DOCSLIB.ORG

Module 2A: Theory of Paramagnetism

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Module 2A: Theory of Paramagnetism Module 2A: Theory of paramagnetism Debanjan Bhowmik Department of Electrical Engineering Indian Institute of Technology Delhi Abstract In our first module, we discussed the problems associated with the current proces- sor and memory technology in the industry, which are transistor based, and briefly mentioned why magnetic devices can replace transistors, non-volatility of magnets be- ing cited as the main motivation. In the second module, we will discuss the theory of magnetism from the basics, keeping in mind the metallic spintronics systems relevant for technology. The metal used to store information in these systems is essentially ferromagnetic, so we need to understand ferromagnetism in this course but to do that first we need to understand paramagnetism, which we do in this part of the module (2A). 1 1 Types of magnetism We start with a familiar equation discussed in Electromagnetics courses: B~ = µ0(H~ + M~ ) (1) ~ ~ M = {mH (2) As a result, ~ ~ ~ ~ B = µ0(H + {mH) = µ0µrH (3) Parameters: µr= relative permeability, {m= magnetic susceptibility Fields: H~ = applied magnetic field, B~ = magnetic flux density and M~ = magnetization Difference between H~ and B~ is that H~ is what is applied, and B~ is the actual field which is the combination of H~ and the response of the material to H~ , which is given ~ ~ by the magnetization M = {mH. For example, in the case of a solenoid coil with an iron bar inserted inside it, H~ = NI (N= number of turns per unit length, I= current), but magnitude of B~ is much higher than magnitude of H~ because of the large magnetic ~ moment of iron, in response to the H, or the high {m of iron. The purpose of this set of lectures, or study of magnetism in general, is to look into the microscopic physics that goes into the determination of {m, which we don't cover in a standard Electromagnetics course. Based on the value of {m that a material exhibits, we have different types of magnetic materials: When {m is small and negative, it is known as diamagnetic. When {m is small and positive, it is known as paramagnetic. When {m is very large and positive, it is known as ferromagnetic. We do not cover diamagnetism in these lectures. Instead, we next discuss the theory of paramagnetism. Here we have to remember that the {m is a bulk parameter of the material, while the origin of the magnetism is microscopic. Hence we have to use quantum mechanics to understand the origin of magnetic moment in an atom, and then use statistical mechanics to understand how magnetic moments of individual atoms act in unison in the bulk, which is an aggregate of a large number of such atoms, to yield a bulk parameter like {m. 2 2 Quantum mechanical origin of magnetism in an isolated atom Let us look at the solution of Schrodinger's equation of an electron in an hydrogen atom. We ml know that the angular part of the wavefunction Yl is an eigen function of the magnitude of orbital angular momentum (L2) and orbital angular momentum in one specific direction (say z) (Lz): 2 ml 2 ml ml ml L Yl = ~ l(l + 1)Yl ; LzYl = ~mlYl (4) where −l 6 ml 6 l. Magnitude of orbital angular momentum of the electron and its value in the z direction are given as follows: 0:5 jµmj = γ~(l(l + 1)) ; µmjz = γ~ml (5) In addition to orbital angular momentum, electron has an intrinsic spin angular mo- mentum. Magnetic moment from the spin is given by: µmjs = γ~ms (6) 1 1 1 where electron being a spin 2 particle has ms equal to 2 or − 2 . Thus a state of an electron in an hydrogen atom can be given by the four quantum num- bers: n; l; ml and ms. Energy only depends on the n. However when we try to calculate the net magnetic moment in other atoms we have to keep in mind that the energy degeneracy is lifted because of electron- electron repulsion in a multi electron system which is not the case with hydrogen atom. Say in Fe atom which has 16 electrons, first 18 electrons form the Ar core, after which 2 electrons go to 4s. Of the 6 electrons which fill up the 3d shell, which electron has what values of quantum numbers (ml and ms) are described by the Hund's rules, as follows: 1. S = Σms is maximized. This is because if electrons have same spin quantum number, from Pauli exclusion principle, their other quantum numbers cannot be identical. So they do not fill up same orbital states and hence electron-electron Coulombic repulsion is mini- mized. 2. L = Σml is maximized. It has been found if orbits have same direction, electron-electron repulsion is lower since electrons have less chance of collision. 3. Due to spin-orbit coupling J becomes a new quantum number because L and S are dependent on each other. Spin orbit coupling energy is given by ξL:~ S~. If ξ > 0 which is the case for less than half filled shell, L and S are anti-parallel and J = jL − Sj. If ξ < 0 which is the case for less than half filled shell, L and S are anti-parallel and J=L+S. The four quantum numbers now are n; J; mj; ms. −J 6 mj 6 J. 3 Now keeping the Hund's rules in mind, let us look at the quantum numbers of the 1 6 electrons of 3d shell of Fe atom. The first five electrons have ms = 2 and the last 1 electron has ms = − 2 following the first Hund's rule, to maximize S = Σms. The first five electrons have to go to different orbitals since their ms is the same. Each of them go to ml = 2; 1; 0; −1; −2 orbitals. The 6th electron goes to ml = 2 orbital to maximize L = Σml, following rule 2, and making L parallel to S, following rule 3, since the 3d shell of Fe atom is more than half filled. J= L+ S = 4. Net magnetic moment of the atom (µ) is the combination of orbital and spin angular momentum of the electrons in the atom. It is given by: µ = gµBmj (7) where −J 6 mj 6 J., µB is the Bohr magnetron and g is the Lande g-factor. 3 Calculation of magnetization of a system of non-interacting atoms using partition function (Curie paramagnetism) So far we have derived the magnetic moment of an isolated atom using quantum me- chanics. Now we need statistical mechanics to find out the average magnetization of a system of non-interacting atoms, which will give us the susceptibility {m of a paramagnet. To calculate a macroscopic quantity like magnetization, we need to identify the number of microstates corresponding to a single value of magnetization. Then we take a ratio of that number of microstates to the total number of microstates, which gives the probability of the system exhibiting magnetization of that value. We do this for every value of magneti- zation. The sum of the product of all possible values of magnetization and the probability of the system to exhibit that value gives the expectation value of the magnetization, or the magnetization of the bulk. E − b We know that probability of a microstate the energy of which is E = e kB T (Refer to "Fundamentals of Statistical and Thermal Physics" by F. Reif for the derivation. Essen- tially we have to count the number of microstates of the bath, with which the system is in equilibrium.) Let there be n states for each atom; h is the applied magnetic field; m is the moment of the atom: State 1 : m = −gµBJ; E1 = gµ0µBJh State 2 : m = −gµB(J − 1); E2 = gµ0µB(J − 1)h ............ 4 State n-1 : m = gµB(J − 1); En−1 = −gµ0µB(J − 1)h State n: m = gµBJ; En = −gµ0µBJh Probability of m1 atoms in state 1, m2 atoms in state 2, Probability of m1 atoms in state 1, m2 atoms in state 2, .......mn atoms in state n= No. of possible microstates x Probability of each microstate = E m E m E m 1 N! − 1 1 − 2 2 − n n e kB T e kB T ::::::e kB T = Z m1!m2!:::::::mn! gµ0µ Jhm1 gµ0µ (J−1)hm1 gµ0µ Jhmn 1 N! − B − B B e kB T e kB T :::::e kB T Z m1!m2!:::::::mn! Sum of all the probabilities = 1 X 1 N! − gµ0µB Jhm1 − gµ0µB (J−1)hm2 gµ0µB Jhmn e kB T e kB T :::::e kB T = 1 (8) Z m1!m2!:::::::mn! m1;m2;:::::::mn ,or, the partition function X N! − gµ0µB Jhm1 − gµ0µB (J−1)hm2 gµ0µB Jhmn Z = e kB T e kB T :::::e kB T = m1!m2!:::::::mn! m1;m2;:::::::mn gµ µ Jh gµ µ (J−1)h gµ µ Jh − 0 B − 0 B 0 B N (e kB T + e kB T + ::::::::::e kB T ) (9) since m1 + m2::::::: + mn = N. Average value of magnetization 1 X N! < m >= (−gµBJm1) + (−gµB(J − 1)m2) + ::::::(gµBJmn) Z m1!m2!:::::::mn! m1;m2;:::::::mn − gµ0µB Jhm1 − gµ0µB (J−1)hm2 gµ0µB Jhmn e kB T e kB T :::::e kB T − gµ0µB Jh − gµ0µB (J−1)h gµ0µB Jh k T d(lnZ) Nk T d(e kB T + e kB T + ::::::::::e kB T ) = B = B µ0 dh µ0 dh d(sinh(J + 1 ) gµ0µB h )) NkBT 2 kB T gµBh = = gµBJNBJ ( ))) (10) µ0 dh kBT 1 1 1 1 1 where BJ (x) = J [(J + 2 )coth((J + 2 )x) − 2 coth( 2 )x)] is called the Brillouin function.
Load more

Recommended publications
  • Solid State Physics 2 Lecture 5: Electron Liquid
    Physics 7450: Solid State Physics 2 Lecture 5: Electron liquid Leo Radzihovsky (Dated: 10 March, 2015) Abstract In these lectures, we will study itinerate electron liquid, namely metals. We will begin by re- viewing properties of noninteracting electron gas, developing its Greens functions, analyzing its thermodynamics, Pauli paramagnetism and Landau diamagnetism. We will recall how its thermo- dynamics is qualitatively distinct from that of a Boltzmann and Bose gases. As emphasized by Sommerfeld (1928), these qualitative di↵erence are due to the Pauli principle of electons’ fermionic statistics. We will then include e↵ects of Coulomb interaction, treating it in Hartree and Hartree- Fock approximation, computing the ground state energy and screening. We will then study itinerate Stoner ferromagnetism as well as various response functions, such as compressibility and conduc- tivity, and screening (Thomas-Fermi, Debye). We will then discuss Landau Fermi-liquid theory, which will allow us understand why despite strong electron-electron interactions, nevertheless much of the phenomenology of a Fermi gas extends to a Fermi liquid. We will conclude with discussion of electrons on the lattice, treated within the Hubbard and t-J models and will study transition to a Mott insulator and magnetism 1 I. INTRODUCTION A. Outline electron gas ground state and excitations • thermodynamics • Pauli paramagnetism • Landau diamagnetism • Hartree-Fock theory of interactions: ground state energy • Stoner ferromagnetic instability • response functions • Landau Fermi-liquid theory • electrons on the lattice: Hubbard and t-J models • Mott insulators and magnetism • B. Background In these lectures, we will study itinerate electron liquid, namely metals. In principle a fully quantum mechanical, strongly Coulomb-interacting description is required.
    [Show full text]
  • Study of Spin Glass and Cluster Ferromagnetism in Rusr2eu1.4Ce0.6Cu2o10-Δ Magneto Superconductor Anuj Kumar, R
    Study of spin glass and cluster ferromagnetism in RuSr2Eu1.4Ce0.6Cu2O10-δ magneto superconductor Anuj Kumar, R. P. Tandon, and V. P. S. Awana Citation: J. Appl. Phys. 110, 043926 (2011); doi: 10.1063/1.3626824 View online: http://dx.doi.org/10.1063/1.3626824 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v110/i4 Published by the American Institute of Physics. Related Articles Annealing effect on the excess conductivity of Cu0.5Tl0.25M0.25Ba2Ca2Cu3O10−δ (M=K, Na, Li, Tl) superconductors J. Appl. Phys. 111, 053914 (2012) Effect of columnar grain boundaries on flux pinning in MgB2 films J. Appl. Phys. 111, 053906 (2012) The scaling analysis on effective activation energy in HgBa2Ca2Cu3O8+δ J. Appl. Phys. 111, 07D709 (2012) Magnetism and superconductivity in the Heusler alloy Pd2YbPb J. Appl. Phys. 111, 07E111 (2012) Micromagnetic analysis of the magnetization dynamics driven by the Oersted field in permalloy nanorings J. Appl. Phys. 111, 07D103 (2012) Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 12 Mar 2012 to 14.139.60.97. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions JOURNAL OF APPLIED PHYSICS 110, 043926 (2011) Study of spin glass and cluster ferromagnetism in RuSr2Eu1.4Ce0.6Cu2O10-d magneto superconductor Anuj Kumar,1,2 R. P. Tandon,2 and V. P. S. Awana1,a) 1Quantum Phenomena and Application Division, National Physical Laboratory (CSIR), Dr.
    [Show full text]
  • Condensation of Bosons with Several Degrees of Freedom Condensación De Bosones Con Varios Grados De Libertad
    Condensation of bosons with several degrees of freedom Condensación de bosones con varios grados de libertad Trabajo presentado por Rafael Delgado López1 para optar al título de Máster en Física Fundamental bajo la dirección del Dr. Pedro Bargueño de Retes2 y del Prof. Fernando Sols Lucia3 Universidad Complutense de Madrid Junio de 2013 Calificación obtenida: 10 (MH) 1 [email protected], Dep. Física Teórica I, Universidad Complutense de Madrid 2 [email protected], Dep. Física de Materiales, Universidad Complutense de Madrid 3 [email protected], Dep. Física de Materiales, Universidad Complutense de Madrid Abstract The condensation of the spinless ideal charged Bose gas in the presence of a magnetic field is revisited as a first step to tackle the more complex case of a molecular condensate, where several degrees of freedom have to be taken into account. In the charged bose gas, the conventional approach is extended to include the macroscopic occupation of excited kinetic states lying in the lowest Landau level, which plays an essential role in the case of large magnetic fields. In that limit, signatures of two diffuse phase transitions (crossovers) appear in the specific heat. In particular, at temperatures lower than the cyclotron frequency, the system behaves as an effectively one-dimensional free boson system, with the specific heat equal to (1/2) NkB and a gradual condensation at lower temperatures. In the molecular case, which is currently in progress, we have studied the condensation of rotational levels in a two–dimensional trap within the Bogoliubov approximation, showing that multi–step condensation also occurs.
    [Show full text]
  • Pauli Paramagnetism of an Ideal Fermi Gas
    Pauli paramagnetism of an ideal Fermi gas The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Lee, Ye-Ryoung, Tout T. Wang, Timur M. Rvachov, Jae-Hoon Choi, Wolfgang Ketterle, and Myoung-Sun Heo. "Pauli paramagnetism of an ideal Fermi gas." Phys. Rev. A 87, 043629 (April 2013). © 2013 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevA.87.043629 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/88740 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW A 87, 043629 (2013) Pauli paramagnetism of an ideal Fermi gas Ye-Ryoung Lee,1 Tout T. Wang,1,2 Timur M. Rvachov,1 Jae-Hoon Choi,1 Wolfgang Ketterle,1 and Myoung-Sun Heo1,* 1MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Received 7 January 2013; published 24 April 2013) We show how to use trapped ultracold atoms to measure the magnetic susceptibility of a two-component Fermi gas. The method is illustrated for a noninteracting gas of 6Li, using the tunability of interactions around a wide Feshbach resonance. The susceptibility versus effective magnetic field is directly obtained from the inhomogeneous density profile of the trapped atomic cloud. The wings of the cloud realize the high-field limit where the polarization approaches 100%, which is not accessible for an electron gas.
    [Show full text]
  • Energy and Charge Transfer in Open Plasmonic Systems
    Energy and Charge Transfer in Open Plasmonic Systems Niket Thakkar A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2017 Reading Committee: David J. Masiello Randall J. LeVeque Mathew J. Lorig Daniel R. Gamelin Program Authorized to Offer Degree: Applied Mathematics ©Copyright 2017 Niket Thakkar University of Washington Abstract Energy and Charge Transfer in Open Plasmonic Systems Niket Thakkar Chair of the Supervisory Committee: Associate Professor David J. Masiello Chemistry Coherent and collective charge oscillations in metal nanoparticles (MNPs), known as localized surface plasmons, offer unprecedented control and enhancement of optical processes on the nanoscale. Since their discovery in the 1950's, plasmons have played an important role in understanding fundamental properties of solid state matter and have been used for a variety of applications, from single molecule spectroscopy to directed radiation therapy for cancer treatment. More recently, experiments have demonstrated quantum interference between optically excited plasmonic materials, opening the door for plasmonic applications in quantum information and making the study of the basic quantum mechanical properties of plasmonic structures an important research topic. This text describes a quantitatively accurate, versatile model of MNP optics that incorporates MNP geometry, local environment, and effects due to the quantum properties of conduction electrons and radiation. We build the theory from
    [Show full text]
  • Superparamagnetic Nanoparticle Ensembles
    1 Superparamagnetic nanoparticle ensembles O. Petracic Institute of Experimental Physics/Condensed Matter Physics, Ruhr-University Bochum, 44780 Bochum, Germany Abstract: Magnetic single-domain nanoparticles constitute an important model system in magnetism. In particular ensembles of superparamagnetic nanoparticles can exhibit a rich variety of different behaviors depending on the inter-particle interactions. Starting from isolated single-domain ferro- or ferrimagnetic nanoparticles the magnetization behavior of both non-interacting and interacting particle-ensembles is reviewed. A particular focus is drawn onto the relaxation time of the system. In case of interacting nanoparticles the usual Néel-Brown relaxation law becomes modified. With increasing interactions modified superparamagnetism, spin glass behavior and superferromagnetism are encountered. 1. Introduction Nanomagnetism is a vivid and highly interesting topic of modern solid state magnetism and nanotechnology [1-4]. This is not only due to the ever increasing demand for miniaturization, but also due to novel phenomena and effects which appear only on the nanoscale. That is e.g. superparamagnetism, new types of magnetic domain walls and spin structures, coupling phenomena and interactions between electrical current and magnetism (magneto resistance and current-induced switching) [1-4]. In technology nanomagnetism has become a crucial commercial factor. Modern magnetic data storage builds on principles of nanomagnetism and this tendency will increase in future. Also other areas of nanomagnetism are commercially becoming more and more important, e.g. for sensors [5] or biomedical applications [6]. Many potential future applications are investigated, e.g. magneto-logic devices [7], [8], photonic systems [9, 10] or magnetic refrigeration [11, 12]. 2 In particular magnetic nanoparticles experience a still increasing attention, because they can serve as building blocks for e.g.
    [Show full text]
  • Magnetic Fields in Matter 2
    Electromagnetism II Instructor: Andrei Sirenko [email protected] Spring 2013 Thursdays 1 pm – 4 pm Spring 2013, NJIT 1 PROBLEMS for CH. 6 http://web.njit.edu/~sirenko/Phys433/PHYS433EandM2013.htm Classical consideration: diamagnetism Can obtain the same formula classically: Consider an electron rotating about the nucleus in a circular orbit; let a magnetic field be applied. Before this field is applied, we have, according to Newton's second law, 2 F0 m0 r F0 is the attractive Coulomb force between the nucleus and the electron, and 0 is the angular velocity. Applied field an additional force: the Lorentz force FL evB 2 eB FL = -eBr F0 – eBr = m r 0 2m Reduction in frequency corresponding change in the magnetic moment. The change in the frequency of rotation is equivalent to the change in the current around the nucleus: ΔI = (charge) (revolutions per unit time) 1 eB I Ze 2 2m The magnetic moment of a circular current is given by the product (current) x (area of orbit) 2 2 1 eB e rxy e r 2 B 2 2m xy 4m 2 2 2 2 Here rxy = x + y . The mean square distance of the electrons from the nucleus is r = x2 + y2+ z2. For a spherically symmetrical charge distribution x2 = y2 = z2 = r2/3 2 2 2 2 2 2 2 e r 0e NZ r r r xy B 3 6m 6m Diamagnetism in ionic crystals and crystals composed of inert gas atoms: they have atoms or ions with complete electronic shells. Another class of diamagnetics is noble metals, which will be discussed later.
    [Show full text]
  • Pauli Paramagnetism in Metals with High Densities of States*L
    JOURNAL OF RESEARCH of the National Bureau of Standards-A. Physics and Chemistry Vol. 74A, No. 4, July-August 1970 Pauli Paramagnetism in Metals with High Densities of States*l S. Foner Francis Bitter National Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (October 10, 1969) Because the Pauli paramagnetic susceptibility of alloys have permitted considerable progress to be a free electron gas, x~ is proportional to the density of made. The properties of these systems are reviewed states N(O) at the Fermi energy, it is expected that mea­ and estimates of N(O) and V are examin ed. Here qualita­ surements of X~ of metals and alloys can yield a tive and quantitative comparisons of various experi­ reasonable measure of N(O). If xg can be measured ments can be made. The possibilities of applying large directly, then N(O) can be determined with high accura­ magnetic fields and thereby independently measuring cy. The major obstacle to this procedure is that the N(O) and small changes of N(O) near the Fermi energy measured static susceptibility Xi, = xV(l-N(O)V)=xgD by such studies are discussed. Recent theories which (for the Stoner model) where V is a measure of the elec­ examine the changes of Xv with alloying are also com­ tron-electron interaction potential. Unfortunately, V is pared with experimental results. not easily determined experimentally, or theoretically, A few useful references for relevant recent papers so that measurements of Xv do not yield accurate mea­ concerning the susceptibility of nearly ferromagnetic sures of N(O).
    [Show full text]
  • Arxiv:1301.1139V1 [Cond-Mat.Quant-Gas]
    Pauli paramagnetism of an ideal Fermi gas Ye-Ryoung Lee,1 Tout T. Wang,1, 2 Timur M. Rvachov,1 Jae-Hoon Choi,1 Wolfgang Ketterle,1 and Myoung-Sun Heo1, ∗ 1MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2Department of Physics, Harvard University, Cambridge, MA 02138, USA We show how to use trapped ultracold atoms to measure the magnetic susceptibility of a two- component Fermi gas. The method is illustrated for a non-interacting gas of 6Li, using the tunability of interactions around a wide Feshbach resonances. The susceptibility versus effective magnetic field is directly obtained from the inhomogeneous density profile of the trapped atomic cloud. The wings of the cloud realize the high field limit where the polarization approaches 100%, which is not accessible for an electron gas. PACS numbers: 67.85.Lm, 67.10.Db I. INTRODUCTION D E E F Ultracold atoms can be prepared with almost complete control over their density, temperature and interactions. They serve as model systems for exploring unsolved prob- lems in many-body physics as well as for demonstrat- E ing well-known textbook physics, such as of ideal, non- interacting gases that do not exist in nature. For example several phenomena related to Pauli blocking of fermions were clearly demonstrated only after the advent of ul- tracold Fermi gases [1]. Here we use the tunability of 'HQVLW\ atomic interactions near Feshbach resonances to create a non-interacting Fermi gas with two components. This is a realization of an ideal, non-interacting free electron gas (FEG) with spin up and down components, as assumed 5DGLDOSRVLWLRQ in the simple theory of metals.
    [Show full text]
  • Magnetic Properties and the Superatom Character of 13-Atom Platinum Nanoclusters
    Magnetochemistry 2015, 1, 28-44; doi:10.3390/magnetochemistry1010028 OPEN ACCESS magnetochemistry ISSN 2312-7481 www.mdpi.com/journal/magnetochemistry Article Magnetic Properties and the Superatom Character of 13-Atom Platinum Nanoclusters Emil Roduner 1,2,* and Christopher Jensen 1,† 1 Institute of Physical Chemistry, University of Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart, Germany; E-Mail: [email protected] 2 Chemistry Department, University of Pretoria, Pretoria 0002, South Africa † Present address: ThyssenKrupp Industrial Solutions AG, Business Unit Process Technologies, Neubeckumer Straße 127, D-59320 Ennigerloh, Germany. * Author to whom correspondence should be addressed; E-Mail: [email protected] or [email protected]; Tel.: +0041-44-422-34-28. Academic Editor: Carlos J. Gómez García Received: 25 October 2015 / Accepted: 18 November 2015 / Published: 26 November 2015 Abstract: 13-atom platinum nanoclusters have been synthesized quantitatively in the pores of the zeolites NaY and KL. They reveal highly interesting magnetic properties like high-spin states, a blocking temperature, and super-diamagnetism, depending heavily on the loading of chemisorbed hydrogen. Additionally, EPR active states are observed. All of these magnetic properties are understood best if one considers the near-spherical clusters as analogs of transition metal atoms with low-spin and high-spin states, and with delocalized molecular orbitals which have a structure similar to that of atomic orbitals. These clusters are, therefore, called superatoms, and it is their analogy with normal atoms which is in the focus of the present work, but further phenomena, like the observation of a magnetic blocking temperature and the possibility of superconductivity, are discussed.
    [Show full text]
  • 23. Magnetic Fields and Materials
    Scott Hughes 10 May 2005 Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005 Lecture 23: Magnetic materials 23.1 Magnetic ¯elds and stu® At this stage of 8.022, we have essentially covered all of the material that is typically taught in this course. Congratulations | you now have a solid grounding in all the major concepts of electricity and magnetism! There remains one major, important subject which we have not discussed in depth: the interaction of magnetic ¯elds and materials. At this point, we don't even have a really good understanding of a bar magnet works | an unsatisfying state of a®airs, given that this is the way in which we normally encounter magnetism in the real world! One reason we have avoided covering this subject is that it is not really possible to discuss it properly within getting into a detailed discussion of the quantum mechanical description of matter. The way in which matter responds to magnetic ¯elds is totally determined by the quantum mechanical nature of their molecular structure, particularly their electrons. Nonetheless, we can make signi¯cant headway in understanding the interaction of mag- netic ¯elds and materials by combining what we have learned so far with a somewhat ap- proximate, qualitative description of how materials respond to magnetic ¯elds. The main concepts we will need are summarized in the following two subsections: 23.1.1 Electron orbitals The electrons in a molecule exist in orbits. Very roughly, we can picture an orbit as a simple loop of current: A loop of current like this of course tends to generate its own magnetic ¯eld.
    [Show full text]
  • Statistical Mechanics of the Magnetized Pair Fermi Gas
    IW1-P-93/-15 Statistical Mechanics of the Magnetized Pair Fermi Gas J. DAICIC, N. E. FhANKEL and V. KOWALENKO School of Physics, University of Melbourne, Parkville., Victoria 3052, Australia Abstract Following our previous work on the magnetized pair Bose gas [1], we study in this paper the statistical mechanics of the charged relativistic Fermi gas with pair creation in d spatial dimensions. Initially, the gas in no external fields is studied, and we obtain expansions for the various thermodynamic functions in both the ///»? — 0 (neu'rino) limit, and about the point fi/m — 1, where /* is the chemical potential. We also present the thermodynamics of a gas of quantum-number con­ serving massless fermions. We then make a complete study of the pair Fermi gas in a homogeneous magnetic field, investigating the behavior of the magnetization over a wide range of field strengths. The inclusion of pairs leads to new results for the net magnetization due to the paramagnetic moment of the spins and the diamagnetic Landau orbits. ' •&* 1 Introduction 1 here are many physical contexts in which the ideal Fermi gas is useful as a model for many-body finite-temperature systems. At low temperatures, where the single-particle energy spectrum can be taken as non-relativistic. the ideal Fermi gas [2] can describe weakly interacting systems such as He3 and electrons in metals [3]. The thermodynamics of this system, both with [4] and without [5] an external magnetic field has been studied by More and Frankel in the region where c ~ 1, where z is the fugaclty, corresponding to temperatures near the Fermi temperature TF- Astrophysically.
    [Show full text]