BRAGA, João Philipe Macedo. Mecânica Quântica Não-aditiva. 2015. 62 f. Tese (Doutorado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2015. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2015-11-05T20:20:37Z No. of bitstreams: 1 2015_tese_jpmbraga.pdf: 564039 bytes, checksum: 2f103f04eaf29689dc72e53d169b3906 (MD5) / Approved for entry into archive by Edvander Pires(edvanderpires@gmail.com) on 2015-11-05T20:20:56Z (GMT) No. of bitstreams: 1 2015_tese_jpmbraga.pdf: 564039 bytes, checksum: 2f103f04eaf29689dc72e53d169b3906 (MD5) / Made available in DSpace on 2015-11-05T20:20:57Z (GMT). No. of bitstreams: 1 2015_tese_jpmbraga.pdf: 564039 bytes, checksum: 2f103f04eaf29689dc72e53d169b3906 (MD5) Previous issue date: 2015 / In this thesis, we study the nonadditive quantum mechanics (NAQM), which is a theory developed from first principles in order to understand the effects of the space metric in the quantum theory. In non-Euclidean spaces, the translation of length ∆x does not necessarily take a particle from the position x to x + ∆x. The result of this translation depends on the metric. This is the starting point for the development of the NAQM. Through a redefinition of the translation operator, we obtain new commutation relations between the position operator and the momentum operator, and a Schrödinger-like equation which describes the time evolution of the state of a particle. We show that this equation, with appropriate boundary conditions, can be seen as a Sturm-Liouville problem, ensuring that the energies of the particle are real and that the eigenstates of the hamiltonian are orthonormal and form a basis in the space of the states. In spite of these modifications, we show the determinism in the time evolution, the superposition principle and the local and global probability conservation remain valid. On the other hand, we generalize the Ehrenfest theorem, showing that, for the average values of the physical quantities, the NAQM is identical to the classical mechanics in a non-inertial reference frame, and we demonstrate the existence of a nonzero minimum uncertainty for the momentum. Besides, we investigate, classically as well as quantically, the dynamical effects of the metric in the time evolution of a free particle. In order to perform the quantum simulation, we adapt the split operator technique to solve numerically the new Schrödinger equation. Lastly, we explore the possibility of mapping of several physical problems of different nature through the arising of an effective potential which appears due to a simple change of coordinates. / Nesta Tese, apresentamos a mecânica quântica não-aditiva (MQNA), uma teoria desenvolvida a partir de primeiros princípios com o intuito de entender quais são os efeitos da métrica do espaço na teoria quântica. Em espaços não-euclideanos, uma translação de comprimento ∆x não leva necessariamente uma partícula de uma posição x para outra x + ∆x. O resultado dessa translação depende da métrica. Esse é o ponto de partida para o desenvolvimento da MQNA. Através de uma redefinição do operador translação, obtivemos novas relações de comutação entre os operadores posição e momentum e uma equação tipo equação de Schrödinger que descreve a evolução temporal do estado da partícula. Mostramos que essa equação, juntamente com certas condições de contorno, pode ser vista como um problema de Sturm-Liouville, garantindo que as energias da partícula são reais e que os autoestados da hamiltoniana são ortonormais e formam uma base no espaço dos estados. Apesar dessas modificações, mostramos que continuam válidos o determinismo na evolução temporal, o princípio da superposição e a conservação local e global da probabilidade. Em contrapartida, generalizamos o teorema de Ehrenfest, mostrando que, para os valores médios das grandezas físicas, a MQNA cai na mecânica clássica em um referencial não inercial, e demonstramos a existência de uma incerteza mínima diferente de zero no momentum. Além disso, investigamos, tanto classicamente como quanticamente, os efeitos dinâmicos da métrica na evolução temporal de uma partícula livre. Para realizar a simulação quântica tivemos que adaptar a técnica split operator para resolver numericamente a nova equação de Schrödinger. Por fim, exploramos a possibilidade de mapearmos diversos problemas físicos de naturezas distintas através do surgimento de um potencial efetivo, consequência de uma simples mudança de coordenadas.

The Feynman path integral formulation of quantum mechanics is a path integral representation for a propagator or probability amplitude in going between two points in space-time. The wave function is expressed in terms of an integral equation from which the Schrodinger equation can be derived. On taking the limit h — 0, the method of stationary phase can be applied and Newton's second law of motion is obtained. Also, the condition the phase vanishes leads to the Hamilton - Jacobi equation. The secondary objective of this paper is to study ways of relating quantum mechanics and classical mechanics. The Ehrenfest theorem is applied to a particle in an electromagnetic field. Expressions are found which are the hermitian Lorentz force operator, the hermitian torque operator, and the hermitian power operator.

Nesta tese de doutorado apresentamos os resultados de quatro tópicos referentes a estudos de propriedades elétricas que são interpretados com o auxílio da Teoria Quântica de Átomos em Moléculas (QTAIM). No primeiro deles, foram calculados momentos de dipolo e suas derivadas através de um novo formalismo de divisão de átomos em moléculas, baseado em campos de forças de Ehrenfest (CFE), sendo que estes dados são comparados com aqueles advindos da QTAIM. Desta forma, um modelo alternativo de partição em carga - fluxo de carga - fluxo de dipolo (CFCFD) é discutido para derivadas do momento dipolar. Os resultados gerais obtidos pelo formalismo CFE foram satisfatórios em termos quantitativos, embora QTAIM ainda fornece uma descrição mais apropriada destes fenômenos das polarizações atômicas e de suas variações durante vibrações. Na sequência, investigamos os Atratores Não Nucleares (NNAs), que são identificados através de uma análise QTAIM da densidade eletrônica. O nosso intuito foi descobrir novas moléculas que apresentam essa peculiaridade, bem como encontrar padrões entre os casos encontrados que permitam contribuir para o entendimento dos fatores que levam ao seu aparecimento. Para isso trabalhamos com moléculas diatômicas homonucleares de elementos representativos com números atômicos que variavam de Z=1 até Z=38 e moléculas heteronucleares formadas pela combinação dos mesmos. Os nossos dados mostram que NNAs podem ser encontrados em alguns pontos dentro da faixa de distâncias internucleares investigada para quase todos os sistemas diatômicos homonucleares, exceto para as moléculas de Hidrogênio, Hélio e Estrôncio. Por sua vez, encontramos trinta casos de NNAs em sistemas heteronucleares, muitos dos quais ainda inéditos na literatura. Descobrimos também que a polarizabilidade atômica aparentemente tem um papel importante na explicação dos casos encontrados. Tratamos também de moléculas contendo interações fracas como as de Van der Waals (moléculas tri-atômicas contendo um gás nobre ligado a um composto diatômico iônico) de modo a investigar os valores de dipolos atômicos QTAIM de uma maneira mais direta, ou seja, via comparação com um modelo simples para estes compostos. Por fim, estudamos moléculas em estados excitados, sendo que nossa análise focou em dois casos peculiares (CO e de CF2N2) que apresentam momento de dipolo nulo no estado fundamental, enquanto valores significativos desta propriedade são observados em seus primeiros estados excitados. Desta forma, QTAIM foi fundamental para compreender como o processo de excitação pode levar à mudanças tão significativas em tais propriedades elétricas. / In this PhD thesis we present the results of four different topics that refer to a study of electric properties interpreted with The Quantum Theory of Atoms and Molecules (QTAIM). First, dipole moments and their derivatives were calculated from a new formalism based on Ehrenfest Force Fields (EFF) and a comparison with data from QTAIM is carried out. Therefore, the Charge-Charge Flux-Dipole Flux (CCFDF) model was discussed for the dipole moment derivatives. The results from EFF were satisfactory in quantitative terms although QTAIM still seems to be better for the description of atomic polarization and its variations during vibrations. In the sequence, we investigated the Non-Nuclear Attractors (NNAs) that could be identified with the QTAIM formalism. Our intention was to discover new molecules that present this peculiarity, as well as to find trends among these cases that allow contributing for the understanding of the factors that lead to their appearance. For this purpose, we selected homonuclear diatomic molecules of elements presenting atomic numbers ranging from Z=1 to Z=38 and heteronuclear diatomic molecules containing these same elements. Our data shows that NNAs could be found in almost every homonuclear molecule expect by the systems formed by Hydrogen, Helium, and Strontium. On other hand, we have found 30 cases of NNAs in heteronuclear molecules, many of them seen for the first time. We also have noticed that the atomic polarizabilities play a main role in the understanding of these cases. We also treated molecules containing weak Van der Waals interactions (triatomic complexes presenting a noble gas bonded to a diatomic ionic molecule) in order to investigate the atomic dipole values obtained with QTAIM in a direct way, that is, by means of a comparison using a simple model for this kind of bonding. Finally, we studied molecules in excited states. Our focus was in two peculiar cases (CO and CF2N2), which present null dipole moments in their ground states but exhibit significant dipole moment values in their first excited states. Therefore, QTAIM was fundamental to understand how the excitation process can lead to important changes in electric properties.

Nesta tese de doutorado apresentamos os resultados de quatro tópicos referentes a estudos de propriedades elétricas que são interpretados com o auxílio da Teoria Quântica de Átomos em Moléculas (QTAIM). No primeiro deles, foram calculados momentos de dipolo e suas derivadas através de um novo formalismo de divisão de átomos em moléculas, baseado em campos de forças de Ehrenfest (CFE), sendo que estes dados são comparados com aqueles advindos da QTAIM. Desta forma, um modelo alternativo de partição em carga - fluxo de carga - fluxo de dipolo (CFCFD) é discutido para derivadas do momento dipolar. Os resultados gerais obtidos pelo formalismo CFE foram satisfatórios em termos quantitativos, embora QTAIM ainda fornece uma descrição mais apropriada destes fenômenos das polarizações atômicas e de suas variações durante vibrações. Na sequência, investigamos os Atratores Não Nucleares (NNAs), que são identificados através de uma análise QTAIM da densidade eletrônica. O nosso intuito foi descobrir novas moléculas que apresentam essa peculiaridade, bem como encontrar padrões entre os casos encontrados que permitam contribuir para o entendimento dos fatores que levam ao seu aparecimento. Para isso trabalhamos com moléculas diatômicas homonucleares de elementos representativos com números atômicos que variavam de Z=1 até Z=38 e moléculas heteronucleares formadas pela combinação dos mesmos. Os nossos dados mostram que NNAs podem ser encontrados em alguns pontos dentro da faixa de distâncias internucleares investigada para quase todos os sistemas diatômicos homonucleares, exceto para as moléculas de Hidrogênio, Hélio e Estrôncio. Por sua vez, encontramos trinta casos de NNAs em sistemas heteronucleares, muitos dos quais ainda inéditos na literatura. Descobrimos também que a polarizabilidade atômica aparentemente tem um papel importante na explicação dos casos encontrados. Tratamos também de moléculas contendo interações fracas como as de Van der Waals (moléculas tri-atômicas contendo um gás nobre ligado a um composto diatômico iônico) de modo a investigar os valores de dipolos atômicos QTAIM de uma maneira mais direta, ou seja, via comparação com um modelo simples para estes compostos. Por fim, estudamos moléculas em estados excitados, sendo que nossa análise focou em dois casos peculiares (CO e de CF2N2) que apresentam momento de dipolo nulo no estado fundamental, enquanto valores significativos desta propriedade são observados em seus primeiros estados excitados. Desta forma, QTAIM foi fundamental para compreender como o processo de excitação pode levar à mudanças tão significativas em tais propriedades elétricas. / In this PhD thesis we present the results of four different topics that refer to a study of electric properties interpreted with The Quantum Theory of Atoms and Molecules (QTAIM). First, dipole moments and their derivatives were calculated from a new formalism based on Ehrenfest Force Fields (EFF) and a comparison with data from QTAIM is carried out. Therefore, the Charge-Charge Flux-Dipole Flux (CCFDF) model was discussed for the dipole moment derivatives. The results from EFF were satisfactory in quantitative terms although QTAIM still seems to be better for the description of atomic polarization and its variations during vibrations. In the sequence, we investigated the Non-Nuclear Attractors (NNAs) that could be identified with the QTAIM formalism. Our intention was to discover new molecules that present this peculiarity, as well as to find trends among these cases that allow contributing for the understanding of the factors that lead to their appearance. For this purpose, we selected homonuclear diatomic molecules of elements presenting atomic numbers ranging from Z=1 to Z=38 and heteronuclear diatomic molecules containing these same elements. Our data shows that NNAs could be found in almost every homonuclear molecule expect by the systems formed by Hydrogen, Helium, and Strontium. On other hand, we have found 30 cases of NNAs in heteronuclear molecules, many of them seen for the first time. We also have noticed that the atomic polarizabilities play a main role in the understanding of these cases. We also treated molecules containing weak Van der Waals interactions (triatomic complexes presenting a noble gas bonded to a diatomic ionic molecule) in order to investigate the atomic dipole values obtained with QTAIM in a direct way, that is, by means of a comparison using a simple model for this kind of bonding. Finally, we studied molecules in excited states. Our focus was in two peculiar cases (CO and CF2N2), which present null dipole moments in their ground states but exhibit significant dipole moment values in their first excited states. Therefore, QTAIM was fundamental to understand how the excitation process can lead to important changes in electric properties.

L'objectif de cette thèse est de traiter trois problèmes relatifs aux réseaux de grande taille. Les outils utilisés à cette fin sont issus des probabilités et plus spécifiquement de la théorie des files d'attente. En plus d'améliorer la compréhension des systèmes étudiés, les travaux réalisés dans cette thèse ont permis de prouver des résultats théoriques nouveaux ainsi que d'illustrer certains phénomènes probabilistes. Dans le Chapitre II, un modèle de réseau à partage de bande passante est étudié. Contrairement à ce qui avait été étudié dans la littérature, les utilisateurs n'utilisent pas de contrôle de congestion. On suppose qu'ils envoient des données avec un débit maximum et protègent leur transmission à l'aide d'un mécanisme basé sur des codes correcteurs d'erreur. Le modèle obtenu est analysé pour deux topologies de réseaux spécifiques : les réseaux linéaires et les arbres montants. A l'aide de limites fluides, les conditions de stabilité de ces réseaux sont établies. Ces limites fluides donnent lieu à un phénomène intéressant de moyennage stochastique. Ensuite, une autre méthode de renormalisation est utilisée pour prouver que la région de stabilité de ces processus converge vers la région optimale lorsque que les débits maximaux des utilisateurs deviennent infiniment petits par rapport à la taille des liens du réseau. Dans le Chapitre III, on se propose d'étudier CSMA/CA, un algorithme d'accès implémenté dans certains standards de réseaux sans fil. Chaque lien est constitué d'un émetteur et d'un récepteur et un graphe d'interférence modélise les collisions potentielles entre les liens. Les arrivées et les départs de ces derniers sont prises en compte. Une approximation est faite en supposant que la dynamique d'accès au canal est infiniment plus rapide que la dynamique des arrivées et départs de liens. Il est alors établi que le CSMA permet une utilisation optimale des ressources radio dans le cadre des réseaux ad-hoc. Cependant, il est également prouvé que ce même algorithme n'est pas efficace pour les réseaux avec une station de base ; dans ce cas, un biais en faveur des transmissions vers la station de base est observé. A la fin du chapitre, l'hypothèse simplificatrice est discutée. Les deux derniers chapitres de la thèse sont consacrés à l'étude d'un grand système distribué de stockage de données avec pertes. L'objectif est d'estimer la vitesse de perte des fichiers ou la durée de vie d'un fichier donné. Dans le Chapitre IV, c'est le premier point de vue qui est adopté. Le système est considéré de manière globale. Le système est constitué d'un grand nombre de fichiers qui peuvent avoir chacun deux copies au maximum. Chaque copie disparaît au bout d'un temps aléatoire. Un mécanisme centralisé de sauvegarde permet alors de restaurer les copies perdues. Un fichier dont les deux copies ont été détruites est définitivement perdu. Le système est étudié dans le cas limite où le nombre de fichiers tend vers l'infini. Afin de décrire correctement le système, trois échelles de temps différentes sont étudiées. Ralentir le temps permet de comprendre le mécanisme de sauvegarde ; laisser le temps inchangé permet de définir la capacité du système ; accélérer le temps permet d'évaluer la vitesse de perte des fichiers. Le principe de moyennage stochastique est également observé à l'échelle de temps la plus rapide. Dans le chapitre V, le point de vue d'un fichier donné est adopté. Des liens sont établis avec les modèles classiques d'Ehrenfest, issu de la physique statistique, et d'Erlang, issu des télécommunications. Des méthodes basées sur les martingales sont utilisées pour calculer la transformée de Laplace des temps d'atteinte de ces deux processus. Ces transformées permettent alors d'estimer le comportement asymptotique de ces temps d'atteinte et notamment le temps de disparition d'un fichier.

In the present study the results of investigations on polycrystalline MgB2 and on single crystals of YNi2B2C and HoNi2B2C are presented. In particular, measurementes of specific electrical resistance, thermal conductivity, thermoelectric power, and of the linear thermal expansion coefficient were performed. Moreover, the specific heat of polycristalline borocarbide samples was evaluated. From the measured data, the temperature dependencies of the Lorenz number and of the Grueneisen parameter can be determined, also the pressure dependence of the superconducting transition temperature using the Ehrenfest relation. At low temperatures a characteristic deviation of the resistivity from the Bloch-Grueneisen law in the normal state for all investigated substances was observed. A reentrant behaviour in resistivity and thermoelectric power occurs at the antiferromagnetic phase transition of HoNi2B2C. The thermal conductivity of MgB2 below 7 K is dominated by the scattering of phonons at grain boundaries. The absence of both, a maximum of thermal conductivity in the superconducting state, and the change of its slope at the superconducting transition temperature points to the validity of the two-band model that also describes the temperature dependence of specific heat. Measurements of thermoelectric power confirm the different normal-state character of the charge carriers of the investigated superconductors. Diffusion thermopower and phonon drag describe the measured data of all investigated compounds ov a wide range of temperature. The thermal expansion of HoNi2B2C below 10 K is dominated by the magnetic contribution. For all investigated substances the Grueneisen parameter features very large values in selected temperature ranges. In the case of MgB2, its temperature dependence is evidently connected with the properties of the relevant phonon mode. For the borocarbides, the electrical resistance depends very weakly on the crystallographic direction, but in contrast the thermal conductivity does in a quite strong manner. Despite of the antiferromagnetic phase transition in the case of HoNi2B2C, thermoelectric power and thermal expansion show minor anisotropy. / In der vorliegenden Arbeit werden Ergebnisse von Untersuchungen an polykristallinem MgB2 sowie an YNi2B2C- und HoNi2B2C-Einkristallen analysiert. Dafür erfolgten Messungen des spezifischen elektrischen Widerstands, der Wärmeleitfähigkeit, der Thermokraft und des linearen thermischen Ausdehnungskoeffizienten. Zudem wurde die spezifische Wärmekapazität polykristalliner Borkarbide bestimmt und aus den erhaltenen Daten die Temperaturabhängigkeit der Lorenz-Zahl und des Grüneisen-Parameters sowie mittels der Ehrenfest-Relation die Druckabhängigkeit der Sprungtemperatur ermittelt. Bei tiefen Temperaturen findet man im normalleitenden Zustand für alle betrachteten Substanzen ein charakteristisches Abweichen des Widerstands vom Bloch-Grüneisen-Gesetz. Bei HoNi2B2C tritt beim antiferromagnetischen Phasenübergang im Widerstand und in der Thermokraft ein reentrant-Verhalten auf. Die thermische Leitfähigkeit von MgB2 wird unterhalb von 7 K durch die Streuung der Phononen an Korngrenzen bestimmt. Das Fehlen eines Maximums in der Wärmeleitfähigkeit im supraleitenden Zustand und einer Anstiegsänderung bei der Sprungtemperatur liefert einen Hinweis auf die Gültigkeit des Zweibandmodells, mit welchem auch der Temperaturverlauf der Wärmekapazität erklärt werden kann. Messungen der Thermokraft bestätigen den unterschiedlichen Charakter der Ladungsträger im normalleitenden Zustand der untersuchten Supraleiter, wobei Elektronendiffusion und Phonon Drag die Messdaten aller betrachteten Verbindungen in weiten Temperaturbereichen beschreiben. Für HoNi2B2C wird die thermische Ausdehnung unterhalb von 10 K durch den Beitrag der magnetischen Ordnung bestimmt. Der Grüneisen-Parameter weist für alle untersuchten Substanzen in Teilbereichen sehr große Beträge auf. Sein Temperaturverlauf hängt bei MgB2 offenbar mit Eigenschaften der maßgeblichen Phononenmode zusammen. Für die Borkarbide ist die Richtungsabhängigkeit des elektrischen Widerstandes sehr schwach, in der Wärmeleitfähigkeit hingegen recht stark ausgeprägt. Abgesehen vom antiferromagnetischen Phasenübergang bei HoNi2B2C weisen Thermokraft und Ausdehnungskoeffizient eine geringe Anisotropie auf.

In the present study the results of investigations on polycrystalline MgB2 and on single crystals of YNi2B2C and HoNi2B2C are presented. In particular, measurementes of specific electrical resistance, thermal conductivity, thermoelectric power, and of the linear thermal expansion coefficient were performed. Moreover, the specific heat of polycristalline borocarbide samples was evaluated. From the measured data, the temperature dependencies of the Lorenz number and of the Grueneisen parameter can be determined, also the pressure dependence of the superconducting transition temperature using the Ehrenfest relation. At low temperatures a characteristic deviation of the resistivity from the Bloch-Grueneisen law in the normal state for all investigated substances was observed. A reentrant behaviour in resistivity and thermoelectric power occurs at the antiferromagnetic phase transition of HoNi2B2C. The thermal conductivity of MgB2 below 7 K is dominated by the scattering of phonons at grain boundaries. The absence of both, a maximum of thermal conductivity in the superconducting state, and the change of its slope at the superconducting transition temperature points to the validity of the two-band model that also describes the temperature dependence of specific heat. Measurements of thermoelectric power confirm the different normal-state character of the charge carriers of the investigated superconductors. Diffusion thermopower and phonon drag describe the measured data of all investigated compounds ov a wide range of temperature. The thermal expansion of HoNi2B2C below 10 K is dominated by the magnetic contribution. For all investigated substances the Grueneisen parameter features very large values in selected temperature ranges. In the case of MgB2, its temperature dependence is evidently connected with the properties of the relevant phonon mode. For the borocarbides, the electrical resistance depends very weakly on the crystallographic direction, but in contrast the thermal conductivity does in a quite strong manner. Despite of the antiferromagnetic phase transition in the case of HoNi2B2C, thermoelectric power and thermal expansion show minor anisotropy. / In der vorliegenden Arbeit werden Ergebnisse von Untersuchungen an polykristallinem MgB2 sowie an YNi2B2C- und HoNi2B2C-Einkristallen analysiert. Dafür erfolgten Messungen des spezifischen elektrischen Widerstands, der Wärmeleitfähigkeit, der Thermokraft und des linearen thermischen Ausdehnungskoeffizienten. Zudem wurde die spezifische Wärmekapazität polykristalliner Borkarbide bestimmt und aus den erhaltenen Daten die Temperaturabhängigkeit der Lorenz-Zahl und des Grüneisen-Parameters sowie mittels der Ehrenfest-Relation die Druckabhängigkeit der Sprungtemperatur ermittelt. Bei tiefen Temperaturen findet man im normalleitenden Zustand für alle betrachteten Substanzen ein charakteristisches Abweichen des Widerstands vom Bloch-Grüneisen-Gesetz. Bei HoNi2B2C tritt beim antiferromagnetischen Phasenübergang im Widerstand und in der Thermokraft ein reentrant-Verhalten auf. Die thermische Leitfähigkeit von MgB2 wird unterhalb von 7 K durch die Streuung der Phononen an Korngrenzen bestimmt. Das Fehlen eines Maximums in der Wärmeleitfähigkeit im supraleitenden Zustand und einer Anstiegsänderung bei der Sprungtemperatur liefert einen Hinweis auf die Gültigkeit des Zweibandmodells, mit welchem auch der Temperaturverlauf der Wärmekapazität erklärt werden kann. Messungen der Thermokraft bestätigen den unterschiedlichen Charakter der Ladungsträger im normalleitenden Zustand der untersuchten Supraleiter, wobei Elektronendiffusion und Phonon Drag die Messdaten aller betrachteten Verbindungen in weiten Temperaturbereichen beschreiben. Für HoNi2B2C wird die thermische Ausdehnung unterhalb von 10 K durch den Beitrag der magnetischen Ordnung bestimmt. Der Grüneisen-Parameter weist für alle untersuchten Substanzen in Teilbereichen sehr große Beträge auf. Sein Temperaturverlauf hängt bei MgB2 offenbar mit Eigenschaften der maßgeblichen Phononenmode zusammen. Für die Borkarbide ist die Richtungsabhängigkeit des elektrischen Widerstandes sehr schwach, in der Wärmeleitfähigkeit hingegen recht stark ausgeprägt. Abgesehen vom antiferromagnetischen Phasenübergang bei HoNi2B2C weisen Thermokraft und Ausdehnungskoeffizient eine geringe Anisotropie auf.