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From Heisenberg, Slater, and Stoner to Van Vleck

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From Heisenberg, Slater, and Stoner to Van Vleck The story of magnetism: from Heisenberg, Slater, and Stoner to Van Vleck, and the issues of exchange and correlation ∗ Navinder Singh Physical Research Laboratory, Ahmedabad, India. [email protected] July 31, 2018 Abstract This article is devoted to the development of the central ideas in the field of magnetism. The presentation is semi-technical in nature and it roughly follows the chronological order. The key contributions of Van Vleck, Dorfman, Pauli, Heisenberg, and Landau are presented. Then the advent of the field of itinerant electron magnetism starting with the investigations of Bloch and Wigner, and more successful formulation by Slater and Stoner is presented. The physical basis of the Slater-Stoner theory is discussed and its problems are summarized. Then, an overview of the debate between itinerant electron view of Stoner and localized electron view of Heisenberg is presented. Connected with this debate are the issues of exchange interactions. The issues related to the origin of exchange interaction in Stoner model are discussed. We review the "middle-road" theory of van Vleck and Hurwitz–the very first theory which takes into account the electron correlation effects in the itinerant model. We close our presentation with the discussion of the very important issue of strong electron correlation in the itinerant picture. This paper is divided into two parts: In started with the crucial contributions of van the first part, an apparent paradox between Vleck in the post quantum era (from 1926 to the Langevin theory of paramagnetism and 1930s). Van Vleck’s key contributions are pre- arXiv:1807.11291v1 [cond-mat.str-el] 30 Jul 2018 the Bohr - van Leeuwen theorem is presented sented: (1) his detailed quantum statistical me- and explained. Then, the problems in the- chanical study of magnetism of real gases; (2) oretical understanding of magnetism in the his pointing out the importance of the crystal pre-quantum mechanical era (1900 - 1926) are fields or ligand fields in the magnetic behav- presented. The resolution of these problems ior of iron group salts (the ligand field the- ∗ ory); and (3) his many contributions to the Cell: +919662680605; Landline: 00917926314457. 1 An overview of magnetism interaction in localized models. We start by discussing issues related to the origin of ex- change interaction in Stoner model. Then we discuss the nature of exchange interaction in the Heisenberg model and an important work- ing rule "the Slater curve" for the sign of this interaction. After highlighting its problems we introduce the contributions of Vonsovsky and Zener which introduce the idea of indi- Figure 1: Through this article we pay homage to John rect s-d exchange interactions. Then Paul- Hasbrouck Van Vleck (March 13, 1899 – Oc- ing’s valence bond theory for the iron group tober 27, 1980) who set the foundation of the- metals is presented. Next comes the famous ories of electron correlation with his "middle- road" theory. debate between the itinerant picture (Stoner model) and the localized picture (Heisenberg model). Pros and cons of both approaches are elucidation of exchange interactions in d elec- discussed. The debate was settled in the fa- tron metals. Next, the pioneering contribu- vor of the itinerant model in the 1960s, when tions (but lesser known) of Dorfman are dis- d-band Fermi surface was observed in iron cussed. Then, in chronological order, the key group transition metals. However, the issue contributions of Pauli, Heisenberg, and Lan- of correlation effects in the itinerant model dau are presented. Finally, the advent of the remained open. The debate still appears in field of itinerant electron magnetism starting its varied avatars in the current literature on with the investigations of Bloch and Wigner, unconventional superconducting strongly cor- and more successful formulation by Slater and related materials. Next, we briefly discuss Stoner is presented. The physical basis of the the well settled issues of exchange interactions Slater-Stoner theory is discussed and its prob- in insulator compounds (direct exchange; su- lems are summarized. perexchange; and double exchange). Then we In the second part an overview of the de- review the "middle-road" theory of van Vleck bate between itinerant electron view (Stoner) and Hurwitz (the very first theory which and localized electron view (Heisenberg) is takes into account the electron correlation ef- presented. Connected with this debate are the fects in the itinerant model). We then intro- issues of exchange interactions. These can be duce Friedel-Alexander-Anderson-Moriya the- divided into two categories: (1) exchange in- ory of moment formation in pure iron group teraction in itinerant models, and (2) exchange transition metals which is a kind of gener- 2 An overview of magnetism alization of the famous Anderson impurity and highly successful branch of physics called problem and further advances the "middle- statistical mechanics which bridged the gap road" ideas of Hurwitz and van Vleck. Finally between the microscopic dynamical laws that the discussion of the very important issue of govern the motion of atoms and molecules strong electron correlation in the itinerant pic- and the macroscopic laws of thermodynamics. ture is presented. One of the first successful application of sta- tistical mechanics is the Langevin theory of i PART A paramagnetism (1905) [refer paper I ]. How- ever, there is one subtlety involved. In 1911, Niels Bohr in his PhD thesis applied the method of statistical mechanics to understand I. Failure of the classical magnetism from atomic point of view. He concluded that within the setting of classical picture: the Bohr-van Leeuwen statistical mechanics it is not possible to ex- theorem plain any form of magnetism of matter! His The 19th century saw two major advance- method yielded zero magnetization. Thus ments in fundamental physics. One is there is an apparent contradiction between the "wedding" of electricity and magnetism Bohr’s approach and Langevin’s approach, as through investigations of Oersted, Faraday, both came in the pre-quantum era. Maxwell and others. The other major devel- The result of zero magnetism in classical sta- opment occurred in the understanding of ther- tistical mechanics was re-discovered and elab- modynamical phenomena from molecular– orated independently in 1919 by Miss J. H. kinetic point of view. Thermodynamical con- Van Leeuwen. The result is now famous as cepts like temperature, pressure, and thermo- Bohr-van Leeuwen theorem. It can be ex- dynamical laws were understood from the mo- plained in the following way[1]. Consider the tion and interactions of atoms/molecule–the case of a material in which all the degrees-of- building blocks of matter. Maxwell for the freedom are in mutual thermodynamical equi- first time used probabilistic or statistical argu- librium including electrons. In statistical me- ments to derive the physical properties like chanics thermodynamical quantities, includ- pressure, viscosity etc of gases starting from ing magnetization, are computed from free en- the molecular–Kinetic point of view. This sta- ergy which can be expressed through partition tistical method was greatly extended by Lud- function which is further expressed as a phase wig Boltzmann (and independently by Gibbs), iHistory of magnetism I: from Greeks to Paul Langevin and they transformed it into a well respected and Pierre Weiss, Navinder Singh, hereafter referred as I. 3 An overview of magnetism integral of the Boltzmann factor (exp(− H )) treated classically by Langevin. He attributed kBT involving the Hamiltonian (H). In an external a permanent magnetic moment to each atom p2 magnetic field, the Hamiltonian ( 2m + V(r)) without worrying about its origin. This state 1 e 2 ii must be replaced by ( 2m (p − c A) + V(r)) of affairs is best explained by J. H. Van Vleck where A is the vector potential and p is the "When Langevin assumed that the magnetic canonical momentum. It turns out that the moment of the atom or molecule had a fixed phase integral (the partition function, Z) be- value µ, he was quantizing the system without comes independent of vector potential when realizing it." the integration over momentum in the phase Assignment of a permanent magnetic mo- ′ e space integration is changed to p = p − c A, ment to an atom is actually an introduction i.e., when momentum variable is changed. So of a quantum mechanical ingredient in to the partition function becomes independent the problem which Langevin did not recog- of vector potential, and resulting free energy nize explicitly. Also, Langevin did not take (F = −kBTlnZ) also becomes independent of into account the space quantization (spin can vector potential and magnetic field. It gives only have discrete quantized values along the zero magnetization when differentiated (M = magnetization direction). In Langevin’s the- ∂F − ∂H ). In conclusion, this theorem raises an ory magnetic moment can point in any di- apparent paradox: how does magnetic effects rection and the phase integral was computed arise in the Langevin theory which also uses for all possible orientations. Thus one can classical statistical mechanics? Quantum me- regard the Langevin theory as semi-classical, chanics was not known when Langevin ad- and the apparent paradox with fully classical vanced his theory (in 1905). Bohr-van Leeuwen theorem is immediately re- moved. As a side remark it is to be noted that II. Reconciling the Langevin when a fixed magnetic moment is assigned to theory with the Bohr-van Leeuwen an atom, one is departing from the principles of classical electrodynamics that an orbiting theorem (i.e., accelerating) electron inside of an atom It turns out that the Langevin theory is not must radiate energy. Permanent magnetic fully classical. It is actually semi-classical or moment implies permanent circling electrons semi-quantum in nature. Langevin did not inside the atom. The Langevin assumption consider all the degrees-of-freedom classically, of fixed magnetic moment directly leads to as considered in the Bohr-van Leeuwen theo- Bohr’s principle of stationery states on which rem.
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