m , so that we can put 0

z m  .
Substituting the components of m in eq. (1.13), after simple transformations we get:

         2 2 2 2 2 2 2 4 2 2 4 2 2 1 1 2 6 1 3 3 3 14 z z x y y z x x y y x y D l l a l l E b d l l l l e l l l l l l            E (1.15)
The first and second terms on the right-hand side of this equation can be written as a sum of an isotropic term,

l l          E E (1.16)
where we have introduced an effective constant of uniaxial anisotropy:

2 eff 1 3 D a a E b    . (1.17) Expressing   sin cos , sin sin , cos l       l , (1.18)
eq. (1.16) becomes:

  2 2 4 3 6 6 iso eff 1 6
3 cos 1 sin cos sin 3 sin cos 6 .

a l d l el            E E (1.19)
We are interested in the orientation of l in a stable equilibrium, i.e. in a minimum of  

                      E (1.
d el l            E . (1.21)
The nature of a critical point, in accordance with the "second partial derivative test" [1.16],

depends on the signs in this point of

                                       E E E E E H . (1.22)
Namely, a critical point is a minimum or a maximum point of

d l el              E E . (1.23) Obviously,   2     E vanishes for sin 3 0   , i.e. 1 3 , 0, 1, ,5 n n      . At these points,       2 2 8 4 2 2 eff eff det 9 4 ; 6 a e d l a el l         E H . (1.24)
In the latter equations as well as in the subsequent ones we have neglected higher than the first power in e terms, as far as such terms are small, cf. (iii) The last possible case is that of cos 3 0   , i.e.

 

1 6 2 1 , 0, 1, , 5 n n       .
In this case, the

                                        (1.25) where 2 1 eff 2 2 eff 2 2 eff 6 5 5 d l a el d l a d l d a        (1.26)
In the latter equations as well as in the subsequent ones, we have neglected terms higher than the second power in d and higher than the first power in e as well as terms in the product d e . Note that for the parameter values quoted in Table 1.5, 2  is well outside the domain of definition of  , therefore only the nature of the critical points

  1 1 2 ;         and   1 1 2 ;         should be determined.

Conclusions

The crystal and magnetic structures of iron borate have been discussed in details. The magnetic energy of FeBO 3 , E has been expanded in a series of products of tesseral harmonics of the components of unit vectors in the direction of and l m up to the sixth order and the equilibrium orientations of and l m have been determined for two possible magnetic states. In the first case, l is directed along one of 2 C and m lies in the symmetry plane perpendicular to this axis, making a negligibly small angle with the basal plane. In the second case, l lies in one of the symmetry planes making a small angle with the basal plane, and m is directed along the symmetry axis perpendicular to l and lies in the basal plane.

Synthesis of iron-gallium borate single crystals 2.1 Introduction

In this chapter we describe the procedure of synthesizing Fe x Ga 1-x BO 3 single crystals with a part (1 -x) of Fe 3+ ions isomorphously substituted by Ga 3+ ions.

As mentioned above in Introduction, the series of Fe x Ga 1-x BO 3 single crystals, synthesized in a wide range of compositions, allows detailed studies of transformation of various physical characteristics under the transition from magnetically ordered to paramagnetic state.

Iron borate crystals can be synthesized by two routes: (i) from gas phase and (ii) from solution in the melt [2.1, 2.2]. Using the gas phase technique, bulk single crystals of iron borate with large non-basal faces of optical quality are obtained. The occurrence of natural non-basal faces has allowed identifying and describing surface magnetism present on such faces and caused by specific surface anisotropy [2.3, 2.4]. Another effect observed only in bulk crystals is magnetic birefringence of sound [2.5].

The solution in the melt technique allows obtaining thin single crystals, 0.05 -0.1 mm along the 3 C axis and up to 10 mm in the basal plane, (0001) in hexagonal coordinate system.

Such crystals are of high structural perfection and are much less costly in manufacturing in comparison with those obtained from gas phase. For the purposes of the present work this technique has proved to be most appropriate.

To the best of our knowledge, the first synthesis of FeBO Using the solution in the melt technique simplifies the crystal synthesis, allows obtaining materials with complex compositions and reduces the crystallization temperature of refractory compounds. Usually, crystallization regimes suitable for obtaining desired crystal phases are determined by trial and error, requiring significant time and financial resources. Hence, express methods of determining the ranges of stability of crystalline phases and temperature ranges of phase formation are desirable. In the present work for this purpose we have used differential thermal analysis (DTA) and probe methods.

After Fe x Ga 1-x BO 3 synthesis, a detailed characterization is required for further studies and applications of the samples. The exact crystal composition and parameters of the crystal lattice have been determined by XRF and XRD, respectively. Besides, EPR has been used to estimate local disorder in the Fe x Ga 1-x BO 3 crystals at low x values.

Crystal synthesis

The synthesis of Fe x Ga 1-x BO 3 crystals by solution in the melt technique is schematically represented in Figure 2  obtaining a homogeneous solution melt in a crucible;

 installing the crucible in a crystallization furnace;

 slow cooling of the solution according to a predetermined temperature mode;

 extracting the crucible from the furnace;

 dissolving the remnants of solution melt in 20 % solution of HNO 3 and extracting the crystals from the crucible.

Differential thermal analysis

Differential thermal analysis (DTA) can be applied to a wide range of substances; meanwhile, there have been only few DTA studies of the solution-melt systems. This can be explained by the fact that for diluted solutions, sensitivity of this technique is greatly reduced because of the smallness of the exothermic peak on the thermograms [2.10].

Therefore, we have developed a highly sensitive setting for rapid analysis of small amounts of the solution melts in the temperature range from 300 to 1150ºС. This setting allows detecting weak temperature change T  caused by crystal formation and determining temperature ranges of crystallization with sufficient accuracy [2.11, 2.12].

As a differential thermocouple for measuring T  , we have used two metallic crucibles of 1.5 cm 3 volume, allowing to detect T  values as small as ca. 0.01 °C.

We have carried out DTA studies of different charge compositions for synthesis of mixed iron gallium borate crystals. As an example, the DTA curve for the composition used to synthesize FeBO 3 is shown in Figure 2.2. As one can see, in the range of 835737ºC a flat exothermic peak is observed. A detailed analysis of this temperature range, see Figure 2.3, has revealed the existence of two separate exothermic peaks, above and below ca. 817ºC. The hightemperature, 835-817ºC, and the low-temperature, 817-737ºC, peaks correspond to the emergence of Fe 3 BO 6 and FeBO 3 phases, respectively.

The obtained DTA thermograms have proved highly efficient for studying phase transitions occurring in the course of crystal synthesis by solution in the melt route. In particular, they have allowed determining temperature ranges of forming Fe x Ga 1-x BO 3 crystals in the whole contents range, 0 1 x   .

Probe method

Certain shortcomings of the DTA, namely, high crystallization rates required and impossibility of in situ control of emerging phases, require a complementary analysis technique. With this aim, the probe method has been applied for the same compositions as DTA. As a probe, a platinum wire of 0.5 mm in diameter, fixed on the lifting device, has been used. The probe was immersed into the melt, then the temperature was modified following a distinct regime in accordance with a predetermined program, and finally, the probe was removed. The microcrystals formed on the probe have been studied by scanning electron microscopies, see Figure 2.4. In this way, the emergence of the same crystalline phases in the same temperature ranges as those found with DTA has been confirmed. Electron microscope pictures showing FeBO 3 single crystals on the probes for two different magnifications.

Charge compositions

Crystallizations were carried out in the Ga

Preparing a solution

After determining a charge composition, all reagents have been dehydrated in a drying chamber at 150 o C during 24 hours. Next, the reagents have been weighted with a high-precision balance and mixed with a laboratory-developed device. An example of charge prepared for synthesis is shown in Figure 2.5.

For obtaining a homogeneous solution melt, small portions, about 5 g, of the charge have been successively adjoined to a metallic crucible and each time kept for 20 min in a muffle furnace at 900 o C. As an example, Figure 2.6 shows a crucible with the solution melt extracted from the furnace. As one can see, the solution used for GaBO 3 crystal synthesis is almost transparent. Crucible with the solution melt used for synthesizing GaBO 3 crystals after extracting from the muffle furnace.

Crystallization

The crucible with the solution melt has been installed in the crystallization furnace with uniform temperature distribution. The crucible in the furnace before starting the crystallization process is shown in Figure 2.7. The crystallization setting consists of the following elements, see Figure 2.8:

(i) crystallization furnace;

(ii) lowering-rotating-lifting device;

(iii) seed holder;

(iv) temperature controller;

(v) managing thermocouple;

(vi) controlling thermocouple;

(vii) power unit;

(viii) computer. (ii) homogenization of the solution melt (at the beginning of this step, the seed holder is immersed in the solution melt and serves as a mixer);

(iii) sharp temperature dropping;

(iv) nucleation and crystal growth (using the seed holder during this step allows monitoring the emergence of FeBO 3 crystal phase; at the end of this step, the seed holder with the synthesized crystals is extracted from the solution melt);

(v) and (vi) cooling the furnace.

After the last step, a crucible has been removed from the furnace and boiled in 20 % solution of HNO 3 in order to dissolve the remnants of the solution melt, so that the crystals can be extracted.

°C (i) (ii) (iii) (iv) (v) (vi) Figure 2.9
Temperature mode of crystallization used in synthesizing Fe x Ga 1-x BO 3 crystals of a definite composition.

Synthesized crystals

The synthesized Fe x Ga 1-x BO 3 crystals are shown in Figures 2.10 to 2.12. All crystals have the shape of hexagonal plates with the dimensions of a few millimeters in the basal plane and ca. 0.1 mm in the perpendicular direction. Gallium borate is colourless while iron borate is green. The aspect of Fe x Ga 1-x BO 3 crystals is gradually modified with the iron contents.

Characterization of synthesized crystals 2.3.1 Crystal composition: X-ray fluorescence analysis

In order to determine exact contents of iron in the synthesized crystals, х crystal , we have carried out XRF analysis, a simple and accurate analytical method of determining the elemental composition of materials. The X-ray beam interacts with the atoms in the crystal in such a way that electrons are ejected from inner shells, leaving behind holes. Then the electrons from outer shells recombine with the holes, and the energy difference between the two states involved is released in the form of a photon. Since this energy difference is specific to each element, it is used to identify the latter. The concentration of a certain element is proportional to the intensity of the corresponding peak in the energy or wavelength spectrum. Figure 2.13 shows the results of XRF analysis; as one can see, х crystal , substantially differ from those in the charge, х charge ; besides, a considerable difference in iron contents occurs for different crystals extracted from the same crucible. For instance, for х charge = 0.01, х crystal is in the range from 0.002 to 0.04. Iron contents in Fe x Ga 1-x BO 3 crystals vs. initial iron contents in the charge. The vertical bars show the spread of iron contents for the samples extracted from the same crucible. The dashed line is a guide for the eyes.

2.3.2

Crystal structure: X-ray diffraction studies

The parameters of crystal lattice in the series of Fe x Ga 1-x BO 3 crystals have been determined by XRD with a RigakuSmartLab diffractometer using copper radiation CuKα in the angle range of 5 Isomorphous substitution of iron with gallium produces change of the lattice parameters manifesting itself in a shift of the XRD peaks from their positions in pure iron borate, see Figure 2.14. The hexagonal lattice parameters a and c can be calculated using the following expression [2.14]: The lattice parameters in Fe x Ga 1-x BO 3 crystals have been calculated using the positions of the most "convenient" peaks, e.g., (300) for a, and (006) and (0.0.12) for c. Next, these parameters have been refined by graphical extrapolation for all diffraction peaks in the angle range of 40 2 100     . The dependences of a and c on the crystal composition are shown in Figure 2.15. According to Vegard's law [2.15], suggesting a linear relationship between the lattice parameters of a solid solution and concentrations of its constituent components, the following relation should hold for Fe x Ga 1-x BO 3 crystals:

Fe Ga B O FeBO GaBO 1 3 3 3 Fe Ga B O FeBO GaBO 1 3 3 3 crystal crystal crystal crystal (1 ) , (1 )
.

x x x x a x a x a c x c x c         (2.
2)

The dashed lines on Figure 2

Crystal quality: EPR studies

For estimating the quality of the synthesized crystals, we have chosen the EPR characterization, particularly sensitive to imperfections of local structure. At low x values, only the EPR of diluted Fe 3+ in GaBO 3 lattice is observed; therefore, this ion can serve as a highsensitive probe for evaluating the degree of structural disorder in crystals. A single crystal with x = 0.003 has been studied by EPR with an X-band Bruker spectrometer at 4 K and magnetizing fields up to 1 T. A detailed account of this study will be given below, cf. Chapter 4, "EPR of irongallium borate single crystals with low x".

The experimental EPR spectrum, curve a in Figure 2.16, is typical of isolated Fe 3+ in oxygen environment. The spin Hamiltonian parameters have been determined by computer simulating the EPR spectra with a laboratory-made code. Besides, no characteristic line splitting has been detected, evidencing the absence of twinning in the crystals.

In order to numerically estimate the degree of local disorder, we have attributed different normal random site-to-site distributions to the Cartesian coordinates of the Fe 3+ ligands, and calculated the spin Hamiltonian parameter distributions by means of the superposition model, see details in Chapter 4, "EPR of iron-gallium borate single crystals with low x". Figure 2.17 shows computer simulations of a chosen EPR line, for different degrees of local disorder estimated from the width,  of the ligand coordinate distributions. Obviously, with increased disorder, the width of this line drastically increases and its amplitude decreases, so that the absorption intensity remains roughly the same. The best agreement between this experimental and computergenerated line profile is attained at 0.0003 0.0005    Å. Besides, Figure 2. 16 shows that with such parameter distributions, a good fitting, curve b, is achieved for the whole experimental spectrum.

Conclusions

We have succeeded in obtaining high-quality Fe x Ga 1-x BO 3 single crystals in the whole range of concentrations, using the solution in the melt technique for isomorphous substitution in iron-gallium borate system. By means of DTA and the probe method we have determined the component ratios in the charge and the corresponding temperature modes optimal for the crystal synthesis for different x.

The XRF analysis has revealed certain difference in Fe 3+ ion contents in different crystals extracted from the same crucible. The XRD analysis has shown that in mixed iron gallium borates the modification of the crystal lattice parameters closely follows the Vegard's law.

The EPR studies have allowed evaluating the degree of local disorder. Its quantitative characteristic -distribution width of the ligand coordinates, 0.0003 0.0005    Å, -confirms the high quality of the synthesised crystals.

Electron magnetic resonance

of iron-gallium borate single crystals with 0.2≤x≤1

Basic formalism

A classical description of AFMR has first been given by Kittel in 1951 [3.1].

In a two-sublattice antiferromagnet the equilibrium orientation of the sublattice magnetizations 1 2 and

M

M minimises the magnetic energy E of this system, i.e., at equilibrium

    1 2 eff 1 eff 2
and

M M H H E E . (3.1)
The equations of precession of

        1 1 eff 1 2 2 eff 2   H H M M M M (3.2)
where  is the gyromagnetic ratio defined as

1 2 ge m   
, e is the electron charge, m is the electron mass and g is the electronic g-factor.

In a magnetic resonance experiment, one applies perpendicularly to H an oscillating magnetic field h (the magnetic component of an electromagnetic wave, typically in the microwave range), of a frequency  and amplitude h H  . In most EMR spectrometers,  is kept constant and H is linearly swept, so that the resonance occurs when  is equal to the frequency of precession of magnetizations.

Considering weak precessions of

1 2
and M M about their equilibrium orientations and solving eqs. (3.2), magnetic modes of the antiferromagnetic substance in the vicinity of the ground state can be found.

The subsequent analysis has been made for rhombohedral antiferromagnets with weak ferromagnetism [3.3]; thus, it is applicable to FeBO 3 as well as to (isomorphous) mixed crystals of the Fe x Ga 1-x BO 3 series. For H contained in the basal plane of the crystals, the expressions of the low-frequency (LF) and high-frequency (HF) AFMR modes are, respectively [3.3-3.5]:

    1 2 2 D E hex 1 2 a E D D 36 cos 6 and 2 H H H H H H H H H H H                      (3.3)
(In what follows, we will consider only the LF AFMR mode.) Different parameters appearing in these equations are specified in Table 3.1, and  is the angle between H and the x-axis. The definitions of the constants given in Table 3.1 are given in Chapter 1, "Crystal and magnetic structure of iron borate".

Clearly, using the AFMR, one can determine these parameters and get information on magnetic properties of the crystals. In this chapter we describe the results of EMR studies of Fe x Ga 1-x BO 3 crystals for 0 1 x   and report in more detail those for x = 0.85; 0.75; 0.65; 0.34 and 0.2.

Experimental results

EMR studies of Fe x Ga 1-x BO 3 single crystals have been carried out with two different spectrometers:

(i) a laboratory-developed spectrometer operating in the frequency range from 7 to 37.35 GHz and recording absorption spectra † , and

(ii) a commercial high-sensitivity X-band (9.464 GHz) Bruker spectrometer recording derivative of absorption spectra.

In both cases, the crystals were studied in the temperature range from 4 to 310 K and magnetizing fields up to 10 kOe applied in the basal plane.

EMR at different iron contents and temperatures

Depending on the iron contents and the temperature, several types of EMR have been observed in Fe x Ga 1-x BO 3 crystals. Figure 3.1 shows the transformation of spectra shapes with x at room temperature. At x = 1 (pure iron borate) only a low-field resonance is observed. In the microwave frequency and temperature ranges used in this work, only this resonance line is observed for x = 1. The EMR in pure iron borate has been earlier identified as AFMR [3.4]. At somewhat lower iron contents, x = 0.75, besides the low-field line, a new broad resonance at higher magnetic fields emerges with an effective g-factor eff 2.0 g  . As far as iron substitution for gallium occurs more or less randomly, such crystals are expected to contain regions with different local iron concentrations, implying different magnetic ordering. The low-field line observed in the mixed crystals, by analogy with iron borate [3.4], can be identified as AFMR line arising from magnetically ordered regions. The high-field line can be ascribed to Cluster Magnetic Resonance (CMR), i.e., EMR arising from only partially magnetically ordered regions, or to superposition of CMR and EPR subsisting in the vicinity of the Néel temperature. At still lower iron contents, x = 0.2, the AFMR line disappears and the high-field line becomes more pronounced. For x = 0.04, the latter line disappears as well, and the EPR spectrum of diluted Fe 3+ ions, broadened by dipole-dipole interactions, comes into view. At still lower iron contents, x = 0.003 this spectrum is spectacularly narrowed. A detailed account of the EPR studies of this crystal will be given below, cf. Chapter 4, "EPR of iron-gallium borate single crystals with low x". As one can see, at T 304 K only the high-field line is present; this temperature can be tentatively identified as an effective Néel temperature for this crystal (this assignment will further be corroborated by SQUID results). Obviously, the nature of this line can be only paramagnetic. At lower T , the AFMR line splits off the high-field one and gradually shifts towards lower fields; simultaneously, the relative intensity of the high-field line decreases. (Above room temperature, a second weak low-field line is also visible for this crystal; the nature of this line is not yet obvious.)

In order to prove or disprove the existence of a high-field line below N T , we have carried out experiments with the high-sensitive Bruker spectrometer. Figure 3.3 shows the EMR spectra for x = 0.85 crystal at different temperatures in the magnetic field range corresponding to this line. Obviously, such a line is present in the whole temperature range; and below N T it can be identified as CMR.

Similar transformations of the EMR spectra in the vicinity of N T occur in x = 0.75, 0.65 and 0.34 crystals. and different temperatures shown alongside the curves. The high-field line is superposed with a narrow signal from a probe with 2.006 g  . For all crystals with 0.34 x  a splitting of the AFMR line in several more or less resolved components is observed. These components have different frequency dependences of their resonance fields (FDF). Figure 3.4 shows this line for x = 0.75 crystal at different frequencies; obviously, at higher frequencies this splitting becomes more pronounced. It can be caused either by a certain mesoscopic inhomogeneity of the crystals (the occurrence of regions with different local iron concentrations resulting in different magnetic properties) or by magneto-elastic effects due to random stresses [3.4]. In the case where such splitting was significant, we have defined the resonance field as the maximum of the most intense component. T , the resonance field, first, sharply decreases, showing a transition from paramagnetic to magnetically ordered state. Next, this decrease is slowed down, and below a certain temperature -ca. 80 K for x = 0.65 crystal and ca. 20 K for x = 0.34 crystal -the tendency is reversed, viz., the resonance shifts upfield. Figure 3.7 shows EMR spectra for x = 0.34 crystal at different temperatures. With lowering T from the Néel temperature (77 K in this crystal) to ca. 20 K, the AFMR line slightly shifts downfield. Meanwhile, a further cooling to 4 K results in a pronounced upfield shift and broadening of this line. In contrast, the high-field line decreases in intensity with lowering T; however, it again becomes obvious at 4 K. All these transformations suggest the occurrence of one more magnetic transition at low temperatures. -Fe O  [3.10, 3.11], similarly as in our case with decreasing iron content the supposed transition temperature decreases.

Thus, we assume that Fe x Ga 1-x BO 3 crystals with x = 0.34 and 0.65 undergo the Morin's transition at temperatures M T below 20 K and 80 K, respectively. Above M T crystals are easy- plane antiferromagnets with a weak in-plane ferromagnetism; the sublattice magnetizations lie in the basal plane and are slightly tilted (as occurs in FeBO 3 in the whole temperature range below N T , cf. Chapter 1, "Crystal and magnetic structure of iron borate"). Below M T the weak ferromagnetism vanishes and the crystals become pure easy-axis antiferromagnets, the sublattice magnetizations lying along the trigonal axis.

Obviously, in order to confirm this assumption, additional measurements, e.g. that of the dependence of the magnetization on the temperature, are required.

The EMR spectra for x = 0.2 crystal, see Figure 3.8, consist of a single line at 2 g  , quite similar to the high-field line observed for higher x; consequently, the antiferromagnetic regions in this case are absent in the whole temperature range. At lower T this line considerably broadens, and Figure 3.9 (left) shows that its intensity does not follow the 1 T  Curie law; therefore it cannot be a usual EPR line. We assume that this line is due to magnetic clusters.

As far as, except at very low x values, the EMR line at 2 g  is present in all crystals with 1 x  , cf. Figures 3.1, 3.3 and 3.7, we can conclude that in such crystals long-range and short- range (cluster-type) magnetic ordering coexist. Figure 3.9 (right) shows the temperature dependence of the CMR line intensity for x = 0.65, confirming that the Curie law for this resonance is not respected.

Relationship between antiferromagnetic resonance frequency and magnetizing field

Field-sweep EMR spectra recordings at different microwave frequencies allow obtaining the FDF. Figure 3.11 shows these dependences at different temperatures for x = 1 crystal. With decreasing T , the FDF for FeBO 3 shifts downfield. FDFs for crystals with 1 x  show somewhat different behaviour. Figure 3.12 and 3.13 illustrate these dependences at different temperatures for x = 0.85 and 0.34 crystals, respectively. With decreasing T , the FDF, first, shifts downfield and next, shows an opposite tendency. In a greater or lesser extent, this behaviour is characteristic of all crystals with 1 x  ; the most pronounced shift is observed for x = 0.34 crystal, see Figure 3.13.

Fitting the FDF with eq. (3.3) using the least-squares method, allows one to determine the D H , as well as the isotropic and anisotropic, E hex H H , energy gaps.

Dzyaloshinskii-Moriya field

Data on D H for different crystals at different temperatures are summarized in Table 3.2. , see Table 3.2. These findings corroborate the idea of the existence of the magnetic transition described above.

3.2.4

Isotropic energy gap

Anisotropic energy gap for iron borate

Using the values of D H from Table 3.2, we have determined hex E H H , see eq. (3.3), from the measurements at a fixed  carried out by rotating H in the basal plane of the crystal.

    2 D H H H     for x = 1
crystal at 77 K. One can see that this dependence can be accounted for by a superposition of hexagonal and uniaxial anisotropies in basal plane. The occurrence of uniaxial anisotropy in this case can be due to a slight deviation of H from the basal plane or caused by mechanical stresses [3.14]. Therefore, the angular dependence of the resonance field has been fitted by the following expression, cf.eq. (3.3):

  2 2 D E hex 36 cos 6 cos 2 H H H H H H p                 (3.

Conclusions

With decreasing x, the EMR spectra show a gradual passage from the low-frequency AFMR mode at x = 1 towards the EPR of diluted iron ions at 1 x  , going through a coexistence of AFMR and CMR arising, respectively, from magnetically ordered and partially disordered crystal regions for 0.34 0.85 x  

, and CMR only for 0.34 x  .

The Néel temperatures for crystals with .

The intensity of the high-field line observed in all crystals with 1 x  except at very low xvalues, does not follow the 1 T  Curie law, suggesting that this line is due to magnetic clusters.

The anisotropic energy gap, determined for x = 1 crystal at 77 K, has allowed to calculate the effective constant of hexagonal anisotropy; the latter will be further used in the analysis of basal magnetocrystalline anisotropy in Chapter 6, "Magnetocrystalline anisotropy of iron borate and iron-gallium borates". In condensed state, free atoms or ions form bonds (typically, ionic or covalent) and thus lose their magnetic moments; as a result, the substance becomes diamagnetic. Paramagnetism in condensed state exists due to some exceptions to this rule, where one or several unpaired electrons are maintained. These exceptions are: structure defects, impurity ions as well as ions with partially filled electron shells, i.e. transition elements and rare-earths; a generic term for all these entities is "paramagnetic species". Each representative of such species has an electronic spin S , either integer or half-integer, its projection along a quantification axis taking 2

L M L L L    
. Finally, in the case of the Russell-Saunders coupling, the resulting angular moment   J L S occurs, taking the values

 , 1, , J L S L S L S    
 , and each of these states is quantified by a corresponding quantum number

 , 1, , J M J J J     .
The magnetic moment associated with a given J is:

J g    J  (4.1)
where  is the Bohr magneton and J g is the Landé factor:

        1 1 1 1 1 2 J J J L L S S g J J         (4.2)
Note that in the case of a pure orbital moment and a pure spin, In a magnetizing field B ,  is quantified along the direction of B , so that the state with a given J splits into 2 1 J  substates called Zeeman sublevels:

J J J g g BM         J B B E  . (4.3)
This splitting in the case 5 2

J  is shown in Figure 4.1.

In EPR, magnetic dipole transitions between the Zeeman sublevels are aroused under the action of an electromagnetic wave. The resonance occurs when the electromagnetic energy quantum h matches the energy difference between the Zeeman sublevels:

J J g h B M       E (4.4)
In accordance with the selection rules, in the first approximation only transitions between adjacent sublevels are allowed, This occurs for the 5 d configuration where 0 L  as well as for other configurations, if L is "frozen". In these cases the g -factor is almost isotropic and its value is close to S g . If L is not completely frozen, in order to describe the spin state an effective spin S taking into account the orbital contribution is introduced.

x y z l l l         , (4.9)
 and  being the polar and azimuthal angles, respectively. and not on the field B. This means that B should be low enough for the splitting between the Zeeman sublevels to be much smaller than that between the spin multiplets of the ground and the first excited state of S.

H should comprise all terms consistent with symmetry, including time reversal symmetry.

Linear in S terms in eq. (4.7) can be expressed as    B S g where g is the g-matrix [4.2]:

1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 xx xy xz x x x yx yy yz y y y zx zy zz z z z g g g B B B g g g B B B g g g B B B                                                 g . (4.10)
Similarly, second-order fine structure terms in eq. (4.7) can be expressed as   S S

D

where the D-tensor

2 0 2 1 2 2 2 2 2 2 0 1 2 2 2 2 1 1 0 2 2 2 1 2 1 2 1 1 2 2 2 xx xy xz yx yy yz zx zy zz D D D B B B B D D D B B B B D D D B B B                                                    D (4.11)
is symmetric and traceless, consequently it can be diagonalized by appropriately turning the , , x y z frame:

0 0 0 0 0 0 x y z D D D                       D (4.12)
Thus, only two parameters are necessary to completely define D :

  3 1 2 2
and

z x y D D E D D    (4.13)
and D E are, respectively, axial and rhombic second-order fine structure constants, and the "rhombicity" parameter is defined as

3 x y z D D E D D     (4.14)
Usually, the , , x y z axes are chosen within the convention that

x y z D D D   [4.3], in which case 1 3 0    (4.15)
The lower and upper limits of  correspond to axial   0 E  and orthorhombic  

1 3 E D  symmetry, respectively.
For the fourth-order fine structure parameters, the following notation is sometimes used: a (cubic), F (axial), H and G (rhombic).

If the local symmetry is not lower the rhombic, in the , ,

x y z frame where D is diagonal, the g - matrix is also diagonal [4.2]:

0 0 0 0 0 0 x y z g g g                       g (4.16)
so that the , , x y z axes can be considered as local magnetic axes.

By diagonalizing the spin Hamiltonian matrix, the Zeeman sublevel energies i E (eigenvalues) and the corresponding wave functions i  (eigenvectors) are obtained.

The transitions between the Zeeman sublevels are of magnetic dipolar nature; therefore they are induced by coupling between the magnetic field 1 B of the electromagnetic wave of a frequency  and the magnetic dipole moment with components   , , ,

i i i i gS g S i x y z      (4.17)
In most commercial EPR spectrometers the two magnetic fields are orthogonal, ) and 1 B (explicitly). These orientations can be described as follows. (We assume for simplicity that the , , x y z frame coincides with the crystallographic frame c c c , , x y z .) In the laboratory frame l l l , , x y z , defined by the configuration of the microwave cavity and its disposition with respect to the poles of the electromagnet of the EPR spectrometer, we choose l z B  and 1 l y B  .

The relative orientations of the laboratory and crystallographic frames can be described by the following matrix [4.4]

                                                         A ( 4
                                        l l (4.21)
and in the crystallographic frame:

cl l 1 cl 1l
sin cos sin cos cos cos sin sin sin and sin cos sin cos cos cos sin sin

                                                               l A l l A l (4.22)
With this definition, the matrix element of the perturbation operator becomes:

      1 1 , , , , p q pq B               l S g (4.23)
In the usual field-sweep EPR spectroscopy, the resonance condition can be expressed as follows:

eff q p pq g B h      E E (4.24)
where eff g is the effective g -factor, pq B is the corresponding resonance field.

In reality, the resonance occurs within a certain field range about pq B , and its intensity is

F B B B                                   . ( 4

.26)

L F is applicable in the case of broadening caused by relaxation processes, whereas G F describes broadening due to static disorder.

Computer treatment of the EPR spectra

An important facet of the EPR studies is numerical analysis (computer simulation) of experimental spectra. Indeed, extracting meaningful physical information from the experimental results requires accurate spectra fitting. The fitting procedure, called spectra parameterization, consists in determining numerical values of the spin Hamiltonian parameters providing a complete description of the paramagnetic species. Such a description will only be satisfactory if the spin Hamiltonian includes all essential interactions contributing to the genesis of the EPR spectrum.

The computer-assisted treatment of the EPR spectra includes the following steps shown in General scheme of numerical analysis of experimental EPR spectra.

Experimental results

Fe x Ga 1-x BO 3 single crystals with 0.01 x  were studied by EPR with an X-band (9.464 GHz) spectrometer (Bruker) in the temperature range from 4 to 290 K and magnetizing field up to 1 T. For all crystal orientations with the exception of the basal plane, strong anisotropy of the resonance fields and pronounced angular dependence of the corresponding linewidths have been observed.

No temperature dependence of the resonance magnetic fields has been found, and the integral spectra intensities at different temperatures closely followed the 1 T  Curie law. Experimental EPR spectra of Fe x Ga 1-x BO 3 crystal with 0.003

x  at 4 K for different orientations of B : 11   , 270    (a); 37   , 300    (b); 66   , 307    (c); 96   , 311    (d); 135   , 318    (e); 164   , 339    (f) and 169   , 30    (g).

Discussion

Conventional spin Hamiltonian: spectra parametrization

In order to determine the EPR parameters, detailed computer simulations of the EPR spectra recorded at different orientations of B have been carried out using a laboratory- developed code based on the conventional spin Hamiltonian of trigonal symmetry, containing Zeeman as well as second-and fourth-order fine structure terms [4.5, 4.6]:

    0 0 3 3 2 1 1 2 4 4 4 3 180 9 sin 3 cos 3 g DO a F O a O O             B S H (4.27)
where the various symbols have been defined previously. In the actual case, This could be achieved by following the temperature dependences of relative intensities of different resonance lines identified with transitions between distinct Zeeman sublevels. Figure 4.7

shows computer-generated spectra corresponding to different temperatures, with a and F parameters set to zero in order to simplify the analysis.

The various resonance lines in Figure 4.7 are identified by the numbers of Zeeman sublevels involved in the corresponding transitions. This identification is straight forward with our simulation code, separating contributions of transitions between each pair of levels. Figure 4.8 shows the energy levels and possible EPR transitions for positive and negative signs of D ; expectedly, with changing the sign the order of levels in low and intermediate magnetizing field is inversed. As one can see, for positive D the transitions occur between lower, occupied, Zeeman sublevels, whereas for negative D the transitions can take place only between higher Zeeman sublevels, which are empty at low temperatures. in cm .

General spin Hamiltonian: superposition model analysis

As far as both spin Hamiltonian parameter sets, see Table 4.1, result in one and the same spin Hamiltonian matrix, they can be considered as equivalent at least, from the mathematical viewpoint. Meanwhile, they may not necessarily be so from the physical viewpoint.

In order to elucidate this issue, we have tested the compatibility of both sets of parameters with the predictions of the Newman superposition model [4.8]. For this purpose, we have put forward a simulation code based on the general spin Hamiltonian for Fe 3+ ; sin 3

; cos 3 B F a B a B a         ( 4
    0 / l t l j l j b r b r r   (4.32)
where the intrinsic parameters l b and the power exponents l t are considered as fitting parameters.

In applying eqs. (4.31) and (4.32), the iron ligand coordinates have been calculated with structural parameters for both FeBO 3 and GaBO 3 , reported by Diehl [4.10] and Seleznev [4.7], respectively. Meanwhile, as far as results of determination of l b and l t obtained in both cases have been very close to each other, we have chosen to give all these results for ligand coordinates averaged between the FeBO 3 and GaBO 3 structures. t . Once again, the set II a and F parameters can be satisfactorily accounted for with close Newman model parameters; on the other hand, no agreement can be found for the set I parameters. One can conclude that the set I fourth-order fine structure parameters are totally incompatible with the predictions of the superposition model. In contrast, the set II parameters can be quite consistently accounted for by this model. Thus, the spin Hamiltonian of eq. (4.27) is seen to be ambiguous; therefore, its use should be avoided. On the other hand, the spin Hamiltonian of eq. ( 4.28) has no such drawback.

Tesseral spin Hamiltonian: local disorder

We return to the discrepancy in relative line intensities in experimental and computergenerated spectra, see Figure 4.10. This effect can be due to local disorder present in the crystals and resulting in statistical site-to-site distributions of the spin Hamiltonian parameters. As a consequence, a more or less pronounced broadening and concomitant amplitude decrease are observed for the lines with a stronger or weaker dependence of their resonance fields on these parameters. From the viewpoint of the magnetic resonance spectroscopist (both electronic and nuclear, vide infra) different degree of local disorder gives rise to more or less broad distributions of relevant spectroscopic parameters. Therefore, in most cases, the degree of disorder can be defined with respect to the ratio distribution width/mean value for the most representative parameters. Typically, low local disorder occurs in high quality crystals possessing low concentrations of structure defects while high local disorder is observed in non-crystalline, in particular glassy solids [4.17] as well as in crystals with highly flawed structure [4.18, 4.19]. In what follows, we shall refer to low and high degree of disorder in accordance with the abovementioned criterion.

In order to provide a more quantitative estimate of line broadening in the experimental EPR spectra, we have assumed that all the ligand coordinates undergo random site-to-site distributions. As a result, the spin Hamiltonian parameters will also be distributed. However, the latter distribution cannot be analyzed in the framework of eq. ( 4.27) because of the fact that in this case the local symmetry is lower than trigonal one. Moreover, strictly speaking, neither can it be analyzed in the framework of eq. The spin Hamiltonian form needed can be adapted from eq. ( 4) in the paper by Tennant et al. [4.21], see the review article by Kliava and Berger [4.9]:

    1 1 0 e 1 2,4 1,3,5 1 , S S S S S l l l l l lm lm lm lm l l m l l m l l l g B                      S n S B T B H T (4.33)
Here the first and second terms in the right-hand side are, respectively, zero-field ( 0 The approach described above has the advantage of automatically taking into account the symmetry lowering brought by fluctuations of the ligand coordinates. We have carried out simulations of the EPR spectra using a laboratory-developed computer code taking into account distributions of the ligand coordinates supposed to be Gaussian with standard deviation  . The distributions have been generated using the Monte Carlo technique [4.22]; 10 5 computergenerated curves have been accumulated for each resulting spectrum. As one can see in Figure 4.15, this number is quite sufficient for producing smooth theoretical spectra; a considerable noise occurs only at the highest distribution width. Figure 4.15 clearly shows the effect of the site-to-site distributions of ligand coordinates: with increasing local disorder (the  value) the spectral features are broadened. Note that this broadening cannot be reproduced by a simple convolution with a certain linewidth, in which case, all features would broaden to more or less one and the same extent. In contrast, explicitly taking into account the parameter distribution, the broadening of the spectral features is determined by the dependence of the corresponding resonance field on the distributed spin Hamiltonian parameters. The latter approach yields a more realistic picture of the real physical situation; therefore, it much better accounts for amplitudes and widths of different resonance lines. In particular, curve b in Figure 4. 15 showing the spectrum generated with 0.0005   Å, represents the best fit to the experimental EPR spectrum of Fe 0.003 Ga 0.997 BO 3 at 4 K, vide supra, Figure 4.10 (curves a and c). We have used the following superposition model parameters, see eq. (4.32): More sophisticated models of local disorder are expected to further improve the quality of fitting to the experimental EPR spectra. In any case, detailed computer simulations of the experimental EPR spectra unambiguously demonstrate the presence of a certain degree of disorder in the environment of Fe 3+ in gallium borate crystals.

Conclusions

The experimental EPR spectra of Fe x Ga 1-x BO 3 single crystals are typical of Fe 3+ in the case where the Zeeman interaction is comparable with the second-order fine structure. The iron ions are located in trigonal symmetry sites and are surrounded by six oxygen atoms.

Detailed computer simulations of the EPR spectra have been carried out in order to determine the spin Hamiltonian parameters. First, we have put forward a simulation code based on the conventional spin Hamiltonian. It has allowed to obtain good fittings to the experimental spectra and to determine the sign of the fine structure parameter D . Meanwhile, we have found that two different best-fit parameter sets are possible, respectively with positive and negative values of the fourth-order fine structure parameter a , both yielding one and the same spin Hamiltonian matrix. This dichotomy has been resolved through testing the consistency of both sets with the predictions of the Newman superposition model. For this purpose we have used the general spin Hamiltonian for trigonal symmetry. The tests have unambiguously shown that only one best-fit parameter set was compatible with the Newman model. Therefore, we highly recommend using the general spin Hamiltonian instead of the conventional spin Hamiltonian.

Applying the best-fit parameter set II, see Table 4.1, we have obtained good fits of the resonance lines positions. Nevertheless, a certain discrepancy in the amplitudes of different lines between the experimental and computer-generated spectra was obvious. To elucidate this issue, we have assumed the existence of local disorder in the crystals. Using the general spin Hamiltonian expressed through the tesseral spherical tensor operators allowed us to establish relations between the distributions of coordinates of the oxygen ligands, on the one hand, and of the spin Hamiltonian parameters, on the other hand. Computer simulations carried out with a code based on this model have yielded high-quality fittings to the experimental spectra, thus confirming our assumption of presence of local disorder in the crystals.

The superposition model parameters for Fe 3+ ion determined by EPR will be further used in the analysis of the crystal field contribution to surface magnetocrystalline anisotropy, see Chapter 7, "Surface magnetocrystalline anisotropy of iron borate single crystals". In an applied magnetizing field B , n m has the energy

NMR studies of iron-gallium borates

n n n I I ђBm m g B          B E m .
(5.

3)

The splitting of the nuclear spin state into nuclear Zeeman sublevels, described by eq. ( 5.3), is shown in Figure 5.1. It allows observing NMR spectra due to transitions between different sublevels induced by the magnetic component of the electromagnetic radiation. The resonance occurs when electromagnetic energy quantum h matches the energy difference between the Zeeman sublevels:

n n I I ђB m g B m h         E . ( 5.4)
In usual NMR conditions, only transitions between adjacent sublevels are allowed, resulting in the selection rule 1 I m    ; hence, the NMR condition on the energy quanta is:

n n ђ g B h B        E . (5.5)
An assembly of non-interacting identical nuclei will resonate at the same frequency. Meanwhile, in the solid state, various a priori anisotropic interactions perturb the spin state energies and shift the resonance lines. Static disorder will therefore manifest itself in NMR spectra broadening. This makes the NMR spectroscopy a powerful research tool allowing to obtain detailed information on structure and physical properties of crystalline and non-crystalline, e.g., glassy materials. In particular, solid-state NMR of quadrupolar nuclei (with In applying the solid state NMR spectroscopy, the major difficulty is spectra broadening mainly caused by static disorder, anisotropic chemical shielding, vide infra, but also by dipole-dipole interactions between different magnetic moments. A number of methods have been put forward in order to minimize the latter. In particular, the dipole-dipole broadening can be effectively removed by magic angle spinning (MAS), first described by Andrew, Bradbury, and Eades [5.2] in 1958 and independently by Lowe [5.3] 1  , and the chemical shielding has anisotropic contributions with the same angular dependence, the former are averaged to zero, and the latter is averaged to its isotropic value.

Besides, in the solid-state NMR spectroscopy, it is essential to remove or, at least, minimize relaxation-induced line broadening. Two relaxation processes are effective: the spin-lattice (or longitudinal) relaxation and the spin-spin (or transverse relaxation) with characteristic time constants T 1 and T 2 , respectively. T 1 is a mean return time of an excited spin system to equilibrium with the thermal reservoir (solid lattice), and T 2 is a mean return time to equilibrium within the spin system itself. As a result of the difference in relaxation mechanisms involved, T 1 is usually longer than T 2 ; therefore, the spin-spin relaxation is the most important cause of broadening of the resonance.

In modern NMR spectroscopy, multiple pulse techniques are employed in order to reduce this broadening. In the experiment series described in this chapter, the Hahn two-pulse echo sequence has been used (i) All magnetic moments are in thermal equilibrium, aligned on the magnetizing field.

(ii) A 90 radiofrequency pulse turns the magnetization through the corresponding angle.

The magnetic moments now are out of equilibrium. Because of inhomogeneity of the local magnetic field, dephasing of moments occurs, leading to a decay of magnetization.

(iii) Next, a 180 pulse is applied; the magnetization is inversed, so that the magnetic moments are refocused (the inhomogeneous dephasing is removed).

(iv) A return to equilibrium of the magnetic moments produces a free induction decay (FID), or "echo", containing useful information on the system. A subsequent Fourier transformation of this signal yields a narrowed frequency-swept NMR spectrum. The MAS NMR of 11 B and 71 Ga has been used to study iron-doped gallium borates, Fe x Ga 1-x BO 3 (x = 0; 0.01; 0.02).

Spin Hamiltonian

The conditions of MAS NMR are determined by interactions between the nuclear spin and the magnetizing field (Zeeman coupling) as well as between the nuclear electric quadrupole moment and the electric field gradient (EFG) produced by the surrounding of the nucleus (quadrupole coupling) [5.6, 5.7]. The nuclear spin Hamiltonian can be expressed as follows [5.6]:

cs iso or z Q ђ ђ               I B I B I I V H H H H H (5.6)
where the first two terms account for the Zeeman coupling, the second term describing the chemical shift, isotropic in the MAS conditions, and the third term corresponds to the quadrupole coupling.

The EFG is proportional to a second rank tensor V with principal components , and

x y z V V V subjected to the restriction [5.6] 0 x y z V V V    .
(5.7)

Therefore, similarly to the case of the D -tensor in EPR, cf. Chapter 4, "EPR of iron-gallium borate single crystals with low x", to fully describe this tensor we need only two parameters: in the actual case, z V and The quadrupole coupling is usually small with respect to the Zeeman coupling; therefore, its contribution to the nuclear Zeeman sublevel energies can be deduced through a perturbation calculation. In the first order, it has an effect only on non-central NMR transitions    yields much narrower and therefore much more conspicuous NMR spectra. The MAS has no "magic" properties on the second-order quadrupole coupling, therefore the latter is not averaged. Thus, in the MAS experiments on powdered samples only the central transition is usually observed [5.7].

x y z V V V    . ( 5

Computer treatment of the MAS NMR spectra

Extracting meaningful physical information from the experimental MAS NMR spectra requires detailed computer fitting.

In order to simulate the experimental spectra, first of all, we need an expression of the resonance frequency r  . For the central transition, the expression of r  as a function of the spin Hamiltonian parameters and the polar angle  with respect to the C 3 axis and the azimuthal angle  with respect to the C 2 axis, is [5.9, 5.10]:

      2 2 r L iso L 4 2 3 1 1 1 6 4 3 5 1 1 cos 1 cos Q I I A B C                            (5.9)
where L  is the Larmor frequency,

  3 2 2 1 Q Q C I I    (5.
A B C                           (5.11)
It should be noted that erroneous expressions of A, B and C are quoted in a number of bibliographical sources, e.g., in the review paper by Freude [5.7].

Tentative simulations of the experimental spectra have allowed estimating chemical shifts and quadrupole parameters; however, relative amplitudes and widths of the spectra features could not be satisfactorily reproduced. Besides, our challenge was to account for the observed broadening of the experimental spectra with increasing iron concentration. Therefore, we have been bound to assume the existence of a certain local disorder in the environment of the paramagnetic nuclei.

Local disorder

In disordered solids, the spin Hamiltonian parameters are expected to be statistically distributed, cf. Chapter 4, "EPR of iron-gallium borate single crystals with low x". The existence of disorder in the environment of paramagnetic nuclei manifests itself in broadening of the NMR spectra. As a consequence, relative amplitudes and widths of spectra features can be satisfactorily reproduced in the simulations only if local disorder is explicitly taken into account in the simulation code, allowing for statistical distributions of the NMR parameters, in particular of the quadrupole parameters. This issue was first raised by Czjzek in the framework of a random-packing model of amorphous materials [5.11, 5.12]. As far as the quadrupole parameters are related to components of the EFG tensor, Czjzek et al. suggested a joint distribution density (JDD) of these parameters. Later, Le Caër et al. [5.13] have provided a more general justification of the Czjzek's JDD; indeed, they have shown that it holds if all components of the EFG tensor (in NMR) or of the quadrupole fine structure tensor (in EPR) tensor are subjected to normal (Gaussian) random distributions. The Czjzek's JDD has been extensively used in both NMR and EPR studies of amorphous materials [5. 14-5.16] and also applied to simulate the NMR spectra of some disordered crystals [5.17]. Yet, the major drawback of this JDD is to completely disregard symmetry and local structure persisting to a certain extent in only partially disordered crystals.

With the aim of describing a randomly distorted structure preserving, to a certain extent, local ordering, a more elaborated JDD of the quadrupole parameters has been suggested by Maurer [5.18] and Le Caër and Brand [5.13]. For brevity, we refer to this JDD as the Maurer's one, although Le Caër and Brand have provided a more detailed theoretical analysis of this model. Interestingly, a similar model has been put forward to describe the JDD of nanoparticle size and shape distribution [5.19].

Below we evaluate the applicability of the Czjzek's and Maurer's JDDs to NMR studies of only slightly disordered crystals.

Czjzek's distribution

In a disordered solid, all components of V are expected to be statistically distributed. In order to satisfy the requirements of diagonal symmetry and tracelessness, as well as of rotational invariance, these components are calculated as linear combinations of five normally distributed random quantities , 1, , 5 2 factor has been introduced in order that the subsequent formulae could be expressed in their habitual form [5.11-5.13, 5.18, 5.20].) Thus, we get:

                 1 5 4 2 4 1 5 3 2 3 1 1 3 1 3 2 3 3 U U U U U U U U U U U V .
(5.12)

Note that with this definition  represents the standard deviation of z V , the principle z-

component of V .
With these assumptions, the Czjzek's JDD takes the form [5.11, 5.13]:

      2 2 2 1 1 2 3 4 2 5 1 1 1 9 2 , 1 e z V z z V P V            (5.13)
with marginal distributions for z V   and 0

1    , respectively:       2 2 5 2 2 2 2 2 m 2 2 2 m 2 1 2 2 3 2 3 4 2 3 1 9 1 3 1 1 e 1 e ; 1 3 . 1 z z V V z z z V V P V P                                                  (5.14)
Alternatively, instead of and z V  the following parameters can be introduced:

1 3 1 ² z V     (5.15)
and

1 1 6 3 1 1 6 3 arctan for 0 arctan for 0 z z V V                     .
(5.16)

For the corresponding form of the Czjzek's JDD one gets [5.12]:

  (5.17)

with the marginal distributions, for 0     and 6

6

      , respectively:     2 2 m m 5 4 1 2 3 1 3 2 2 3 ; co e s P P            . ( 5.18)
The major drawback of the Czjzek's distribution is that it completely disregards symmetry and local structure existing in the perfect crystal. Not surprisingly, this distribution yields good simulation results when describing magnetic resonance spectra of amorphous and heavily disordered materials.

Numerical Maurer's distribution

In the Maurer's and Le Caër's et al. approach [5.13, 5.18, 5.20], the EFG tensor is represented as a sum of two tensors, 0 + V V where V is the random tensor defined above, and 0

V is a fixed traceless tensor describing a "perfect crystal" and characterized by parameters 0 0 and z V  . In the coordinate frame where 0

V is diagonal,     0 0 0 0 1 2 1 2 1 0 0 0 1 0 0 0 1 z V                           V (5.19)
where (Note that in the Maurer's approach all components of the V -tensor are assumed to be normally distributed with zero mean values and equal standard deviations [5.18]. However, such an assumption does not satisfy the above-mentioned requirements for this tensor.)

0 0 0 0 x y z V V V    . ( 5
We have put forward a simulation code implementing the above model. This code (i) generates the normal random quantities , 1, , 5 i U i   , vide supra;

(ii) using eqs. (5.12) and (5. 19) computes and diagonalizes the 0 + V V -tensor;

(iii) for each set of main values of the latter, respecting the above-mentioned convention The graph of  vs.  is shown in Figure 5.4 (left). For 0 0   (this corresponds to the Czjzek's JDD; obviously, in this case 0 0   as well) the increase of  with an increase of  is strictly linear, corresponding to the relationship obtained using the marginal distribution   m P  , see eq. (5.18):

Analytical Maurer's distribution

Besides, Maurer [5.18] has introduced an empirical analytical JDD by associating the marginal distribution of    

, derived from a non-central 2  distribution with five degrees of freedom, with a semi-heuristic marginal distribution of  and allowing for a certain correlation between both random variables. Here we prefer using   instead of  ; indeed, such a choice seems more appropriate for a bivariate JDD   , P   . After amending for a clerical error, this JDD becomes:

           2 2 0 0 0 0 1 2 2 2 1 2 1 ² cos 3 , e P g x                                            (5.25)
where

    2 0 2 1 1 and x g x x x e x          . ( 5

.26)

Of course, the use of an analytical JDD considerably simplifies the analysis of the experimental results; yet, the limits of its applicability should be carefully evaluated. With this aim in view we have examined relations between the "input" parameters 0 0 , and    occurring in eq. ( 5.25) and the corresponding "output" parameters cal cal cal 0 0 , and    , i.e., the characteristics of As one can see, for relatively low disorder, 0 30     , the "input" and "output" parameters are in good agreement; however, at higher disorder, i.e., already at 0 10     these parameters considerably differ from each other. In Figures 5.7 and 5.8 we show the results for 0 0.5   as far as in the low disorder case those for 0 0   and 0 0.5   are almost the same.

For higher disorder, the results for 0 0.5   are still very close to each other and for 0 0   the discrepancy between the "input" and "output" parameters becomes even more pronounced, but in any case we are outside the limits of applicability of the analytical Maurer's JDD.

One can conclude that the analytical Maurer's JDD satisfactorily accounts for local disorder only at low local disorder. The Maurer's JDD, eq. ( 5.25) has relatively simple form; meanwhile, its disadvantage is not to include "physical" parameters featuring in the spin Hamiltonian, eq. (5.6). Therefore, we have developed an alternative form of the Maurer's JDD in the variables z V and  which is much more complicate:

        2 2 0 2 2 1 1 0 0 3 3 2 2 1 2 2 2 2 0 1 1 0 0 3 3 3 2 arctan arctan 1 1 3 3 1 1 1 1 arctan arctan 3 3 2 2 1 9 2 1 3 1 1 ( ) , e V z z z z z V V V V z z g V P V                                                                                         (5.27) where 2 2 1 1 0 0 3 3 2 1 1 2 2 2 2 1 1 1 1 2 0 0 0 0 2 2 3 3 3 3 1 1 1 1 1 1 e V z z z V V z z z z V V g V V                                       and       2 2 2 2 2 1 1 0 0 0 3 0 3 0 0 2 2 2 1 0 3 2 1 0 3 0 0 2 1 0 3 2 2 2 2 2 1 1 0 0 0 3 0 3 0 0 2 2 1 3 1 3 1 1 9 6 1 1 3 1 1 1 9 6 z z z z z z z z z V z V z V V V z V z V V V V V V V V                                                          
(5.28)

Unfortunately, the marginal distributions of the Maurer's JDD cannot be analytically evaluated; they can be computed only numerically. Nevertheless, one can see from eq. (5.28) that the quantities The relation between the distribution widths and correlation coefficients featuring in the two forms of the Maurer's JDD can be found from the general formula relating the characteristics of different interrelated JDDs [5.21, pp. 64 ff]. We get:

        2 0 2 0 2 2 2 2 2 2 0 0 0 0 0 2 3 0 0 0 2 0 0 1 1 1 3 3 3 1 3 3 3 3 sin cos sin 2 cos z z z z z V V z V z V V V V                                                      (5.29)
In the actual case, the EPR studies described in Chapter 4, "EPR of iron-gallium borate single crystals with low x" show that Fe x Ga 1-x BO 3 crystals are sufficiently well-ordered. Let us compare the appropriateness of using the Czjzek's and Maurer's JDD for our crystals.

Comparison with experimental data

The spectra are computed by integrating the resonance absorption over distributed values of the spin Hamiltonian parameters and random orientations of crystallites:

      2 6 r 0 0 0 0 , , sin d d d d P F                       A .
(5.30)

Here   r , F     
is an intrinsic lineshape with a linewidth   . In the case of line broadening due to distributions of the spin Hamiltonian parameters, one gets [5.22]: Experimental MAS NMR spectrum of 11 B in Fe x Ga 1-x BO 3 with x = 0.01 (continuous, blue) and computer-generated spectra for randomly oriented crystallites (dashed, red) and for chosen orientations indicated (dotted, green). The Czjzek's JDD, as expected from eq. (5.18), regardless of the  value, always gives one and the same, extremely broad marginal distribution of  . Figure 5.11 illustrates this behaviour for two  values used to calculate the distributions shown in Figure 5.10. Obviously, such a distribution is incompatible with only slightly perturbed axial site symmetry in the crystal. On the other hand, as one can also see in Figure 5.11, the Maurer's JDD can produce quite narrow marginal distributions of  , describing weak random distortions from the perfect structure. Figure 5.12 shows three-dimensional JDD for the Czjzek's model, and Figure 5.13 shows computer simulated NMR spectra for both Czjzek's and Maurer's models. Clearly, the spectra calculated within the former model are unable to describe the experimental spectrum. In contrast, the Maurer's model provides quite satisfactory fits to the experiment using reasonable parameter values. One can conclude that the Czjzek's model should not be used to fit NMR spectra of materials with weak local disorder.   corresponding to axial local symmetry. At higher   , the JDD not only broadenes but also shifts to lower  , manifesting a certain rhombic distortion. The 0   case corresponds to "orthorhombic" distortion when the EFG tensor has two eigenvalues equal in absolute value and opposite in sign. The transformation of the JDD with  can be accounted for as a rotation about its vertical axis.

2 2 2 r r r r ² ² 2                                                    (5.31) where               L 2 2 2 3 2 2 3
                                                                                            (5.32) and               L 2 2 3 2 2 2 3
                                                         . ( 5

Experimental details

The 11 B MAS NMR spectra of Fe x Ga 1-x BO 3 ground to powders were measured at 128.384 MHz frequency in 9.4 T magnetic field using a Bruker Avance-400 NMR spectrometer. A 4 mm in diameter zirconia (ZrO 2 ) rotor cell equipped with boron nitride, BN stator filled with powdered sample was spinning under magic angle with 10 kHz frequency.

In the actual case, the FID signal from the BN stator was much more intense than the signal produced by the sample [5.23]. Therefore, a " 90 180 Acquisition       " Hahn two pulse echo sequence with echo delay  , as shown in Figure 5.3, has been used to record NMR signals from 11 B nuclei [5.4, 5.6, 5.7]. This sequence for  = 1 ms significantly reduced the signal from the stator and allowed clear detection of the sample signals.

For 11 B with Because of a large second-order quadrupole coupling, the 71 Ga ( 32

I  ) NMR lines were very broad, so that special acquisition conditions were required in order to obtain comprehensible spectra. The 71 Ga MAS NMR spectra were measured at 122.0564 MHz with 4 mm rotors at the spinning rate of 14 kHz. These spectra were recorded by classical direct acquisitions under single pulse FID excitations with an acquisition of 1000 pulse signals repeated with 2.4 μs RF pulse length. The NMR spectra were obtained by Fourier transformation of the FID signals.

The spin-lattice relaxation time T 1 for 71 Ga nuclei in Fe x Ga 1-x BO 3 was measured by saturation-recovery method. For samples with x = 0.01 and 0.02, T 1 was 23 and 2 ms, respectively. One can see that T 1 drastically decreased with an increase in iron concentration. Because of a spurious 11 B NMR signal from the BN stator, similar measurements for 11 B could not be carried out. .17 shows 11 B MAS NMR spectra for Fe x Ga 1-x BO 3 powders with x = 0, 0.01 and 0.02 together with the best-fit computer simulations. The spectra display a characteristic twopeak MAS quadrupole powder pattern for a single boron site [5.8, 5.24]. Clearly, an increase of iron concentration results in broadening of the spectra. 5.1

Experimental results and discussion

C Q and  are related to  and  as:

  6 cos Q z C V       and   6 3 tan      .
The values of Q C and iso  extracted from the simulations allow to determine the coordination of boron [5.7, 5.24

MAS NMR spectra of 71 Ga

Figure 5.18 shows the MAS NMR spectrum of the central transition of 71 Ga nuclei for Fe x Ga 1-x BO 3 powders with different iron contents. The behaviour of the spectra is qualitatively similar to that described above for 11 B isotope. In particular, the two-peak quadrupole powder pattern characteristic of axially symmetric EFG tensor is clearly seen. The simulation procedure for 71 Ga NMR spectra has been similar to that for 11 B; meanwhile, in order to obtain closer fittings, we had to take into account a Gaussian distribution of iso  with a standard deviation   . Nevertheless, the best-fit computer-generated spectra displayed in Figure 5.18 show a certain discrepancy with the experimental ones. This discrepancy can be ascribed to an underlying signal arising from 71 Ga in heavily disordered environment or to a superposed contribution from the non-central NMR transitions.

The best-fit simulation parameters for 71 Ga NMR spectra given in Table 5.2 are consistent with sixfold-coordinated gallium [5.8].

Conclusions

The MAS NMR spectroscopy of the 11 B and 71 Ga allows to determine the local symmetry and to identify surrounding structural units for these atoms in Fe x Ga 1-x BO 3 crystals. The best-fit parameters obtained for 11 B and 71 Ga confirm the threefold coordination and C 3 intrinsic symmetry for the former and the sixfold coordination with lower-than-cubic symmetry for the latter. For both nuclei, broadening of the MAS NMR spectra with the increase in iron contents has been related to variations of quadrupole parameters and chemical shift (in the case of 71 Ga) caused by disorder in the local environment. This assumption is corroborated with accurate computer simulations of Fe 3+ EPR spectra, showing the existence of local disorder in iron-doped gallium borate crystals.

A comparative analysis of the characteristics of the Czjzek's and Maurer's JDD has been carried out in order to elucidate the suitability of using them in the case of low or moderate local disorder. Moreover, for the Maurer's model we have considered in detail both the numerical and the analytical JDD. For the former case we have put forward a simulation code in order to obtain dependences of the mean values ,   , standard deviations ,

    of  and  ,
respectively, and a correlation coefficient  on the input value of  for different 0  and 0 . 

The obtained relationships have allowed to determine the limits of applicability of the Maurer's analytical JDD. The Czjzek's JDD is well adapted to the case of heavily disordered solids; meanwhile it does not provide for local ordering in crystals, partially preserved in the presence of a certain degree of disorder. Indeed, it does not include mean values of the quadrupole parameters; besides, it contains only one parameter describing the distribution widths. As a result, the marginal distribution of the parameter  related to the asymmetry parameter  becomes too broad to be incompatible with the existence of short range ordering. In contrast, the Maurer's JDD has no these drawbacks, therefore it is expected to provide satisfactory fits to experimental NMR spectra in crystals with low degree of local disorder.

The above considerations are fully corroborated by applying the Czjzek's and Maurer's models to computer simulations of NMR spectra of 11 B isotope in gallium borate. With the former distribution no adequate description can be obtained while the latter one provides quite satisfactory fits. The present study shows that the Czjzek's model should not be used in computer-assisted analysis of NMR spectra of materials with low or even moderate local disorder. On the other hand, the Maurer's model is conceptually well adapted to this situation; therefore it is not surprising that it provides quite adequate fits to the experimental NMR spectra.

Magnetocrystalline anisotropy of iron borate and iron-gallium borates 6.1 Introduction

The anisotropic part of the density of the magnetic energy for FeBO 3 , can be expressed as follows, cf. eq. (1.19) in Chapter 1, "Crystal and magnetic structure of iron borate":

  eff FeBO FeBO 3 3 2 3 6 A 1 6 3cos 1 sin cos sin 3 sin cos6 a d e           E (6.1)
where eff a is the effective uniaxial anisotropy constant, As far as for Fe 3+ ( 5 3d electronic configuration) the orbital moment equals zero, the exchange energy in a good approximation is isotropic [6.1], (cf. Chapter 1, "Crystal and magnetic structure of iron borate") so that for FeBO 3 the constants FeBO FeBO The crystal field contributions to these constants have been reported by Seleznev [6.2]; here, for clarity, we give a more comprehensive analysis. These contributions can be calculated in perturbation theory using the spin Hamiltonian for isolated Fe 3+ ions in a diamagnetic crystal isomorphous to iron borate, e.g. gallium borate [6.3]. Thus, cf cf cf , and a d e can be expressed through the spin Hamiltonian parameters.

The dipole-dipole contributions to these constants are usually calculated using the lattice-sum method. The value of Earlier, the occurrence of the dipole-dipole contribution to hexagonal basal anisotropy in iron borate had been ruled out on the grounds of symmetry. Indeed, the dipole-dipole interaction energy is usually considered for "point dipoles" having a negligible size, in which case only uniaxial anisotropy is accounted for. Meanwhile, taking into account higher-order terms in the expansion of the dipole-dipole interaction energy in the Taylor series in the small parameter dipole size/interdipole distance would provide the possibility of describing more sophisticated symmetries, in particular, the hexagonal magnetocrystalline anisotropy. Clearly, such terms can be significant only in the case of "extended dipoles" having non-negligible dipole size. Thus, taking into consideration extended dipoles opens the possibility to reasonably account for the dipoledipole contribution to the basal anisotropy constants and, subsequently, to estimate effective dipole dimensions in iron borate.

With this aim in mind, we have developed a theoretical description of the following models of an extended magnetic dipole: (i) a uniformly magnetized sphere, (ii) an Ampérian current, i.e. a circular current loop of a radius R delimiting an area 2 S R   and (iii) an assembly of two fictitious "magnetic charges" q  a distance d apart.

In the present chapter, in the framework of these models we describe the evaluation of the dipole sizes in FeBO 3 as well as of dipole-dipole contributions to the magnetocrystalline anisotropy constants of Fe x Ga 1-x BO 3 .

Crystal field contribution to magnetocrystalline anisotropy

In order to calculate the crystal field contribution to the magnetic energy, we consider a Fe 3+ ion in a diamagnetic GaBO 3 crystal. The conventional spin Hamiltonian in this case is [6.6, 6.7], cf. eq. (4.27) in Chapter 4, "EPR of iron-gallium borate single crystals with low x":

    0 0 3 3 2 1 1 2 4 4 4 3 180 9 sin 3 cos 3 g DO a F O a O O             B S H (6.4)
where g is close to the free electron g -factor e 2 g  ,  is the Bohr magneton, B is the magnetizing field, S is the electron spin of Fe 3+ , 5 2

S  , D is the second-order axial fine- structure constant, and a F are, respectively, the fourth-order cubic and axial fine-structure constants and 0 0 3 As far as Fe 3+ ions in FeBO 3 are in a strong exchange field, in order to obtain the spin Hamiltonian for a j th Fe 3+ ion in FeBO 3 , 1, 2 j  numbering two non-equivalent iron sites, we substitute an effective exchange field E j B for the magnetizing field B . We get:

    0 0 3 3 2 4 2 1 1 E 3 1 0 9 4 8 4 sin 3 cos 3 j j j j j j g DO F O O O a a             B S H . (6.5)
As far as in this equation the "Zeeman" term, 0

E j j g   B S H
, is much larger than the remaining terms

    2 1 1 3 180 9 0 0 3 3 2 4 4 4
sin 3 cos 3

j j j j O O O D a F a O           H
, we have calculated the energies of the spin levels using the perturbation theory. With this aim in view, we have followed the approach of "correct" zero-order wave functions, described by Landau and Lifshitz [6.9]. The zero-order energies and wave functions have been calculated by solving the secular equation

  0 0 0   H 1 E (6.6)
where 0 H is the matrix of 0 H and 1 is a unit 6 6

 matrix. The eigenvalues of eq. (6.6) are:

      0 0 0 5 3 1 2 2 2 5 3 1 2 2 2 ; ; g B g B g B             E E E . ( 6.7)
The "correct" zero-order wave functions of the spin levels,   0 M  , are expressed as linear combinations of the electron spin eigenfunctions M  where

, , , M     . The latter relationships can be written in the form of scalar products

    0 0 M M M    C  (6.8)
where

                5 3 1 1 3 5 2 2 2 2 2 2 0 0 0 0 0 0 0 M C C C C C C     C and   5 3 1 1 3 5 2 2 2 2 2 2 M            . The vectors   0 M
C can be found from the equation:

      0 0 0 M M    C H 1 E 0 (6.9)
where 0 is a column 6-vector. Solving the latter equation, we get

  M 0
C for each spin level:

             
i i i i i                 C                                                                           ,                     3 2 3 2 1 2 1 2
i i i i i                     C                                                                           ,                           1 2 1 2 1 2 1 2 1 2
i i i i i                                                                                       C                 . ( 6.10)
The first-and second-order corrections to the energies of the spin levels are, respectively [6.9]:

                2 0 0 1 0 0 2 0 0 and p q p pp p p p q p p q              H E E H E E H (6.11)
where in our case and p q take the values

5 3 5 2 2 2
, , , M    . Substituting in these equations

    0 0 p M    with   M 0
C given in eqs. (6.10), we get the energies of the spin levels up to the second order in D and up to the first order in and a F :

            1 2 2 21 2 22 2 2 4 2 1 2 E 2 2 2 4 3 1 1 34 18 1 5 2 1 2 1 3 25 6 1 3 1 5 30 M j j j j j j M M M M M S S S g S S h h h S S S S S M g B g B M M M h S S                                            B E (6.12) where         1 2
j j j j j j j j j j j j j h D D D a a h h h F                           (6.13)
In the strong exchange case, the spins of the two non-equivalent iron ions are antiparallel:

1 1 2 2 , , , and .
               
In our case, the third and fourth terms on the right-hand side of eq. (6.12), i.e. the secondorder corrections in D can be neglected; indeed, for E 300

        cf cf f 5 2 c 5 4 6 5 c 0 K 2 0 K 0 K 0 os 3 a d N D a F e Na             (6.14)
where N is the number of Fe 3+ ions per unit volume (for FeBO 3 28 -3

10 m N 

).

In order to calculate these contributions at elevated temperatures, we express the partition function of a magnetic sublattice of FeBO 3 using eq. (6.12):

e M j E kT j M Z    (6.15)
where k is the Boltzmann constant and T is the absolute temperature, and calculate the crystal field contribution to the density of the magnetocrystalline anisotropy energy for FeBO 3 as follows [6.9]:

cf 1 2 ln j j NkT Z    E (6.16)
Taking into account that the first term on the right-hand side of eq. (6.12) is much larger than the remaining ones, we expand cf E in a Taylor series to first order in the small parameters

E 1 2 j j h g M  B and E 1 4 j j h g M  B :         E 2 2 1 2 cf 2 2 4 2 1 4 1 2 1 3 1 ln e 1 2 3 3 5 5 6 3 1 0 1 j g j B M k M j T j S S h NkT S S S S S M M M M S h                                                            E . (6.17)
Summing over M we get

    1 2 cf 1 4 5 3 1 2 2 2 1 1 2 2 5 3 1 2 2 2 sh 3 4 5ch ch 4 ch sh ln 120 ch 3ch 2 ch j j h NkT h                              E (6.18) where E g B kT   
. Expanding the latter expression in a bivariate Taylor series in 1 2 j h and 1 4 j h and retaining only anisotropic terms results in

    1 1 2 f 4 c 1 2 2 24 j j j N h t h r              E (6.19)
where

        3 5 1 1 2 2 2 2 3 5 1 1 2 2 2 2
8 ch 2 ch 10 ch sh sh 3 and

2 ch 3ch ch sh 5 . sh 3 t r                     (6.20) Summing eq. (6.19) over two non-equivalent Fe 3+ sites   1, 2 j 
, we get the anisotropic part of the crystal field contribution to the density of the magnetocrystalline anisotropy energy for FeBO 3 :

            2 4 f 3 2 c 7 1 2 12
3 cos cos cos 3 sin cos sin 3

N D t a F r N a F r Na r                     E (6.21)
Comparing matching symmetry terms in eqs. (6.21) and (6.1) (the Dzyaloshinskii-Moriya term should not be considered in this comparison), we obtain the crystal field contributions to the magnetocrystalline anisotropy constants of FeBO 3 as a function of  :

        cf cf cf 2 1 2 3 cos 0 3 2 a d N D t a F r Na r e               (6.22) At 0 K T 
this equation reduces to eq. (6.14). Thus, with the spin Hamiltonian parameters

Dipole-dipole contribution to magnetocrystalline anisotropy 6.3.1 Models of extended dipoles

In electrostatics, the primary source of the electric field is an electric charge. In contrast, in magnetostatics, insofar as "magnetic charges" -magnetic monopoles -have not been found in nature, the same fundamental role of primary source of the magnetic field is played by the magnetic dipole. Therefore, adequate modelling of the magnetic dipole is of paramount importance in scientific research.

As far as a point dipole is only an abstract idea, it is useful to consider dipole models -more realistic physical systems yielding the same magnetic field as the point dipole, at least, at distances much larger than their own size. Most often, as such a model in magnetostatics one considers a circular current loop or, by analogy with electrostatics, a pair of fictitious magnetic charges of opposite sign. Meanwhile, here we also consider the magnetic dipole as a uniformly magnetized three-dimensional body of a simple, e.g., spherical shape. Usually, the uniformly magnetized sphere is considered in a different context, viz., as an illustration of a boundary-value problem in magnetostatics [6.11, p. 198 ff], or an example of application of the vector potential [6.12, p. 236], thus overlooking the opportunity of using it as one more model of the magnetic dipole. Below we shall compare in detail all three models of the magnetic dipole: Of course, the magnetic field produced by a dipole model, as well as the dipole-dipole interaction energy at intermediate and shorter distances will differ from that of the point dipole; moreover, the predictions of different models can be quite different. This issue is of importance, e.g., in studying magnetic dipole-dipole interactions between paramagnetic ions in solid state, in which case a comparison with experimental observations allows choosing the most adequate description of a given magnetic source.

In one form or another, the models considered below have been described in a number of textbooks and/or research papers. Meanwhile, we have systematically considered them within the same formalism and have compared exact analytical expressions with Taylor expansions to higher-than-first order, providing simple expressions of potentials, fields and dipole-dipole interaction energy valid not only at large but also at intermediate distances. A computer code has been put forward, allowing to visualize magnetic field lines computed using both exact expressions and Taylor expansions.

Point dipole: an overview

According to the Biot-Savart law of magnetostatics, the magnetic field   B r produced in a point of space  

, , x y z  r by an arbitrary distribution of steady currents in a volume V  is (e.g., see ref. [6.11, pp. 175 ff; 6.12, pp. 215 ff]):

      0 3 d 4 V V             j r r r B r r r . ( 6.24)
The same expression is obtained with the help of the relation

       B r
A r ,   A r being the corresponding vector potential:

    0 d 4 V V          j r A r r r . ( 6.25)
In eqs. (6.24) and (6.25) 0  is the permeability of vacuum,    j r is the current density in a point

  , , x y z      r
of the magnetic source, and the integration is performed over the whole distribution of currents. The corresponding analysis, outlined below, can be found, e.g., in Jacksons' and Landau and Lifshitz's textbooks [6.11, pp. 184 ff, 6.13, pp. 103 ff].

Equation (6.25) can be expanded in powers of 1 r  (the multipole expansion):

      0 1 0 1 cos d 4 n n n n V r P V r              A r j r (6.26)
where n P are the Legendre polynomials and  is the angle between r and  r . The first term of this development (

0 n  ),   0 m d 4 V V r        j r A , (6.27)
is the vector potential of the magnetic monopole, m A , and it is shown to vanish. The second term in eq. (6.26), 1 n  , the magnetic dipole term,

    0 d 3 d 4 V V r          r A r r j , (6.28)
can be put in the following form:

0 d 4 1 r       A m (6.29)
where m is the magnetic dipole moment:

  1 2 d V V        r j r m . (6.30)
Taking the curl of d A , applying the product rule and keeping in mind that m is a fixed vector, for the magnetic field produced by a point dipole one gets:

  0 d 5 4 3 ² r r      r r B m m . (6.31)
According to the Curie symmetry principle [6.14], the effects generated by a cause can have only higher and not lower symmetry than the cause itself. We put the dipole at the origin O (in subsequent sections, O will be chosen in the centre of the dipole model). As far as the dipole field is invariant with respect to rotation about its axis denoted as Oz, the use of cylindrical coordinates , , z   and the corresponding unit vectors , , z

  e e e is the most appropriate, and all calculations can be restricted to a plane containing Oz. Meanwhile the results of calculations often are simpler in spherical coordinates , ,

r  
e e e . Therefore, we use both coordinate systems; however, for the sake of uniformity, vector components in the cylindrical system are expressed in the spherical variables by substituting sin r    and cos z r   , the polar angle of the position vector r of the point of observation is denoted as .

Thus, eq. ( 6.29) in both cylindrical and spherical coordinates becomes:

d 2 0 sin 4 r      A e m . ( 6.32)
The magnetic field becomes in cylindrical coordinates:

  3 2 0 d 3 2 sin 2 3 c s 4 o 1 z r               e e B m . (6.33)
and in spherical coordinates:

  0 d 3 cos sin . 2 4 r r        e e B m (6.34)
The dipole-dipole interaction energy can be obtained as the energy of a dipole n°2 placed in the magnetic field produced by a dipole n°1 (placed in the space origin):

   1 2 1 2 1 2 2 1 5 0 4 3 ² r r             r r B E m m m m m . ( 6.35)
In the case of interaction between identical parallel or antiparallel dipoles, this expression reduces to

    2 2 2 2 2 5 0 0 3 3 3cos 1 4 4 r r r             r E m m m (6.36)
where the -and + signs correspond to parallel and antiparallel dipoles, respectively, and  is the angle between r and the common dipole axis.

Uniformly magnetized sphere

The magnetic field produced by a (uniformly) magnetized sphere, see Figure 6.3, has been considered in detail in the literature, e.g., see [6.11, p. 198 ff]. A related model, that of a spinning spherical shell carrying a uniform surface charge, has been treated in the Griffith's textbook see [6.12, p. 236 ff].

Figure 6.3

Magnetic dipole as a uniformly magnetized sphere.

The magnetic moment of a uniformly magnetized sphere of radius R and volume V can be calculated as

3 4 3 V R    M M m (6.37)
where M is the magnetization vector.

For a distribution of dipoles, A is obtained from eq. (6.25) as the second term of the multipole expansion in Legendre polynomials [6.12, p. 242 ff]. (The first term in this expansion vanishes.) One gets:

        0 0 3 1 d d 4 4 V V V V                         M r r r A r
M r r r r r (6.38) where   operates over the coordinates of  r . Applying the identity

      1 1 r r                     M r M r M r r r r r (6.39)
and taking into account that

  0      M r
for a uniform M , we rewrite eq. ( 6.38) as follows:

    0 d 4 V V              M r A r r r . (6.40)
According to a well-known theorem of vector analysis, the volume integral in this expression can be transformed to an integral over the closed surface S  delimiting the volume V  :

  0 d 4 S S          M n A r r r  (6.41) where r    r n
is the unit vector normal to S,

2 d sin d d S R        
and the integration is performed over the whole surface of the sphere. A comparison between eqs. (6.41) and (6.25) shows that the latter expression represents the vector potential of a surface current of density

  S    j r M n . (6.42)
We choose the space origin in the centre of the sphere and the z axis parallel to r ,

 

sin cos sin sin cos

x y z R               r e e e and
 

1 2 2 2 2 cos r R rR        r r
. Because of the rotational symmetry about M , without loss of generality M can be confined in the xz-plane, forming an angle  with r ,   sin cos

x z M     M e e ,
. M                       M n (6.43)
Making all these substitutions in eq. ( 6.41), we notice that integrating over the range 0 2      will eliminate contributions of all  -dependent terms in eq. (6.43). The remaining integral over   is: where the Cartesian y-axis is identified as a radial cylindrical  -axis, and the magnetic dipole moment is introduced through eq. (6.37). Equation (6.44) is equivalent to eq. ( 5.111) in Jackson's textbook [6.11]. Taking the curl of A inside and outside the magnetized sphere, we get the corresponding magnetic fields int B and ext B in cylindrical coordinates:

  0 2 0 1 2 2 2 0 2 1 3 1 1 2 sin cos d 2 cos sin , , cos 4 sin , 4 r MR r R Rr r r R r R R                                 
  One can see that inside the uniformly magnetized sphere the magnetic field is uniform, as known from magnetostatics. Most interestingly, outside this sphere, the magnetic field at any distance coincides with that of the point dipole, cf. eqs. (6.45) and (6.46) with eqs. (6.33) and (6.34), respectively.

Obviously, in this model the interaction energy between two identical mutually nonpenetrating dipoles is the same as that between two point dipoles, cf. eq. ( 6.36).

Circular current loop

Most often, in magnetostatics one takes a circular loop of electric current (Ampérian current) as a basic model of the magnetic dipole. Let us consider a loop of radius R and area In performing the integration of dA given in eq. ( 6

r R r A A R R                      e e A
  0 2 8 4 R k k r        A e m (6.49)
where

    1 2 1 ² K( ) E( ) k k k k     , (6.50)
and K( ) and E( ) k k are, respectively, complete elliptic integrals of the first and the second kind.

Equation (6.49) coincides with eq. ( 5.37) in Jackson's textbook [6.11]. From this equation we derive the magnetic field in cylindrical coordinates:

            2 2 2 0 2 2 2 2 2 0 2 2 2 cot E K , 4 2 E K 4 z B r R k k R B r r r r r r R r k k R                                 m m (6.51)
and in spherical coordinates:

        0 2 2 2 2 0 2 2 4 cos E , 4 2 c sin os 2 E K . 4 r r r r r B k k R r B r R k                          m m (6.52)
Next, we calculate the interaction energy between two identical and parallel Ampérian currents of the same radii R , see Figure 6.5. In order to obtain the correct sign of the dipole-dipole interaction energy E , we express its relation to the mutual inductance M of two equal currents flowing in the same direction in two parallel coaxial loops, as follows:

² I   E M . (6.53)
Indeed, the potential energy of attracting currents should be negative, and it is more convenient to define M as positive.

Using the expression of M given by Grover [6.15] and Akyel et al. [6.16], in the actual case we get:

  2 0 2 3 3 2 0 8 1 sin cos d 4 r R R v                 E m (6.54)
where the -and + signs correspond to parallel and antiparallel dipoles, respectively,  has been defined in eq. ( 6.50) and

  2 2 2 2 2 4 1 cos v r v R      (6.55) with 1 2 2 2 2 sin 2 sin cos 1 r r v R R                   . ( 6
        2 2 2 2 4 2 1 1 2 2 2 1 !! 1•3 1 2•4 2 !! K n n n                                                   (6.57) and         2 2 2 4 2 2 1 1 2 2 2 1 !! 1•3 1 2•4 3 2 !! E 1 2 1 n n n n                                                    . ( 6
                                    . (6.59)
In the model of a circular current loop, the vector potential, the dipole magnetic field and the dipole-dipole interaction energy can be analytically expressed only through the elliptical integrals, see eqs. (6.49), (6.51), (6.52) and (6.54). In order to obtain more simple expressions valid at intermediate distances, we have used expansions in Taylor series to the sixth order in the small parameter R r   . Expanding eq. ( 6.49), we get:

 

                   m m (6.62)
Here for brevity we are using the Legendre polynomials

P                                          2
P                                            2
P                 E m (6.65)
where the -and + signs correspond to parallel and antiparallel dipoles, respectively.

Pair of fictitious magnetic charges

The third model represents the magnetic dipole as an assembly of two fictitious magnetic monopoles, or "magnetic charges" q  , a distance d apart, see Figure 6.6. By analogy with electrostatics, the magnetic dipole moment is defined as q  d m , so that in order to calculate the magnetic field produced by such a dipole, one is tempted to introduce a scalar magnetic potential, e.g., see [6.11, p. 196 ff]. However, for the sake of consistency, we prefer using here a vector potential, and, in accordance with the superposition principle, it can be taken as a sum of the vector potentials of two magnetic monopoles of opposite signs. yields a correct expression of the magnetic field expected to be produced by a magnetic monopole. However, it is not quite satisfactory from both mathematical and physical standpoints, as far as it exhibits a singularity along the half-line    (the so-called Dirac string); while for a magnetic monopole the direction of this half-line is completely arbitrary. Meanwhile, it can be readily shown that the vector potential of a pair of magnetic monopoles of opposite signs:

p m m 0 p p m 1 cos 1 cos sin si 4 n d r r                         A e m (6.67)
has no more such singularities. The latter equation can be rewritten as:

0 m p 1 1 2 2 cos cos s 4 in r d r d r d r r                       A e m (6.68)
where the connotation of different symbols is shown in Figure 6.6. Obviously, the following relations hold: .

For the magnetic field we get: in cylindrical coordinates,

                                      m m (6.69)
and in spherical coordinates,

                                       m m (6.70)
For the interaction energy between two parallel/antiparallel dipoles, we get: System of two interacting magnetic-charge dipoles.

2 0 2 m p p m 2 1 1 4 r r r d                    E m (6.
The expressions for the field and the energy in the model of a pair of magnetic charges are simpler in comparison with those obtained with the model of a circular current loop. Nevertheless, we still have provided the corresponding expansions in the Taylor series, useful for a direct comparison between these two models. In the same approximation as in the previous section, redefining the small parameter as d r   , eqs. (6.67) to (6.70) become:

 

                   m m (6.74)
An analogous expansion for E yields:

P P P                 E m (6.75)
where the -and + signs correspond to parallel and antiparallel dipoles, respectively. A still better insight in the behaviour of different models at small and intermediate distances can be achieved by visualizing magnetic field lines. By definition, the elementary vector of the tangent to the field line, dL , in each point of this line is parallel to the field vector. The vector product for parallel vectors vanishes, so, for the magnetic field lines we get d   L B 0 . In cylindrical coordinates this reduces to

Comparison between the dipole models

d d z B z B   
, and we get the following equation for the magnetic field lines:

      sup inf , d , z y B y z z y B y z          (6.76)
where inf  and sup  are, respectively, the smallest and the largest value of  for a given field line,

and inf sup , z      
 is a dummy variable.

We have put forward a FORTRAN 77 computer code for calculating the magnetic field lines for different dipole models according to both the exact expressions and their Taylor expansions, see [6.19]. Visualization of the field lines calculated using the Taylor expansions allows to estimate contributions of various expansion terms and the general convergence of the Taylor series for different models.

In the model of a circular current loop the exact expressions of the magnetic field components, eqs. (6.51), are very complicated, so, we have chosen to compute them by numerical integration over the angle   of the elementary field components expressed through the Biot- Savart law, see eq. (6.24) and Figure 6.4:

   
z z R R z R R R z B R B R                                      m m (6.77)
The corresponding exact expressions for the model of a pair of magnetic charges have been given in the previous section, see eqs. (6.69).

The

B B                                    , (6.78)
and for that of a pair of magnetic charges we get:

      2
B B                                 
. (6.79) We remind the reader that in eqs. (6.78) and (6.79)  has different meaning. In all cases, the numerical integration over  in eq. (6.76) has been performed using the Runge-Kutta method [6.20, pp. 420

ff].
With the aid of this program we have visualized the field lines for all models, see Figure 6.9. The calculations have been made using the exact expressions, vide supra, and the modelling parameters have been chosen in such a way that all the field lines ostensibly merge at the maximal distance max r from the source. As one can see, at distances comparable with the model size, the appearance of the field lines predicted by each model is totally different. The lines produced by a uniformly magnetized sphere are parallel to the dipole axis inside the sphere and coincide with those of the point dipole outside the sphere. The lines due to a current loop close on themselves inside the loop while those of a pair of magnetic charges diverge from the positive charge and converge towards the negative one. In all cases, at small distances the behaviour of the field lines has nothing in common with that expected for the point dipole, in which case the field lines close on themselves in the space origin. In certain applications, e.g. in calculating the interaction energy between magnetic moments embedded in a condensed matrix, one needs a good approximation for the magnetic field produced at intermediate distances from the magnetic source. Figure 6.10 compares the magnetic field lines calculated using the exact expressions and Taylor expansions to different orders in the corresponding small parameter for the models of a circular current loop and a pair of fictitious magnetic charges. One can see that in the model of a current loop the expansion up to the 6 th order in R r   provides a good approximation for

Applying the models of extended dipoles to calculate dipole-dipole contribution to magnetocrystalline anisotropy

In the model of uniformly magnetized sphere, vide supra, the dipole-dipole interaction energy is the same as for the point dipoles; therefore it is entirely described by the   2 cos P  Legendre polynomial, see eq. ( 6.36). Hence, this model can describe only uniaxial magnetocrystalline anisotropy and not the hexagonal basal magnetocrystalline anisotropy in iron borate. Thus, in what follows we consider only the two remaining models. We have seen that, starting from the second term in the Taylor series, the expressions of the dipole-dipole interaction energy in these models totally diverge. Interestingly, this divergence goes in opposite directions, so that the characteristics of the magnetic dipole at small and intermediate distances become quite sensible to the choice between these two models.

In order to calculate the dipole-dipole contribution to the magnetocrystalline anisotropy constants of FeBO 3 we shall consider a more general case where the dipole axis does not necessarily coincide with 3 С -axis, cf. Chapter 1, "Crystal and magnetic structure of iron borate".

Two circular current loops

Here we consider the interaction energy E between two identical and parallel circular current loops (Ampérian currents) of the same radii R and areae being the unit vector in the direction of m . For definiteness, we choose the loops centred at the space origin O p and at an arbitrary point O s as the primary and secondary loops, respectively.

The dipole-dipole interaction energy in this model is straight related to the mutual inductance M as in eq. (6.53). By definition,

I   M (6.80)
where  , the magnetic flux induced by the current in the primary loop and passing through the secondary loop, can be calculated as follows: In order to evaluate the closed curve integrals in eq. ( 6.81) we express p s M M , p dl and s dl as follows:

s s p d d l l      A l   . ( 6
            2
M M r R Rr Rr Rr                                              , ( 6
d R R                                        l ( 6
d R R R                                        l . ( 6
.85)

In eqs. (6.83) to (6.85) and   are polar angles of arbitrary points of the primary and secondary loops, respectively. Putting these expressions in eq. ( 6.81), for the interaction energy, cf. eq. ( 6.53), we get: where the integrations are over and   .

Since we need only an approximate expression of E for 1   , we can first expand the integrand in eq. ( 6.86) in a Taylor series up to the fourth order in the small parameter R r   and then integrate the result. We get:  

P P P r         E m (6.87)
where n P are Legendre polynomials [6.21] of the scalar product r  e e m , r e being the unit vector in the direction of r, and the -and + signs correspond to parallel and antiparallel dipoles, respectively. Obviously, eq. ( 6.87) can be directly obtained by substituting the scalar product r  e e m for cos in the Legendre polynomials given above in eq. ( 6.63).

Two pairs of fictitious magnetic charges

.12 shows a system of two interacting dipoles implemented as a pair of "magnetic charges" q  spaced a distance d apart. For the dipole-dipole interaction energy we get: in eq. ( 6.88) one gets:

2 0 2 m p p m 2 1 1 4 r r r d                    E m (
2 0 2 2 2 2 1 1 4 1 2 1 2 r r r d r r                               
P P P r         E m (6.90)
where the -and + signs correspond to parallel and antiparallel dipoles, respectively.

Dipole-dipole contribution to magnetocrystalline anisotropy constants of iron borate

In order to calculate the dipole-dipole contribution to the magnetocrystalline anisotropy constants for FeBO 3 , we have put forward a computer code implementing the lattice-sum method. We have chosen to do the summation in the volume of a rhombohedron congruent to the primitive rhombohedron shown in Figure 6.13.

         cos 2 cos cos z x k i x y z i j x y z k k                                            r l (6.91) where  1 cos cos 2 k i     ,  1 cos cos 6 i j     and  1 2 cos cos 3 k k    
are the cosines of angles between corresponding axes of two coordinate systems. Substituting eq. ( 6.91) in the expressions for the dipole-dipole interaction energy, eqs. (6.87) and (6.90) for the models of a circular current loop and a pair of magnetic charges, respectively, we express this energy in terms of integers numbering the iron sites along the edges of the rhombohedron.

The calculation of the density of dipole-dipole interaction energy now is reduced to computing the following sum:

dip , , 1 2
( 1) ( , , )

m n k m n k N m n k      E E (6.92)
where the factor( 1) m n k    takes into account antiferromagnetic ordering and ( , , ) m n k E is the dipole-dipole interaction energy between ions at the origin (numbered 0, 0, 0 ) and at a site numbered , , m n k .

Henceforth, the magnetic dipole moment at 0K T  will be expressed as

g S   m (6.93)
where , and g S  have the same meanings as in eq. (6.4).

The dipole-dipole contributions at 0 K, together with those of the crystal field, vide supra, are listed in Table 6.1. One can see that the models of a circular current loop and a pair of magnetic charges result in substantially different expressions for the dipole-dipole interaction energies.

Estimation of the dipole size for FeBO 3

The effective hexagonal anisotropy constant for FeBO 3 is cf. eq. (1.28) in Chapter 1, "Crystal and magnetic structure of iron borate": The calculation of the dipole-dipole interaction energy has been done using a model of a circular current loop in the same way as for FeBO 3 , i.e. computing the sum in eq. ( 6.92) taking into account only the sites occupied by iron.

The dependence of dip a on x is shown in Figure 6 R values (dashed-dotted, red) at 0 K.

Conclusions

Possible contributions to the magnetocrystalline anisotropy of mixed iron-gallium borates Fe x Ga 1-x BO 3 , namely, crystal field and dipole-dipole, have been considered in detail. The former contribution has been calculated in perturbation theory using the spin Hamiltonian parameters for isolated Fe 3+ ions in (diamagnetic) gallium borate. The latter contribution has been evaluated under the assumption that the ratio dipole size/ interdipole distance is non-negligible, i.e. that we are dealing with the extended dipoles.

We have developed a theoretical description of three models of an extended magnetic dipole: (i) a uniformly magnetized sphere, (ii) an Ampérian current and (iii) an assembly of two fictitious "magnetic charges". We have shown that all three models yield identical results at large distances with respect to the dipole size. Indeed, the first terms of Taylor expansions of the magnetic fields produced by the models of a circular current loop and of a pair of magnetic charges coincide with the exact expression of the magnetic field outside the uniformly magnetized sphere, the latter being the same as for the point dipole. Therefore, the uniformly magnetized sphere can describe only the uniaxial magnetocrystalline anisotropy and not the hexagonal basal magnetocrystalline anisotropy in iron borate.

The dipole-dipole interaction energy has been calculated for two extended dipole models, viz., a pair of magnetic charges and a circular current loop. The dipole-dipole contribution to the magnetocrystalline anisotropy constants for FeBO 3 has been calculated by the lattice-sum method. A comparison between the experimental and calculated values of the effective basal anisotropy constant has shown that the model of a pair of magnetic charges fails to explain the experimental results. In contrast, the model of a circular current loop provides consistent evidence in support of the dipole-dipole contribution to the basal magnetocrystalline anisotropy of iron borate and, incidentally, yields two more or less realistic estimates of the dipole sizes associated with Fe 3+ ion. In spite of the fact that the approach based on the models of extended dipoles, put forward in this work, provides new insight in the nature of the basal magnetocrystalline anisotropy of iron borate, it is certainly oversimplified. More sophisticated (ab initio) calculations are necessary in order to obtain a more detailed picture of spatial distribution of the magnetic field produced by paramagnetic ions at short and intermediate distances.

The dipole-dipole interaction energy for mixed borates Fe x Ga 1-x BO 3 has been calculated by the same technique as for FeBO 3 . In order to simulate a diamagnetically diluted crystal lattice we have used the Monte Carlo technique. The dipole-dipole contributions to the magnetocrystalline anisotropy constants for crystals with different x have been obtained; dip dip and e d have been calculated for the dipole sizes determined for FeBO 3 .

Surface magnetocrystalline anisotropy of iron borate single crystals 7.1 Theoretical background

The surface magnetism -a specific magnetic state of a thin near-surface layer of magnets -is caused by lowering of symmetry in the environment of near-surface magnetic ions in comparison with those in the volume. Néel was the first to specify the existence of surface magnetocrystalline anisotropy energy, due to this effect [7.1]. However, the manifestation of this anisotropy in conventional ferromagnets is usually obscured by the demagnetizing field and large volume magnetocrystalline anisotropy. In contrast, in iron borate the surface magnetocrystalline anisotropy can be observed due to the fact that the demagnetizing field, proportional to the (weak) magnetization of the antiferromagnetic crystal with weak ferromagnetism, is small and the basal anisotropy is weak [7.2]. Therefore, magnetic characteristics of a thin (0.01-0.1 µm) nearsurface layer of iron borate drastically differ from those of the volume.

The gas-phase deposition technique allows growing bulk single crystals of iron borate with large non-basal faces of optical quality [7.3]. Such crystals have made possible finding out the surface magnetism in iron borate by the magneto-optical Kerr effect [7.4].

The surface magnetocrystalline anisotropy energy can be calculated as the difference of the magnetic energies of ions in the near-surface layer and in the crystal volume. This difference is due to two causes: (i) the occurrence of the crystal surface per se (without modification of the oxygen environment of near-surface iron ions) and (ii) structural distortions in the near-surface layer.

As we have already indicated, cf. Chapters 1 and 6, "Crystal and magnetic structure of iron borate" and "Magnetocrystalline anisotropy of iron borate and iron-gallium borates", respectively, in iron borate the exchange energy in a good approximation is isotropic; therefore, the density of the surface magnetocrystalline anisotropy energy,  is expected to include only dipole-dipole, dip  and crystal field, cf  contributions. Obviously, the cause (i) intervenes only in the calculation of the dipole-dipole contribution whereas the cause (ii) is expected to modify both the dipole-dipole contribution -as a result of alteration of mutual disposition of iron ions in the near-surface layer -and the crystal field contribution -in as much as the oxygen environment of iron ions in the near-surface layer undergoes additional distortions in comparison with the volume.

In the paper by Zubov et al. [7.4] and later in the Strugatsky's thesis [7.5] a theoretical description of  has been put forward taking into account only the cause (i), i.e. neglecting structural distortions in the near-surface layer. In this approximation,  will include only dipoledipole contribution, dip  vide supra, and the maximal value is obtained for the   As one can see, l in the near-surface layer has an orientation different from its equilibrium orientation in the volume, see Chapter 1, "Crystal and magnetic structure of iron borate". As a consequence, a transition layer between the volume and the near-surface layer will form, similar to a domain wall, where l will gradually turn from the equilibrium orientation in the volume to that in the near-surface layer. The density of the magnetocrystalline anisotropy energy in this layer can be expressed as [7.4, 7.5]:

  eff 1 sin a A      (7.3)
where eff a is the effective uniaxial anisotropy constant, cf. Chapter 1, "Crystal and magnetic structure of iron borate", A is a constant in the expression describing exchange interaction in the non-uniform transition layer [7.5]. For FeBO 3 at 0 K Indeed, a considerable deviation of l from the basal plane would result in a significant increase of   , of the order of magnitude of eff a A , cf. eq. (7.3). This situation is energetically unfavorable.

Thus, l in the near-surface layer will lie in the basal plane or make a small angle with it. Its azimuthal angle will be determined by surface magnetocrystalline anisotropy. Putting 2    in eq. ( 7.1), we get [7.4, 7.5]: C axes, see eq. (7.4); therefore, the reduced ferromagnetic vector  m l , see Chapter 1, "Crystal and magnetic structure of iron borate", is perpendicular to this axis, so that 2 C is the hard magnetization axis in the basal plane. This result has been confirmed by experimental observations for   10 14 face of the crystal [7.4].

The saturation field along the hard magnetization axis in the near-surface layer, called the critical field c H , is considered as the measure of the surface anisotropy. In fact, an application of such a field would totally erase the transition layer. Applying a magnetizing field H in the basal plane, the direction of l in the transition layer will gradually change from that in the volume, determined by H , to that in the near-surface layer, determined by both H and the surface magnetocrystalline anisotropy. The density of the magnetocrystalline anisotropy energy in this layer will be [7.4-7.6]   From the preceding analysis we can suppose that this discrepancy could be removed if we allow for structural distortions in the near-surface layer, ("surface reconstruction"). Earlier, an attempt in this direction has been made; however, only changes in the positions of iron ions have been considered, but concomitant distortions in their oxygen environment has been neglected [7.6]. (In such an approximation, only the dipole-dipole contribution to the surface anisotropy is accounted for.) In this case, the experimental c H value could be obtained only for relative extensions as large as 7 12 %  [7.6].

Meanwhile, it is evident that in the near-surface layer, the oxygen environment of iron ions undergoes additional distortions in comparison with the crystal volume, giving rise to the crystal field contribution to the surface magnetocrystalline anisotropy energy. In the present chapter, we provide a model of these distortions and give a complete description of the surface anisotropy of iron borate, including both the dipole-dipole and crystal field contributions.

Surface reconstruction

Previously, in order to calculate the dipole-dipole contribution to the magnetocrystalline anisotropy constants for FeBO 3 , we have used a rhombohedron congruent to the primitive rhombohedron, with faces of   0112 type, cf. This rhombohedron is face-centred, i.e. it contains iron ions in all vertices and face centres, see Figure 7.1; the faces parallel to the crystal surface (and perpendicular to yz planes) are coloured. On the basis of symmetry, we can assume that the distortions of interatomic distances in the near-surface layer occur in yz planes. In the crystal volume each iron is surrounded by six oxygens forming a nearly perfect octahedron, see Figure 1.3 in Chapter 1, "Crystal and magnetic structure of iron borate". In the near-surface layer the oxygen octahedra are distorted. Consider the positions of two nonequivalent iron sites and their oxygen ligands with respect to the crystal surface. Figure 7.2 shows a projection of the rhombohedron of Figure 7.1 on yOz plane, the crystal surface and the basal planes being orthogonal to this plane. For simplicity, we show only two non-equivalent sites, Fe 1 and Fe 2 and their oxygen environments in the absence of distortions.

We assume that the distortions occur only in the near-surface layer of the oxygen octahedra.

The crystal surface (denoted by F in Figure 7.2) passes through O and O oxygens. We can divide the near-surface layer into four parallel sub-layers separated by planes passing through oxygen and iron ions, see Figure 7.2. The B plane is supposed to remain immobile, the distortions occur along the AD edge of the rhombohedron, and the displacements of ions lying at different distances from B are supposed to vary in proportion to these distances.

Crystal field contribution to the surface anisotropy energy

Previously, we have carried out EPR studies of Fe 3+ in GaBO 3 single crystals and determined the spin Hamiltonian parameters using crystallographic data and the Newman's superposition model. In order to calculate the crystal field contribution to the surface magnetocrystalline anisotropy energy, we have followed the procedure described in detail in Chapter 4, "EPR of iron-gallium borate single crystals with low x".

For near-surface iron ions, the superposition model parameters are expected to remain the same as for those in the volume, at least as far as distortions in their environment remain weak.

We have used the following values of and numbering the non-equivalent iron sites). For the type of distortions described above, the spin Hamiltonian takes the form B B  are, respectively, the second and fourth- order fine structure parameters. In eq. (7.10) as well as in the subsequent equations the  signs before certain terms correspond to 1, 2 k  , respectively.

Assuming the "Zeeman" term

k k g 0    H S H
in eq. (7.10) to be much larger than the remaining terms, we have calculated the energies of the spin levels mk E using the perturbation theory in the same way as in Chapter 6, "Magnetocrystalline anisotropy of iron borate and irongallium borates". In the first order of perturbations, we get:

       
k k k k k k k k k k k k k k k k k k k B B B B B B h h B B                                 (7.12)
In the strong exchange case, the spins of two non-equivalent iron ions are antiparallel: m  ), so that the anisotropic part of the right-hand side of eq. ( 6.12) for this level provides the crystal-field contribution to the density of the magnetocrystalline anisotropy energy of the near-surface layer:

 

                            (7. 13
)
where N is the number of Fe 3+ ions per unit surface (

V V V V V B B N B B                    (7.14)
where the V subscript refers to the parameter in the volume, and N is the same as in eq. (7.13).

Thus, cf  , defined as cf

cf S V   
, can be expressed as:

    

Comparison with experimental data

The parameters featuring in eq. (7.15), calculated for 3%   , are given in Table 7.1.

Obviously, the Jm .

, %  Now we can determine the equilibrium orientation of l in the near-surface layer in the absence of H taking into account the crystal field contribution. For this purpose, we minimize the sum dip cf       with respect to and   , cf. eqs. (7.1), (7.3), (7.15) and Table 7.1. As far as for 3%   dip  only slightly changes with  [7.6], for this quantity we can take its value in the absence of structural distortions, cf. eq. (7.1). These results are given in Table 7.2. As one can see, in the case of an extension, in the near-surface layer in equilibrium l becomes perpendicular

   

, which contradicts the experimental data [7.4]; therefore, the eventuality of an extension can be ruled out. In contrast, in the case of a contraction, in equilibrium l remains parallel to 2 C . Indeed, for ionic crystals, contractions rather than extensions are expected in the near-surface layer [7.9]. Taking into account the data in Note that the crystal surface in itself constitutes a structure defect; nevertheless, other types of structure defects can occur in the near-surface layer, for instance, vacancies of magnetic ions or their substitutions by diamagnetic ions (diamagnetic dilution), and such defects are also expected to contribute to cf S a . In particular, in mixed iron-gallium borates Fe x Ga 1-x BO 3 , cf in Fe x Ga 1-x BO 3 crystals at 0 K.

Conclusions

The theory of the surface magnetism of iron borate has been extended to include, besides the dipole-dipole contribution, the crystal field contribution to the surface magnetocrystalline anisotropy energy.

A model of structural distortions in the near-surface layer for   10 14 face of iron borate has been developed, allowing all ions located in this layer to be displaced proportionally to their distance from a reference plane assumed to remain immobile. In order to account for the lowering of symmetry in the near-surface layer, the generalized spin Hamiltonian expressed through the tesseral spherical tensor operators has been applied, and the spin Hamiltonian parameters have been calculated within the superposition model. The crystal field contribution to the surface anisotropy energy at 0 K has been calculated in perturbation theory. Taking into account the distortions of the iron environment produces a significant crystal field contribution to the surface anisotropy constant; indeed, the experimental results can be satisfactorily described assuming relative contractions in the near-surface layer ca. 1% .

  Néel pour les cristaux avec 0.34 0.85 x   ont été déterminées par des RME et SQUID (Superconducting Quantum Interference Device) techniques. Avec la diminution de x, N T sensiblement diminue. Les dépendances en température du champ de RAFM suggèrent une présence de transitions magnétiques aux environs de 80 et 20 K pour x = 0.65 and 0.34, respectivement, les températures de Néel correspondantes proches de 244 et 77 K. Nous supposons qu'à basses températures ces cristaux subissent la transition de Morin à la température M T . Au-dessus de M T , la structure magnétique de ces cristaux est la même que celle de borate de fer pur, c'à-d. antiferromagnétique avec un plan de facile aimantation. En dessous de M T , les cristaux sont antiferromagnétiques avec C 3 pour l'axe de facile aimantation ; par conséquent, le faible ferromagnétisme disparaît. Avec la diminution de x, à la fois le champ de Dzyaloshinskii-Moriya, D H et l'écart d'énergie isotrope, 2 H  diminuent. Pour les cristaux avec 1 x  , en abaissant la température en dessous de N T , D H d'abord augmente, puis passe par un maximum et enfin décroît. Pour les cristaux avec x = 0.65 et 0.34, ce maximum se produit aux alentours de. 80 et 20 K, respectivement, ce qui corrobore l'hypothèse de la transition du Morin dans ces cristaux.

Figure 1 . 1 3 С

 113 Figure 1.1 Stereographic projection of iron borate structure. The Cartesian coordinate axes are directed as follows: 2 С , m x y   and

Figure 1 . 2

 12 Figure 1.2 Rhombohedral unit cell of FeBO 3 . The Cartesian coordinate axes are directed as in Figure 1.1: 2 С , m x y   and

Figure 1 . 3

 13 Figure 1.3 Two non-equivalent sites of Fe 3+ . The Cartesian coordinate axes are directed as in Figure 1.1. The z -axis is perpendicular to the plane of the figure and points towards the reader. Full and empty circles represent ions located above and below this plane, respectively.

Fe ions have equal norms, 1 2 

 2 m m . Within the accuracy of measurements[1.4], these vectors lie in the basal plane and are almost antiparallel. However, because of a slight tilt of occurs, as shown in Figure1.4. Obviously, 

Figure 1 . 4

 14 Figure 1.4Scheme of orientation of magnetic moments of two non-equivalent iron ions, illustrating the emergence of antiferromagnetism and weak ferromagnetism.

Figure 1

 1

  reasonably assume that constants describing the same-order magnetocrystalline anisotropy are of the same order, , , b a c d f e    . It should be noted that Curély et al. have shown that negative sign of D would produce a helical-type magnetic structure [1.13-1.15].

  plane, cf. eqs. (1.18) and (1.14) and Figure 1.1.

Figure 2 . 2

 22 Figure 2.2 DTA curve for the composition used for the synthesis of FeBO 3 in wide temperature range.

Figure 2 . 3

 23 Figure 2.3 DTA curve for the composition used for the synthesis of FeBO 3 in the range of formation of FeBO 3 and Fe 3 BO 6 crystal phases.

Figure 2 . 4

 24 Figure 2.4 Electron microscope pictures showing FeBO 3 single crystals on the probes for two different magnifications.

Figure 2 . 5

 25 Figure 2.5 Prepared charge used in synthesizing Fe x Ga 1-x BO 3 crystals of a definite composition.

Figure 2 . 6

 26 Figure 2.6Crucible with the solution melt used for synthesizing GaBO 3 crystals after extracting from the muffle furnace.

Figure 2 . 7

 27 Figure 2.7Crucible with a solution melt in the furnace.

Figure 2 . 8

 28 Figure 2.8 Scheme of the crystallization setting.

Figure 2 .

 2 Figure 2.9 shows a typical temperature regime used in synthesizing crystals of a definite composition. It includes the following steps: (i) heating of the furnace;

Figure 2 . 10

 210 Figure 2.10 GaBO 3 (a) and FeBO 3 (b) single crystals.

Figure 2 . 11

 211 Figure 2.11 Fe x Ga 1-x BO 3 single crystals with different x.

Figure 2 . 12

 212 Figure 2.12 GaBO 3 crystals on the seed holder.

  Figure 2.13Iron contents in Fe x Ga 1-x BO 3 crystals vs. initial iron contents in the charge. The vertical bars show the spread of iron contents for the samples extracted from the same crucible. The dashed line is a guide for the eyes.

  h, k and l represent the lattice planes.

Figure 2 .

 2 Figure 2.14 XRD patterns of FeBO 3 (bottom curve) and Fe 0.05 Ga 0.95 O 3 (top curve) [2.12].

Figure 2 .

 2 Figure 2.16 Experimental EPR spectrum of Fe 3+ in Fe x Ga 1-x BO 3 crystal with 0.003 x  at 4 K (a) and corresponding computer-generated spectrum (b).

Figure 2 .

 2 Figure 2.17 Experimental EPR line at ca. 0.486 T and the corresponding part of computer-generated spectrum for different distribution widths of the ligand coordinates.

Table 3 . 1 M

 31 Parameters in eqs.(3.3) and their experimental values for FeBO 3 , taken or calculated from the corresponding references. is the sublattice magnetization at a temperature T :

Figure 3 . 1

 31 Figure 3.1 Normalized X-band room-temperature EMR spectra of Fe x Ga 1-x BO 3 crystals with different x: 1 (a), 0.75 (b), 0.2 (c), 0.04 (d) and 0.003 (e).

Figure 3 .

 3 Figure 3.2 shows EMR spectra for x = 0.85 crystal at different temperatures. As one can see, at T 304 K only the high-field line is present; this temperature can be tentatively identified as

Figure 3 . 2

 32 Figure 3.2 Normalized EMR absorption spectra series for x = 0.85 crystal at  17 GHz and different temperatures shown alongside the curves. The high-field line is superposed with a narrow signal from a probe with 2.006 g  .

Figure 3 . 3 X

 33 Figure 3.3 X-band EMR derivative of absorption spectra for x = 0.85 crystal at different temperatures shown alongside the curves.

Figure 3 . 4

 34 Figure 3.4 Normalized AFMR line for x = 0.75 crystal at different frequencies and T = 77 K.

Figures 3 .

 3 Figures 3.5 and 3.6 show the temperature dependences of the resonance field of the AFMR line for two crystals with x = 0.65 and 0.34. Cooling the crystals down from N T , the resonance

Figure 3 . 5

 35 Figure 3.5 Temperature dependence of the resonance field of the AFMR line for x = 0.65 crystal measured at different microwave frequencies. The inset shows a vertical zoom in the low temperature region, with different curves arbitrarily shifted along the vertical axis.

Figure 3 . 6

 36 Figure 3.6Temperature dependence of the resonance field of the AFMR line for x = 0.34 crystal measured at different microwave frequencies.

Figure 3 . 7 3 -

 373 Figure 3.7EMR absorption spectra at  7.7 GHz  for x = 0.34 crystal at different temperatures shown alongside the curves.

Figure 3 .Figure 3 . 8 Figure 3 . 9

 33839 Figure 3.10 shows the decrease of N T , determined by AFMR, with the decrease in x. The AFMR data have been corroborated by SQUID. For comparison, data on N T in similar crystals obtained by magnetometry and Mössbauer spectroscopy [3.12] are included ‡ . Obviously, the results of different determinations are in good agreement with each other.

Figure 3 . 10

 310 Figure 3.10 Néel temperature vs. x determined from AFMR and SQUID measurements. For comparison, magnetometry and Mössbauer spectroscopy data [3.12] are included.

Figure 3 . 11

 311 Figure 3.11 FDFs for x = 1 crystal at different temperatures. The solid curves are fittings according to eq (3.3).

Figure 3 . 12

 312 Figure 3.12 FDFs for x = 0.85 crystal at different temperatures. The solid curves are fittings according to eq. (3.3).

Figure 3 . 13

 313 Figure 3.13 FDFs for x = 0.34 crystal at different temperatures. The solid curves are fittings according to eq. (3.3).

Figure 3 . 14

 314 Figure 3.14 Temperature dependence of the effective Dzyaloshinskii-Moriya field for x = 0.65 crystal.

2 H 2 H 2 H

 222  for different crystals at different temperatures, determined from fitting to eq. (3.3) of the corresponding FDFs, are shown in Table3.3.With increasing T from 4 K up to the Néel temperature,  decreases, see Figure3.15. In the temperature range from 4 to ca. 80 K a slow decrease is observed, while at higher temperatures this decrease is drastically accelerated.The fact that  becomes negative in the vicinity of the Neel temperature may be an artefact caused by non-validity of eq.(3.3) in this temperature range[3.13].

Figure 3 . 15

 315 Figure 3.15 Isotropic energy gap vs. temperature for x = 0.65 crystal.

Figure 3 .

 3 Figure 3.16 shows the angular dependence of the quantity   



  are fitting parameters, the term in p describing the uniaxial component. The best-fit parameters in this equation are listed in Table3.4.

Figure 3 . 16  2 D

 3162 Figure 3.16 Dependence of   

J g becomes: 1 L g  and 2 Sg

 12  , respectively. (The exact value of S g in the latter case, 2.002319 S g  , is explained by quantum electrodynamics.)

J

  Figure 4.1 Zeeman sublevels for

CH

  gives non-trivial contributions to the Zeeman sublevel energies only for 1 S . The spin Hamiltonian parameters m ni B and m n B are assumed to only depend on local structure

  .20) where , ,      are the Euler angles shown in Figure 4.2. Obviously, with such a definition, in the c c c , , x y z frame and   are spherical angles of B and  describes the orientation of 1 B in the plane perpendicular to B .

Figure 4 . 2

 42 Figure 4.2 Euler angles between the crystallographic frame c c c , , x y z and the laboratory frame l l l , , x y z .

F

  is assumed to have either Lorentzian, L F or Gaussian, G F shape:

Figure 4 . 3 :

 43 Figure 4.3General scheme of numerical analysis of experimental EPR spectra.

Figure 4 .

 4 Figure 4.4 shows the EPR spectra for Fe 0.003 Ga 0.997 BO 3 at 4 K for different orientations of B , described by the polar angle  with respect to the 3 C axis and the azimuthal angle  with respect to 2 C . Because of small crystal sizes, exact orientations could be determined only through careful trial and error computer fittings to the experimental spectra.Figure 4.5 shows the actual relation between  and  in rotating the crystal.

Figure 4 .

 4 5 shows the actual relation between  and  in rotating the crystal.

  Figure 4.4Experimental EPR spectra of Fe x Ga 1-x BO 3 crystal with 0.003 x  at 4 K for different orientations of B : 11   , 270    (a); 37   , 300    (b); 66   , 307    (c); 96   , 311    (d); 135   , 318    (e); 164   , 339    (f) and 169   , 30    (g).

Figure 4 . 5

 45 Figure 4.5 Relationship between the spherical angles in the course of rotation of the crystal when recording the spectra shown in Figure 4.4. Circles: results of spectra simulations; full line: the best-fit curve described by the equation   tan 0.12 / 0.57 sin 0.68 cos      .

Figure 4 . 6

 46 Figure 4.6Normalized experimental room-temperature EPR spectra of Fe 0.003 Ga 0.997 BO 3 (a, blue) and Fe 0.042 Ga 0.958 BO 3 (b, red). In both cases 87   , 285   °.

4 O 4 O 4 O 4 

 4444 O O O O  are defined as in the textbook by Al'tshuler and Kozyrev [4.1]. Note that instead of 3  the notations  3 [4.6], 3* [4.5] and 3 [4.1] have also been used. The  signs in eq. (4.27) refer to two non-equivalent Fe 3+ sites with local magnetic axes rotated through the angle   about the 3 C axis [4.5, 4.6], see Figure 1.3 in Chapter 1, "Crystal and magnetic structure of iron borate". Note that in the spin Hamiltonian in the paper by Lukin et al. [4.6] the coordinate system is turned in such a way that x -axis lies in the symmetry plane m .First, we have focused on determining the sign of D , the leading parameter in eq.(4.27).

Figure 4 . 7

 47 Figure 4.7 Computer-generated EPR spectra at 4 K with 0 D  (a, red) and 0 D  (b, blue) and at 300 K (c, green). The parameters a and F are set to zero and the spectra intensities are multiplied by the absolute temperature. The numbering from 1 to 6 of the Zeeman sublevels involved is in ascending order for 0 D  and in descending order for 0 D  .

  Figure 4.9 (top) shows a zoom of computer- generated spectra for positive D at two different temperatures. The comparison with the corresponding experimental spectra, see Figure 4.9 (bottom), allows concluding that the sign of D is, indeed, positive. (The splitting in Figure 4.9 (bottom), instead of the single line in

Figure 4 .

 4 Figure 4.9 (top), is observed because of the presence of two non-equivalent iron sites.)

Figure 4 . 8

 48 Figure 4.8 Energy diagrams for 0 D  (top) and 0 D  (bottom) for 10   and o 278   , showing various resonance transitions for energy quanta of

Figure 4 . 10

 410 Figure 4.10 Normalized experimental EPR spectrum of Fe 0.003 Ga 0.997 BO 3 at 4 K (a, red), the best-fit computer generated spectra for 80   and 309   ° without (b, blue) and with parameter distributions (c, green).

Figure 4 . 11

 411 Figure 4.11 Normalized experimental room temperature EPR spectra, see curves a, b in Figure 4.6 (full lines) and corresponding computer-generated spectra (dashed lines). The linewidth B , as deduced from the simulations, is 0.001 T for 0.003 x  (left) and 0.0097 T for 0.042 x  (right) . .

  .30) In the framework of the Newman model, the i l B parameters are evaluated on the basis of a structural model of the paramagnetic site, as follows [4.8, 4.9]: eq. (4.31) j enumerates the nearest neighbours of the paramagnetic ion (ligands) with spherical coordinates , and are coordination factors proportional to tesseral harmonics with the corresponding l and m indices. The radial functions are expressed as power functions of the metal-to-ligand distances:

Figure 4 .

 4 13 shows the dependence of a and F on 4 Fe 3+ in calcite[4.12].) Obviously, within the framework of the Newman model the set I parameters cannot be fitted to in the whole range of 4 t values tested (Figure4.13, top). In contrast, for the set II parameters, consistent results can be obtained with 4 5 t  for a and 4 8t  for F (Figure4.13, bottom).

Figure 4 . 13

 413 Figure 4.13 Dependences of a (continuous, blue) and F (continuous, red) on the power exponent 4 t for set I (top) and set II parameters (bottom). The experimental a and F values, see Table 4.1, are shown by the dash-dotted (light blue) and the dashed lines (orange), respectively.

Figure 4 . 14

 414 Figure 4.14 Dependences of a (top) and F (bottom) on the intrinsic parameter 4 b for different 4 t values (4, 8, 12 and 16 from right to left, respectively) calculated with set II spin Hamiltonian parameters. The horizontal lines show the set II values of a (dash-dotted, light blue) and F (dashed, orange).

Figure 4 .

 4 Figure 4.14 shows the dependences of a and F on the intrinsic parameter 4 b for

  (4.28) because the spin Hamiltonian expressed by means of the extended Stevens operators do not possess sufficient symmetry[4.20]. In order to overcome this difficulty, we have used the general spin Hamiltonian expressed by means of two-vector tesseral spherical tensor operators 20]. Here n is the unit vector of the direction of B , B l and S l are powers of B and of the spin operators, respectively.

B l  ) and linear Zeeman 1 BlBFigure 4 .

 14 Figure 4.15 Normalized EPR spectra computed with 0   (a), 0.0005   (b), 0.001   (c), 0.003   (d) and 0.01   (e) Å.

3 b

 3 parameters are very small and have not been taken into account.

5. 1 2 I

 12 Basic formalismNuclear magnetic resonance (NMR) was first observed in molecular beams by Isidor Rabi in 1938. In 1946, Felix Bloch and Edward Mills Purcell extended NMR to liquids and solids.Individual neutrons and protons possess a nuclear spin 1  ; certain atomic nuclei isotopes also have an overall spin of integer or semi-integer quantum number I. The angular momentum associated with nuclear spin is quantified by a quantum number having 2 the nuclear magneton, e , p m and ђ being, respectively, the elementary charge, proton mass and reduced Planck constant.

1 2 I

 12  ) has become a helpful accurate technique of structural analysis [5.1].

Figure 5 . 2 .

 52 Figure 5.2. As far as the dipole-dipole interactions vary as

Figure 5 . 1 2 I

 512 Figure 5.1Zeeman sublevels for the nuclear spin

Figure 5

 5 Figure 5.2 Magic-angle sample spinning.

Figure 5 . 3

 53 Figure 5.3 Scheme of the Hahn two-pulse echo sequence.

. 8 )V

 8 Here  , called asymmetry parameter, is confined in the range 0 is proportional to the coupling constant Q C between the nuclear electric quadrupole moment and the EFG[5.7, 5.8].

2 

 2 i U i   , with zero mean values and equal standard deviations 1[5.11, 5.13, 5.20]. (Here the1

. 20 )

 20 The parameters 0  and 0  are introduced by means of eqs. (5.15) and (5.16), respectively,

  .(5.8), (5.15) and (5.16) calculates  and  ; (iv) builds marginal distribution densities of  and  as well as the JDD   , P   for the 0 + V V -tensor; (v) computes the mean values ,   , the standard deviations ,     and the correlation coefficient  of and   .

Figures 5 .Figure 5 . 4

 554 Figures 5.4 to 5.6 illustrate variations of the latter parameters with  for different 0 

JDDFigure 5 . 5

 55 Figure 5.5 Dependences on  of   (left) and   (right) for different 0  and 0  . The dashed lines in the left and right parts correspond to eqs. (5.22) and (5.23), respectively. 0  , 

Figure 5 Figure 5 . 6

 556 Figure 5 in the Maurer's paper suggests that the correlation between  and  tends to decrease with increasing departure from axial symmetry (i.e., increasing 0  ) and with lowering the

  Figures 5.7 and 5.8 show the results of this analysis.

Figure 5

 5 Figure 5.7 Relationships between 0  and cal 0  for 0 0.5   ; 0.102    a.u.,

Figure 5

 5 Figure 5.8 Relationships between 0  and cal 0  for 0 3.0   a.u.; 0.102    a.u.,

  .33)To illustrate the contribution of different crystallite orientations in the "powder" NMR spectra, we have computed and displayed in Figure5.9 the spectra for the best-fit simulation of a whole experimental spectrum (vide infra for a detailed account of experimental results). One can see two sharp features corresponding to the "magic" and equatorial orientations; the contribution of the polar orientation o 0   is weak because of small relative number of such crystallites.

Figure 5 . 9

 59 Figure 5.9Experimental MAS NMR spectrum of11 B in Fe x Ga 1-x BO 3 with x = 0.01 (continuous, blue) and computer-generated spectra for randomly oriented crystallites (dashed, red) and for chosen orientations indicated (dotted, green).

Figure 5 .

 5 Figure 5.10 shows the marginal distributions of  for the Maurer's and Czjzek's models. For the former, the set of parameters corresponds to the best fit to the experimental spectrum, see Table 5.1 below, and for the latter two different  values have been used. As one can see from Figure 5.10 (left), in order to obtain  values close to 0 2.8 Hz 6 M   , in the Czjzek's model one should choose an unrealistically broad distribution of  . On the other hand, taking the same distribution widths of  as those determined with the Maurer's JDD, 0.0 MHz 4    ,

Figure 5 . 10 Figure 5 . 11 2 

 5105112 Figure 5.10 Marginal distributions of  obtained from the Maurer's (points, green) and Czjzek's (squares, red) models. The simulation parameters are: iso 0 24.5 ppm, 2.86 MHz     , 0.04 MHz    , 0 6 , 0.015 

Figure 5 .Figure 5 .

 55 Figure 5.12 Normalized JDD for the Czjzek's model computed with iso 24.5 ppm   and 1.42   (left) and 0.04 MHz   (right).

Figure 5 . 14 Figure 5 .Figure 5 .

 51455 Figure 5.14 Normalized JDD for the Maurer's model computed with simulation parameters: iso 24.5ppm   ,

3 2 I 2 2

 322  nuclear spin, the optimal pulse duration for selective excitation of the central transition equals to the duration of a non-selective 90 pulse divided by 1 I   [5.4, 5.24]. In our experiments, the duration of this pulse was 1.75 i t  µs. The recycle delay between acquisitions was 0.5 s, and a total of 1024 acquisitions was sufficient to resolve characteristic spectral features of 11 B.

Figure 5

 5 Figure 5.17 shows11 B MAS NMR spectra for Fe x Ga 1-x BO 3 powders with x = 0, 0.01 and 0.02 together with the best-fit computer simulations. The spectra display a characteristic twopeak MAS quadrupole powder pattern for a single boron site[5.8, 5.24]. Clearly, an increase of iron concentration results in broadening of the spectra.

Figure 5 . 17

 517 Figure 5.17 Experimental (continuous, blue) and computer-generated (dashed, red) MAS NMR spectra of 11 B in Fe x Ga 1-x BO 3 with x = 0.00 (a), 0.01 (b) and 0.02 (c).

Figure 5 .

 5 Figure 5.18 Experimental (continuous blue line) and computer-generated (dashed red line) MAS NMR spectra of 71 Ga in Fe x Ga 1-x BO 3 with x = 0.00 (a), 0.01 (b) and 0.02(c).

  anisotropy constants for FeBO 3 .

  field (cf) and dipole-dipole (dip) contributions:

  for FeBO 3 has been obtained previously[6.4].FeBO 3 a has been determined by means of AFMR in a wide temperature range, see[6.2, 6.5]; its value, extrapolated to 0

  O  are extended Stevens operators defined, e.g., in the textbook by Al'tshuler and B. M. Kozyrev [6.8]. The  signs refer to two non-equivalent iron sites with local magnetic axes rotated through the angle   about the C 3 axis, see Figure 1.3 in Chapter 1, "Crystal and magnetic structure of iron borate".

only occupied spin level is that with 5 2 M

 2  . Comparing matching symmetry terms in eqs. (6.12) for this level and in eq. (6.1) (the Dzyaloshinskii-Moriya term should not be considered in this comparison) we get the crystal field contributions cf

Figure 6 .Figure 6 . 2

 662 Figure 6.1 shows the temperature dependences of cf

  fictitious magnetic charges.

Figure 6

 6 Figure 6.3 that

.Figure 6 . 4

 64 Figure 6.4 Magnetic dipole as an Ampérian current.

Figure 6 . 5

 65 Figure 6.5System of two interacting Ampérian currents.

.

Figure 6 . 6

 66 Figure 6.6Magnetic dipole as an assembly of two "magnetic charges".

1 

 1 6.7 for the notation. The choice of  signs refers to parallel and antiparallel dipoles, respectively. Note that eqs. (6.69) to (6.71) can be immediately obtained from the corresponding expressions for electric dipoles by substituting the electric dipole moment and the permittivity of vacuum 0  by the magnetic dipole moment and 0 , respectively.

Figure 6 . 7

 67 Figure 6.7System of two interacting magnetic-charge dipoles.

Figure 6 .

 6 Figure 6.8 compares radial dependences of z B in the equatorial plane

Figure 6

 6 Figure 6.8

Figure 6 . 9

 69 Figure 6.9 Magnetic field lines for different dipole models scaled in rdu. The external line (red) corresponds to a pair of magnetic charges 0.75 d  apart. The intermediate dashed (green) and continuous (grey) lines refer to a point dipole and uniformly magnetized sphere of radius 0.20 R  , respectively. The internal line (blue) is due to a current loop of radius 0.40 R  . All the lines merge at the maximal distance max 2.0 r  .

Figure 6 . 10

 610 Figure 6.10 Magnetic field lines in the models of a current loop (a) and of a pair of magnetic charges (b) calculated using the exact expressions of the magnetic fields (continuous lines) and the Taylor expansions to the 0 th , 2 nd , 4 th and 6 th order in the corresponding small parameters. All the lines merge at the maximal distance  max 2.0 r rdu. With increasing the expansion order, in (a) each subsequent field line remains confined inside the previous one, while in (b) it passes alternately from inside to outside of the exact field line profile.

Figure 6 . 11 .

 611 Figure 6.11. By definition, the magnetic moment of a loop is

  of the primary and secondary loops, dA is a differential element of the vector potential at a point s M on the secondary loop, produced by the primary loop: the distance between two points on the primary and secondary loops, and p dl and s dl are differential elements of the corresponding loops.

Figure 6 . 11

 611 Figure 6.11 System of two interacting Ampérian currents.

  , see Figure6.12 for the notation. The choice of the  signs refers to parallel and antiparallel dipoles, respectively.

Figure 6 .

 6 Figure 6.12 System of two interacting dipoles in the model of a pair of magnetic charges.

Figure 6 .

 6 Figure 6.13 A primitive rhombohedron with the edge length 3.6 Å  l

lMM

  In order to get the dipole-dipole contributions to the magnetocrystalline anisotropy at different temperatures, these contribution at 0 K should be multiplied by   at a temperature T , cf. Chapter 1, "Crystal and magnetic structure of iron borate". 0 T for FeBO 3 have been tabulated[6.2]. The temperature dependences of the crystal field contributions are given in eqs.(6.22) and in Figure6.1.

Figure 6 . 2 Figure 6 . 14

 62614 Figure 6.14 shows the spatial distribution of dia-and paramagnetic ions in mixed borates in a layer parallel to the basal plane. The probabilities of iron or gallium occupying a given site, are x or 1-x, respectively. One can see that nanoscopic iron domains -nanoclusters -occur at intermediate x. Such nanoclusters are expected to possess magnetic properties similar to those of magnetic nanoparticles[6.24].

. 15 . 3 )Figure 6 . 15 1 Figure 6 . 16 Concentration dependence of dip d calculated with 1 R 2 R 1 Figure 6 . 17

 1536151616121617 Figure 6.15 Concentration dependence of dip a at 0 K.

.

  Minimizing the sum of dip and   with respect to and   , we get the equilibrium orientation of l in the near-

  5]. In equilibrium l is parallel to one of 2

  magnetization of the crystal, T M being the sublattice magnetization at a temterature T , and D E being the Dzyaloshinskii-Moriya and exchange constants, cf.

2 

 2 will align on C 2 , and  l m will make with C 2 an angle  , so that eq. (7.7) will give:

8 )

 8 Zubov et al. have found that on the   1014 face of iron borate c 1kOe H  [7.4]; meanwhile, the c H value calculated with eq. (7.8) is much lower, c 0.2 kOe H  [7.5].

Figure 6 .

 6 13 in Chapter 6, "Magnetocrystalline anisotropy of iron borate and iron-gallium borates". Here we are interested in the smallest rhombohedron with faces of   1014 type.

Figure 7

 7 Figure 7.1 A rhombohedron with edge length  r 5.862 Å a and apex angle 104.2 used to calculate the density of surface anisotropy energy. The Cartesian coordinate axes are directed as follows: 2 С , x  y lies in the symmetry plane m and

Figure 7 . 2

 72 Figure 7.2Two non-equivalent Fe 3+ sites and their oxygen environments in the absence of distortions (full red and empty blue circles for Fe 1 and Fe 2 , respectively). The Cartesian coordinate axes are directed as in Figure7.1. The x -axis is perpendicular to the plane of the figure and points towards the reader. The nearest-neighbouring "top" and "bottom" oxygen triangles for both iron sites have the same z-coordinate and different x-coordinates.

and 5 b

 5 cf. Chapter 4, "EPR of iron-gallium borate single crystals with low x": parameters are very small, see Chapter 4, "EPR of iron-gallium borate single crystals with low x", so, here they are neglected. On the other hand, the spin Hamiltonian for iron ions in the near-surface layer should include terms describing the concomitant lowering of symmetry. Besides, as far as Fe 3+ ions in FeBO 3 are subject to a strong exchange field, we can use the mean field approximation and substitute an effective exchange field k

  the only occupied spin level is the lowest one (with 5 2

  the S subscript refers to the parameters for the near surface layer in the presence of distortions. The parameters in eq. (7.13) depend on relative distortions, the rhombohedron shown in Figures 7.1 and 7.2 and r a  is an absolute distortion of a r . The positive and negative  values correspond to contractions and extensions, respectively.In the absence of structural distortions, i.e. at 0   , the crystal-field contribution to the density of the magnetocrystalline anisotropy energy of the near-surface layer would be the same as for an analogous layer in the crystal volume:

  as the difference between the parameters of matching symmetry terms in eqs.(7.13) and(7.14).

Figure 7

 7 Figure 7.3 Values of cf S a , dip S a [7.6] and S a vs.  at 0 K.

Figure 7 . 4

 74 Figure 7.4 Calculated

2 Figure 7

 27 Figure 7.5 Calculated dependence of

  Synthesis of FeBO 3 single crystals and determination of their crystal structure were first reported by Bernal et al.[1.1]. Iron borate has rhombohedral calcite-type structure with point

	1. Crystal and magnetic
	structure of iron borate
	1.1 Crystal structure
	group symmetry 3d D and the space group 6 3d D in Schönflies notation. The stereographic
	projection of its structure is shown in Figure 1.1. Two-fold axes, perpendicular to three-fold
	axis 3 С , and planes of symmetry are denoted as 2

С and m , respectively. Structural parameters have been later refined by Diehl et al.

[1.2, 1.3]

. This space group has the R-type Bravais lattice; thus, lattice parameters can be specified in both hexagonal and rhombohedral settings, see Table

1

.1. The relations between hexagonal and rhombohedral lattice parameters are:

Table 1 . 1

 11 Lattice parameters of FeBO 3[1.3].

	Hexagonal set	Rhombohedral set
	h a 	  4.626 1	Å	r a 	5.52	Å
	h c 	  14.493 6	Å	r  	49.54 
	h V 	268.596	3 Å *

Table 1 . 2

 12 Bernal et al. [1.1] and Diehl et al. [1.3] quote h 0.2900 Fractional coordinates of atoms in the unit cell of FeBO 3

	x 	while Diehl [1.2]

Table 1 . 3

 13 Interatomic distances and bond angles in FeBO 3[1.3].

	Fe-O	2.028(1) Ǻ
	B-O	1.379(2) Ǻ
	Fe-Fe	3.601(0) Ǻ
	O-Fe-O	91.82(5) and 88.18(5)°

Table 1 . 4

 14 Definition of various magnetic vectors.

  The relevant condition of occurrence of critical points of this function is vanishing of the gradient E . The components of the latter are as follows:

								E	,  
	function.  2 eff cos a dl	4 cos	2	 1 sin sin 3	4 6 sin cos cos 6 4 el	l	2	sin

  E, if in this point

																		det	  H		0	and
		2	E			2		0 or		2	E			2		0	, respectively.
				With eq. (1.19) only three different cases can occur where	  ,   E 		0 :
	(i)				0
	  H . Obviously, (ii) 2 4 eff det a l  while 2 2 2 eff a l      E 1 2    : l is contained in the basal plane, so that eqs. (1.20) and (1.21) reduce to
																	4	sin 3	;	1	6 6 sin 6
																		sin

  : in this point, l would be parallel to 3 С , and m vanishes, cf. eqs.

(1.18)

and (1.14), i.e., we would deal with an easy-axis antiferromagnet. However, iron borate is an easy-plane antiferromagnet

[1.3]

; therefore, this case does not occur. Note that in this point, E would be at a minimum at this point, if eff a were negative, which is not the case in iron borate, cf. Table

1

.5.

Table 1

 1

	.5. E will be at a minimum at

3 single crystals was reported by Bernal et al. [2.6]. Later, FeBO 3 single crystals were obtained by cooling from 860 to 670 °C a solution melt with molar component ratio suggested by Bernal et al. It was concluded that the crystal

  growth should be carried out in the temperature range 860670 °C, as a further decrease of temperature leads to decreasing crystal quality[2.7]. Boron-lead solvent B 2 O 3 -PbO-PbF 2 for synthesizing FeBO 3 was first used by Le Craw et al.[2.8]. Mixing of the solution melt and crystal synthesis on the seed holder were used by Bezmaternykh et al.[2.9].

  .1.

	It includes the following steps:
	 determining suitable charge compositions and temperature modes (by DTA and probe
	methods);
	 dehydrating the reagents;
	 weighting the charge;
			Determining charge
			compositions
		Determining temperature modes
			Dehydrating the reagents
			Weighting the charge
			Obtaining a homogeneous
			solution melt in a crucible
			Installing the crucible in a
			crystallization furnace
		Crystal synthesis according to the
		predetermined temperature mode
		Extracting the crystals from the Dissolving the remnants of
			crucible solution melt
		─	Qualitatively
			characterizing the
			samples
			+
			Synthesis methodology
	Figure 2.1	General scheme for the synthesis of Fe x Ga 1-x BO 3 single crystals by
	solution in the melt technique.

Table 2 . 1

 21 2 O 3 -Fe 2 O 3 -B 2 O 3 -PbO-PbF 2 system. Here the crystal-forming reagents are Ga 2 O 3 , and Fe 2 O 3 , PbO and PbF 2 serve as solvents and B 2 O 3 is both crystal-forming reagent and solvent. Examples of charge compositions used in the synthesis are shown in Table 2.1. Charge compositions used for synthesizing Fe x Ga 1-x BO 3 crystals with different x values.

		Ga 2 O 3	Fe 2 O 3	B 2 O 3	PbO	PbF 2
	x
				mass %
	0.00	18.60	0.00
	0.02	18.52	0.08
				42.40	27.30	11.70
	0.20	16.99	1.61
	0.30	14.49	4.11
	1.00	0.00	5.73	51.23	29.31	13.73

Table 3 . 2

 32

				x
	T, K	0.34	0.65	0.75	0.85	1.0
	4	6.50±0.15	55.55±0.60 72.85±0.45	78.4±0.5	101.2±0.1 [3.4, 3.6] 99.3±0.5
	45	12.25±0.25
	65	10.85±0.15
	77		59.2±0.4	74.10±0.25	88.2±3.0	99.3±0.2 [3.4, 3.6] 98.7±0.5
	130		56.50±0.25
	170		51.6±0.2		74.55±1.50
	190		47.1±0.1	64.0±0.2
	200		44.20±0.15			89.7±0.1 [3.4, 3.6]
	230		29.9±0.2
	250				54.0±0.5
	260			42.5±0.5
	273					72.0±0.1 [3.4, 3.6]
	290			28.5±0.5
	293				24.9±0.2
	300				15.1±0.3	62.0±0.5 [3.4, 3.6]

D H (in kOe) for different x at different temperatures. A detailed study of the temperature dependence of D H has been carried out for x = 0.65 crystal, see Figure 3.14. In cooling from the Néel temperature, D H increases, passes through a maximum at ca. 80 K and then decreases. A similar behaviour is observed for all crystals with

Table 3 . 3

 33

				x
	T, K
		0.34	0.65	0.75	0.85	1.0
	4	0.75±0.45	5.20±0.45	6.02±0.45	4.3±0.5	4.9±0.2 [3.4, 3.6]
	45	-0.55±0.65
	65	-0.25±0.50
	77		4.90±0.30	5.00±0.15	3.9±1.2	4.25±0.25 [3.4, 3.6] 6.543 ±0.032
	130		4.2±0.1
	170				1.65±1
	190			0.7±0.2
	200		1.65±0.10			1.6±0.2 [3.4, 3.6]
	230		-0.3±0.5
	250				0.70±0.55
	260			-1.7±1

2

H  (in kOe 2 ) for different x at different temperatures.

Table 3 . 4

 34 Best-fit parameters in eq.(3.4)

							Parameter	Value, in kOe 2
								H 	2		6.55±0.05
							E hex H H	0.0542  0.0013
								p		1.13 ±0.05
				From E hex H H and E H , cf. Tables 3.1 and 3.4, we get	H	hex 1.826 10  	5 	kOe; then, using
	the definition of hex H	given in Table 3.1, we get the effective constant of hexagonal anisotropy
	e	exp eff		0.936 Jm	3 	. The	H	hex	0.9 10	5

   

kOe value at 77 K, reported earlier by Doroshev et al.

[3.5], is negative and about twice as small in absolute value as our result. Such a discrepancy is mainly due to different definitions of hex H and  used by these authors.

  After specifying the sign of D , its absolute value as well as exact values of a and F could be determined by accurate computer simulations of the experimental EPR spectra at different orientations of B , e.g., see Figures 4.10 and 4.11. As the result, two different best-fit parameter sets have been obtained, as given in Table4.1. The set I parameters are in a reasonably good agreement with those obtained by Lukin et al.[4.6] and Seleznev[4.7] for a crystal of nominal composition Fe 0.005 Ga 0.995 BO 3 studied by EPR at Q-and V-bands (ca. 36 and 75 GHz, respectively).

				0.564		0.568	0.572	0.576	0.58
							Magnetic field (T)
			0.58		0.584	0.588	0.592
								Magnetic field (T)
	Figure 4.9	Zooms of computer-generated (top) and experimental (bottom) EPR
	spectra for	 	10	and	 	278	° at 4 (continuous, red) and 30 K (dashed, green). The
	spectra intensities have been multiplied by the absolute temperature.

Table 4 . 1

 41 Spin Hamiltonian parameters for Fe 3+ ions in GaBO 3 * .

						This work	Lukin et al. [4.6],
		Parameter
						Set I	Set II	Seleznev [4.7]
			D	**	0.10320.0005	0.09890.0049
			a	**	0.01580.002	0.01580.002	0.01460.002
			F	**	0.00520.002	0.03680.002	0.00520.003
			, deg	24	36	24
	*	g 	2.0023
	**	-1

  For Fe 3+ in sixfold coordination, several second-rank intrinsic parameter values are quoted in the literature. For the reference distance 0 2.101 -1 value has been assumed byAcıkgöz et al. [4.13, 4.14]. The latter value has been adopted in the present case, as far as taking 2 0 b  would result in a negative D value, in disagreement with the experimental results. Figure4.12 shows that for this choice the agreement between the experimental and theoretical D values is attained with the second-rank power exponent 2 8.7 t  , in reasonably good accord with the value 2 8 t  usually quoted for Fe 3+ in oxygen environment[4.11, 4.12]. The calculated E values are close to zero, in accordance with the experimental results. Figure 4.12 D values vs. the power exponent 2 t . The dashed line is the D value determined from the experimental data, see Table 4.1.As only few references to the fourth-rank intrinsic parameter values could be found in the literature[4.12, 4.15, 4.16], a large range of values has been tested for both4

	distance in MgO [4.11]), most often 2 b  	0.412	cm -1 is taken [4.11, 4.12]. Meanwhile, a positive
	2 b 	0.412
			0.12
			0.1
		cm -1	0.08
		D,
			0.06
			0.04
			-10	-5	0	5	10
					t 2
						b and 4 t parameters.
	No consistent results could be obtained with negative 4 b values.
					r 	Å (corresponding to the average Fe-O

cm

Table 4 . 2

 42 Tesseral coordination factors used in eq. (4.34); , and x y z are the corresponding components of n (the unit vector of the direction of B ).

	l	m												C	lm				l	m	C	lm
		0	6 2	2 z  (3	1)						0	4 231	(231 z	6		315 z	4		105 z	2		5)
		1	3 2zx																			1	6 11(33 z	4		30 z	2		5) zx
	2	1	3 2zy																			1	6 11(33 z	4		30 z	2		5) zy
		2	3 2	(	x	2				y	2	)					2	8 21 110	(33 z	4		18 z	2		1)(	x	2		y	2	)
		2	3 2xy																			2	4 21 110	(33 z	4		18 z	2		1) xy
		0	70 4	(35 z	4			30 z	2		3)	3	4 21 110	(	x	2		2 3 )(11 y z	2		3) zx
		1	5 7(7 z	2		3) zx			6	3	4 21 110	(3 x	2		y	2	)(11 z	2		3) zy
		1	5 7(7 z		2		3) zy			4	4 21 33	(	x	4		6	2 2 x y		y	4	)(11 z	2		1)
		2	7 5 (7 2 z	2		1)(	x	2		y	2	)	4	21 33(	x	2		y	2	)(11 z	2		1) xy
	4	2	5 14(7 z	2		1) xy		5	4 231 6	(	x	4		10	2 2 x y		4 5 ) y zx
		3	35(	x	2		2 3 ) y zx			5	4 231 6	(5	x	4		10	2 2 x y		4 y zy )
		3	35(3 x	2		2 y zy )			6	8 231 2	(	x	4		14	2 2 x y		4 5 )( y x	2		y	2	)
		4	35 2 4	(	x	4		6	2 2 x y		y	4	)	6	4 21 110	(	x	2		2 3 )(11 y z	2		3) zx
		4		35 2(	x	2			2 y xy )

Table 5 . 1

 51 Best-fit NMR Parameters for11 B isotope in Fe x Ga 1-x BO 3 .

	x								0.00	0.01	0.02
	iso , ppm 	24.70.1	24.50.1	27.60.1
	0 	z , MH	2.840.0	2.860.02	3.160.02
	 	Hz , M	0.040.0	0.040.01	0.080.01
	0 				0.510 to 0.523
						0.0110.	0.0150.0	0.0270.005
								0.10.1	0.20.1	0.30.1
	C	Q	Hz , M	2.840.0	2.860.02	3.160.02
		C	Q	,	MHz	0.040.0	0.040.01	0.080.01
								0.0 to 0.023
	 				0.0190.	0.0260.	0.0470.009
		Q C 	0.10.1	0.20.1	0.30.1

Table 5 .

 5 2 NMR Parameters for 71 Ga isotope in Fe x Ga 1-x BO 3 .

							x	0.00	0.01	0.02
	iso , ppm 	6.900.1	6.100.1	2.800.1
					 	0.0 to 0.4	0.0 to 0.8	6.51.0
	0 	z , MH	4.770.02	4.780.02	4.820.02
	 	Hz , M	0.020.005	0.060.005	0.060.005
					0 	0.510 to 0.523
							0.0130.005	0.0180.005	0.0500.005
								00.1	0.10.1	0.20.1
	C	Q	Hz , M	4.770.02	4.780.02	4.820.02
		C	Q	,	MHz	0.020.005	0.060.005	0.060.005
								0.0 to 0.023
					 	0.0220.005	0.0310.005	0.0870.005
					Q C 	00.1	0.10.1	0.20.1

  Taylor expansions of the field line equation in the computer code have been obtained by expanding in the Taylor series the ratio z B over B  . For the model of a current loop this

	results in
	z	2	1	3cos	2	1	1 4	 5 cos	2	 3	2	5 32	 7 cos	4	6 cos	2	 1	4
		3	sin 2	5 512		203cos	6	273 cos	4		65 cos	2	 5	6

  As in the previous case, the approximate expression of E can be expressed in terms of the Legendre polynomials of the scalar product r  e e m up to the fourth order:

	E	m										.	(6.89)
								e e m				e e m
		2	4	0	3 ²		2	4	2	6	4	

Table 6 . 1

 61 Crystal field and dipole-dipole contributions to the magnetocrystalline anisotropy constants of iron borate at 0 K.

				Dipole-dipole contribution
	Constants,	3 Jm 	Crystal field contribution	Circular current loop	Pair of fictitious magnetic charges

  Ga 1-x BO 3 , we have put forward a computer code implementing the lattice-sum method as for FeBO 3 . With this aim in mind, we have modelled a diamagnetically diluted crystal lattice using the Monte Carlo technique[6.23].

	With the 1 R value we get:
				e	dip	-3 12.2 Jm ; 	d	dip	3 7.57 10 Jm  	-3	,	(6.98)
	and with 2 R value we get:
					e	dip	FeBO 3 -3 e 0.215 Jm ; eff e  	 d	FeBO 3 2 eff a d dip 1 4   1 10 Jm 3	-3	.	(6.94) (6.99)
	The experimental value exp eff e 6.3.2.5 Dipole-dipole contribution to magnetocrystalline 3 0.936 Jm   has been determined from the AFMR experiments
	at 77 K cf. Chapter 3, "Electron magnetic resonance of iron-gallium borate single crystals with anisotropy constants of iron-gallium borates
	0.2≤x≤1". In the following, we assume that	e	eff		e	exp eff	. Substituting exp eff e	in (6.94) and taking
	into account eqs. (6.2) and (6.3), we get: In order to calculate the dipole-dipole contribution to the magnetocrystalline anisotropy
	constants dip e	and	exp eff d for mixed borates Fe x cf dip e e e    dip	1 4	a	 FeBO 3 cf d	 	FeBO 3 FeBO 3 E dip 2 d D  2	.	(6.95)
	For FeBO 3 a	we have also used the experimental value FeBO 3 exp a	5 3.2 10 Jm  	3 	determined by
	AFMR at 77 K [6.2, 6.5]. FeBO 3 D	and	FeBO 3 E	at 77 K have been calculated from experimental values
	of corresponding effective fields and sublattice magnetization, see Table 3.1 in Chapter 3,
	"Electron magnetic resonance of iron-gallium borate single crystals with 0.2≤x≤1". Substituting
	to eq. (6.95) these values as well as those given in Table 6.1, for the model of a circular current
	loop we get:
							3 7.47 10 	R	4		405.4 R	2	3.2 0.936  	,	(6.96)
	yielding two positive solutions:	1 R		0.2189 and	2 R		0.0797Å.
	For a pair of fictitious magnetic charges we get:
									853	d	4		135	d	2	3.2 0.936  	.	(6.97)
	Obviously, the latter equation can have only complex solutions; therefore, this model is not
	applicable in the actual case.
	In order to assess the plausibility of 1 R	and	2 R values, they should be compared with the
	ionic radius i R of Fe 3+ ; indeed, we can reasonably infer that the effective dipole size should be of
	the same order of magnitude as the size of the physical object producing the corresponding
	dipole moment. For high-spin Fe 3+ in sixfold coordination i 0.645 R 	Å [6.22]; therefore the 1 R
	value seems to be a more realistic estimate than 2 R , the latter value being an order of magnitude
	smaller than i R . Nevertheless, we have chosen to calculate the dipole-dipole contributions to the
	magnetocrystalline anisotropy constants of FeBO 3 at 77 K with both values.

   and  are, respectively, the polar angle with respect to C 3 and the azimuthal angle with respect to 2 C of the antiferromagnetic vector l , see Chapter 1, "Crystal and magnetic structure of iron borate", in the near-surface layer.

																							10 14 face:
				dip			1 dip	2 sin cos 	2	  	dip	2	2 sin sin 	2	  	dip	3	cos	2	  	dip	4	sin cos sin   	(7.1)
	where		1 dip	5 2.2 10 ,    		dip	2	5 0.8 10 ,   		3 dip	5 1.6 10 and    		4 dip	3.9 10  		5	in	-2 Jm (at 0 K),
	Minimizing dip  with respect to				and	 , the equilibrium orientation of l in the near-
	surface layer is [7.5]:
														0 		2.64 and	0 		2 		in rad		.	(7.2)

  Table 1.4 and eq. (1.14) in Chapter 1, "Crystal and magnetic structure of iron borate"; and  is the angle between H and 2 C . Taking into account that 0

								M	300 M		1.47	[7.7] and
	the values of	D	and	E quoted in Table 1.5, we get	M 	12G	at room temperature. The
	equilibrium orientation of l in the near-surface layer will be determined by the minimum of the
	sum	   	dip	, and as far as this quantity is independent of , see eqs. (7.5) and (7.6), we just
	have to solve the equation:
						dip				dip

Table 7 . 1

 71 Crystal field contribution to the surface anisotropy constants, in

-3

Table 7 . 2

 72 Equilibrium orientation of l in the near-surface layer in the absence of H .

Table 7

 7 .2, eqs.(7.13) and(7.14), for cf  , defined as Comparing eqs.(7.16) and (7.5), we get the crystal field contribution to the surface anisotropy constant:  dependence for the layer containing Fe 3+ ions (D in Figure7.2) has been previously calculated at 300 K[7.6]. As the dipole-dipole interaction energy is proportional to the square of magnetization, e.g. see eq. (6.36) in Chapter 6, "Magnetocrystalline anisotropy of iron borate and iron-gallium borates",

	cf 	S		cf 	V	, we get:
									cf 	10  	2 2 NB	S	sin	2		.	(7.16)
									a	S	cf	10  	2 2 NB	S	,	(7.17)
	and the total surface anisotropy constant can be expressed as
									a	S		a	S	dip		a	S	cf	.	(7.18)
			The	a	S	dip	 

† These results have been obtained in collaboration with A. Drovosekov and N. Kreines in the Kapitsa Institute for Physical Problems, Moscow.

Acknowledgements

First of all, I express my cordial appreciation to both my scientific supervisors, Professors Janis Kliava and Mark Strugatsky. You have directed my research with a great efficiency, kindness and competence. You have given me more than knowledge, more than skills and experienceyou have taught me to think. Owing to you, I fell in love with science, and all years of my PhD studies were interesting, enjoyable, exciting, sometimes intriguing and very fruitful. Thank you for

Summary of results

The most significant results of this thesis are:

1. Working out a solution in the melt synthesis route for iron-gallium borates Fe x Ga 1-x BO 3 , and preparing and characterizing high-quality single crystals in the whole range of compositions.

2. Electron magnetic resonance monitoring of iron-gallium borates in the transformation from magnetically ordered to paramagnetic state: antiferromagnetic resonance (AFMR)coexistence of AFMR and cluster magnetic resonance (CMR) -CMR -electron paramagnetic resonance (EPR). Plausible observation of the Morin's transition in irongallium borates.

3. Determining temperature and concentration dependences of magnetic characteristics of iron-gallium borates, namely, the Néel temperature and Dzyaloshinskii-Moriya field.

4. Full parametrization of EPR spectra of Fe 3+ ions in crystals with low iron contents by means of detailed computer simulations using laboratory-developed codes for the conventional, general and "tesseral" spin Hamiltonians, taking into account parameter distributions caused by local disorder.

Computer modelling of Magic Angle

Spinning NMR spectra of 11 B and 71 Ga in Fe x Ga 1-x BO 3 with low iron contents, applying different distribution functions of quadrupole parameters and chemical shift in order to describe local disorder in the crystals.

6. Calculating magnetocrystalline anisotropy, uniaxial and basal, of iron-gallium borates as a sum of crystal field and dipole-dipole -in the model of extended dipoles -contributions; estimating the size of magnetic dipole related to Fe 3+ ion.

7. Extending the theory of surface magnetism of iron borate by including, besides the dipoledipole contribution, the crystal field contribution to surface magnetocrystalline anisotropy; modelling structural distortions in the near-surface layer of the crystals.

Author contribution

Articles in international journals and conference proceedings: