Magnetic ordering, magnetic anisotropy and the mean-field theory

Size: px
Start display at page:

Download "Magnetic ordering, magnetic anisotropy and the mean-field theory"

Transcription

1 Magnetic ordering, magnetic anisotropy and the mean-field theory Alexandra Kalashnikova Ferromagnets Mean-field approximation Curie temperature and critical exponents Magnetic susceptibility Temperature dependence of magnetization and Bloch law Antiferromagnets Neel temperature usceptibility Magnetic anisotropy Magnetic-dipole contribution Magneto-crystalline anisotropy and its temperature dependence Ferro- and antiferromagnets in an external field 1

2 Magnetic ordering, magnetic anisotropy and the mean-field theory Ferromagnets Mean-field approximation Curie temperature and critical exponents Magnetic susceptibility Temperature dependence of magnetization and Bloch law

3 Mean-field approximation: Weiss molecular field μ J m wm Weiss molecular field - field created by neighbor magnetic moments Alignment of a magnetic moment μ in the total field +wm? 3

4 A brief reminder: independent magnetic moments in external field μ Thermal energy: U T kt At room temperature: J Potential energy of a magnetic moment in the field : U cos In a field of 1MA/m: J U / U T ~

5 Paramagnetism: independent magnetic moments in external field Induced magnetization Probability for a magnetic moment to orient at the angle θ: U exp U M T exp kt Full magnetization along the field: 1 Ncoth cos kt Langevine function L 5

6 Paramagnetism: independent magnetic moments in external field Full magnetization along the field: M 1 Ncoth M kt At RT in the field of 1МА/м: M Paramagnetic susceptibility: NL( ) para N N( / 3...) 3kT M N 3kT Curie law 6

7 Paramagnetism: independent magnetic moments in external field Quantization of the magnetic moment g J z Resulting magnetization along the field: M NgJ B B J z B J z J, J 1... J gj B kt Gd 3+ Fe 3+ Brillouin function Cr 3+ At small : M J NgJ B... 3J1 Curie law para Ng J ( J 1) B 3kT 7 [enry, PR 88, 559 (195)]

8 Ferromagnets: Weiss molecular field J Averaged projected magnetic moment: z g B z B B () () g ( kt B m) Weiss field due to exchange interactions J with N nearest neighbors: m NJ g z B z B NJ g B gb kt z elf-consistent solution 8

9 Back to ferromagnets: Weiss molecular field J =0 z B () < z >() z g ( kt B m) kt NJ gb NJ =0 non-zero solutions: z opposite orientations of the order parameter 9

10 Ferromagnets (=1/): Curie temperature J =0 T 1 T C >T 1 B () T >T C Variations of these curves with T: z B () z kt NJ At T>T C z 0 gb NJ =0 Ordering temperature: z T C nd order phase transition JN k changes continuously with T 10

11 Ferromagnets (=1/): Magnetization in a vicinity of T C J =0 T 1 T C >T 1 B () T >T C z B () z kt NJ gb NJ =0 At T T C z 0 1/ 0 3 T 1 C z 4 T 11

12 Ferromagnets (=1/): Curie temperature J Ordering temperature: nd order phase transition T C JN k Mean-field approximation predicts: 1D case: D case: 3D case: T C J 0 k J k J T C 4 T C 6 k Reality: No magnetic ordering T C. 69 T C J k J k Agreement gets better for the 3D case Reason: fluctuations were neglected 1

13 Ferromagnets (=1/): at low temperatures J =0 T 1 T C >T 1 B () T >T C z B () At T 0 g ( kt B m) z Prediction: z Reality: M 1 T 1 e C / T 3/ ( T ) M 0 1 ( T / TC ) 13

14 Ferromagnets (=1/) in applied field J 0 T C z B () g ( kt B m) < z >() Non-zero magnetization above T C usceptibility Above T C z in small fields k( T TC ) B T T C k( T 1 T C ) Curie-Weiss law 14

15 Ferromagnets Weiss molecular field theory: 1 M M ( T 0) 1 M 0 T C z JN k 0 Temperature Curie T 1 C T 1/ Critical exponent β=1/ T C Curie-Weiss law T T C T 1 k( T T C ) Critical exponent γ=-1 Landau theory of phase transitions gives the same values of critical exponents 15

16 Ferromagnets Weiss molecular field theory: 1 M M ( T 0) 1 0 T C T M T C T ( T ) T C ) ( T C T Landau theory pin-spin correlations 16 and MF

17 Mean-field approximation: Bethe mean-field theory 0 j J From uncorrelated spins to uncorrelated clusters of spins 0 interaction with its 4 neighbors is treated exactly (cluster) j are subject to effective (Weiss) field T J C k ln( N /( N )) Approximation predicts: Reality: 1D case: D case: 3D case: T C 0 T C. 885 T C J k J k No magnetic ordering T C. 69 T C J k J k Agreement is improved 17

18 Temperature dependence of magnetization At T=0 z M(r)=M 0 Minimum of the exchange energy At T>0 Thermal fluctuations M(r) uperposition of harmonic waves thermal spin waves (introduced by Bloch) Nonparallel i Increase of the exchange energy The magnitude of the gradient of M is important! 18

19 Magnetization at T>0 M(r)=M 0 +ΔM(r) Characteristic period of magnetization variation λ λ>> a (inter-atomic distance) λ<< L (sample size) ΔM(r) <<M 0 mall deviation of M from M 0 (M(r)) =M 0 =const Length of M is conserved Plane waves in a continuous medium (some analogy with sound waves) Energy of a ferromagnet as a function of the spatial distribution of magnetization? Increase of the Zeeman energy: -(M(r)-M 0 ) Increase of the exchange energy: Exchange constant 19

20 Magnetization at T>0 Energy of a ferromagnet as a function of the spatial distribution of magnetization: Increase of the Zeeman energy -(M(r)-M 0 ) Increase of the exchange energy 0

21 Magnetization at T>0 z mall deviations of M: M x, M y << M 0 M 0 1

22 Magnetization at T>0 z Introduce complex combinations:

23 Magnetization at T>0: Exchange waves z A sum of the energies of plane waves spin waves 3

24 Magnetization at T>0: magnons Energy of a ferromagnet: number of quasiparticles - magnons in the state with an energy: Quasi-momentum of magnon Effective mass of a magnon Bose-Einstein distribution of thermal magnons 4

25 Magnetization at T>0: magnons Each magnon reduces the total magnetic moment of the sample by μ Quadratic dispersion Bose-Einstein distribution Bloch (3/-) law: correct temperature dependence of magnetization at low T 5

26 Magnetic ordering, magnetic anisotropy and the mean-field theory Antiferromagnets Neel temperature usceptibility 6

27 Antiferromagnets μ 1i μ i Magnetic sublattices: M 1 =Σμ 1i /V M =Σμ i /V M=M 1 +M L=M 1 -M Magnetization (ferromagnetic vector) Antiferromagnetic vector) Equivalent magnetic sublattices same type of magnetic ions in the same crystallographic positions Number of magnetic sublattices is equivalent to the number of magnetic ions in the magnetic unit cell 7

28 Antiferromagnets: Neél temperature A B =0 Equivalent sublattices: Molecular (Weiss) fields for the case of two sublattices: w AA =w BB =w 1 w AB =w BA =w A =w AA M A +w AB M B B =w BB M B +w BA M A M A =-M B Magnetization of a sublattice in the presence of the molecular fields: Neél temperature: B () =0 For each sublattice: T 1 T N >T 1 T >T C 8

29 Antiferromagnets: susceptibility Equivalent sublattices: afm par w AA= w BB =w 1 >0 w AB= w BA =w <0 A =w AA M A +w AB M B + B =w BB M B +w BA M A + M A =-M B Asymptotic Curie temperature: 1 afm 1 k ( T TA) ) A para 1 kt usceptibility above T N : ferro 1 k( T T C ) TA T N T C T 9

30 Antiferromagnets: susceptibility afm par Equivalent sublattices: w AA =w BB =w 1 >0 w AB =w BA =w <0 M A =-M B usceptibility in the Neél point: T N usceptibility at T=0: χ=0 30

31 Antiferromagnets: some examples T N [P. W. Anderson, Phys. Rev. 79, 705 (1950)] 31

32 Magnetic ordering, magnetic anisotropy and the mean-field theory Magnetic anisotropy Magnetic-dipole contribution Magneto-crystalline anisotropy and its temperature dependence 3

33 Magnetic anisotropy Magnetic anisotropy energy energy required to rotate magnetization from an easy to a hard direction Without anisotropy net magnetization of 3D solids would be weak; in D systems it would be absent [Mermin and Wagner, PRL 17, 1133 (1966)] M Magnetization axial vector ources of anisotropy: Deformation Intrinsic electrical fields hape etc. - polar impacts Magnetic anisotropy sets an axis, not a direction 33

34 Magnetic anisotropy: phenomenological consideration Uniaxial anisotropy M E a K u 4 1 sin Ku sin... K u1 0 (001) easy plane Cubic anisotropy K u1 0 Effective anisotropy field a [001] easy axis E a Ku1 cosn... M M 3 w a K K K 1 0 K 1 0 easy axes- {100} easy axes- {111} 34

35 Magnetic anisotropy Compound erg cm -3 erg cm -3 Origins of magnetic anisotropy ingle-ion anisotropy Anisotropic exchange Magnetic-dipolar interactions pin-orbit interaction Interactions between pairs of magnetic ions 35

36 Magnetic-dipolar contribution to the anisotropy Cubic crystal E a Nq x M For a pair of neighboring ions: E( ) w l cos ϕ=α 1 4 q... ex const Dipolar interaction Quadrupolar interaction fcc: K1 Nq N number of atoms per volume unit cubic anisotropy constant vcc: K1 16/ 9Nq K Nq 1 36

37 exagonal crystal θ... sin sin 4 1 u u a K K E Cubic crystal ) ( q l E K E a cos ϕ=α Magnetic-dipolar contribution to the anisotropy θ u1 u a q l K u ~ 1 Contributes to the shape anisotropy This films demonstrate in-plane anisotropy 37

38 ingle-ion anisotropy (crystal-field theory) pinel CoFe O 4 [001] [111] Co + in the octahedral coordination + trigonal distortion along [111] O - [100] plitting of the 3d shell In the ligand field: Co + 3d 7 x -y ; z xy; xz; yz 38

39 ingle-ion anisotropy (crystal-field theory) [001] [111] Co + in the octahedral coordination + trigonal distortion along [111] [100] plitting of the 3d shell In the ligand field + trigonal distortion x -y ; z 3d 7 xy; xz; yz Maxima of electronic density in the plane (111) Maxima of electronic density along the axis [111] 39

40 ingle-ion anisotropy (crystal-field theory) [001] [111] Co + in the octahedral coordination + trigonal distortion along [111] [100] plitting of the 3d shell In the ligand field + trigonal distortion x -y ; z 3d 7 xy; xz; yz Orbital moment L [111] pin-orbit interaction 3 rd unds rule: w O L Co + (3d 7 ) L 0 cos -angle between и [111] 40

41 ingle-ion anisotropy (crystal-field theory) [001] [111] Co + in the octahedral coordination + trigonal distortion along [111] w O L L cos -angle between и [111] [100] Co + (3d 7 ) 0 Averaging over 4 types of Co + positions with distortions along different {111} axes: w a NL( x y y z z x ) K [lonczewski, JAP 3, 53 (1961) ]

42 Temperature dependence of magneto-crystalline anisotropy Classical theory gives: E K a n Cubic anisotropy K K n n n ( T ) M ( ) T ( 0) (0) M Uniaxial anisotropy K K n n ( T ) (0) M M ( T ) (0) 10 3 K K n n Fe ( T ) (0) M M ( T ) (0) n( n1) [Zener, Phys. Rev. 96, 1335 (1954)] Exp: Theory: Competition between different anisotropies gives rise to spin-reorinetation transitions 4

43 Magnetic ordering, magnetic anisotropy and the mean-field theory Ferro- and antiferromagnets in an external field 43

44 Ferromagnets in applied magnetic field: domain walls displacement Cubic anisotropy (K 1 >0) [100] 44

45 Technical Ferromagnets magnetization in applied magnetic processes: field: magnetization rotation of magnetization rotation e.a. M M e.a. Reversible rotation towards the field Irreversible rotation towards the field 45

46 Magnetization process in a multisublattice medium: spin-flop and spin-flip transitions in an antiferromagnet Χ max =-1/w M =-w M 46

47 Magnetization process in a multisublattice medium: spin-flop and spin-flip transitions in an antiferromagnet If the anisotropy is weak: M Condition for the spin-flop (spin-reorienetation transition) C 47

48 Magnetization process in a multisublattice medium: spin-flop and spin-flip transitions in an antiferromagnet If the anisotropy is weak: M If the anisotropy is strong: M pin-flip C C 48

Luigi Paolasini

Luigi Paolasini Luigi Paolasini paolasini@esrf.fr LECTURE 5: MAGNETIC STRUCTURES - Mean field theory and magnetic order - Classification of magnetic structures - Collinear and non-collinear magnetic structures. - Magnetic

More information

Electromagnetism II. Instructor: Andrei Sirenko Spring 2013 Thursdays 1 pm 4 pm. Spring 2013, NJIT 1

Electromagnetism II. Instructor: Andrei Sirenko Spring 2013 Thursdays 1 pm 4 pm. Spring 2013, NJIT 1 Electromagnetism II Instructor: Andrei Sirenko sirenko@njit.edu Spring 013 Thursdays 1 pm 4 pm Spring 013, NJIT 1 PROBLEMS for CH. 6 http://web.njit.edu/~sirenko/phys433/phys433eandm013.htm Can obtain

More information

Chapter 6 Antiferromagnetism and Other Magnetic Ordeer

Chapter 6 Antiferromagnetism and Other Magnetic Ordeer Chapter 6 Antiferromagnetism and Other Magnetic Ordeer 6.1 Mean Field Theory of Antiferromagnetism 6.2 Ferrimagnets 6.3 Frustration 6.4 Amorphous Magnets 6.5 Spin Glasses 6.6 Magnetic Model Compounds TCD

More information

Ferromagnetism. Iron, nickel, and cobalt are ferromagnetic.

Ferromagnetism. Iron, nickel, and cobalt are ferromagnetic. Ferromagnetism Technische Universität Graz Institute of Solid State Physics Ferromagnetism elow a critical temperature (called the Curie temperature) a magnetization spontaneously appears in a ferromagnet

More information

Magnetic ordering of local moments

Magnetic ordering of local moments Magnetic ordering Types of magnetic structure Ground state of the Heisenberg ferromagnet and antiferromagnet Spin wave High temperature susceptibility Mean field theory Magnetic ordering of local moments

More information

Magnetism at finite temperature: molecular field, phase transitions

Magnetism at finite temperature: molecular field, phase transitions Magnetism at finite temperature: molecular field, phase transitions -The Heisenberg model in molecular field approximation: ferro, antiferromagnetism. Ordering temperature; thermodynamics - Mean field

More information

An introduction to magnetism in three parts

An introduction to magnetism in three parts An introduction to magnetism in three parts Wulf Wulfhekel Physikalisches Institut, Karlsruhe Institute of Technology (KIT) Wolfgang Gaede Str. 1, D-76131 Karlsruhe 0. Overview Chapters of the three lectures

More information

Transition Elements. pranjoto utomo

Transition Elements. pranjoto utomo Transition Elements pranjoto utomo Definition What is transition metal? One of which forms one or more stable ions which have incompletely filled d orbitals. 30Zn? Definition Zink is not transition elements

More information

WORLD SCIENTIFIC (2014)

WORLD SCIENTIFIC (2014) WORLD SCIENTIFIC (2014) LIST OF PROBLEMS Chapter 1: Magnetism of Free Electrons and Atoms 1. Orbital and spin moments of an electron: Using the theory of angular momentum, calculate the orbital

More information

CHAPTER 2 MAGNETISM. 2.1 Magnetic materials

CHAPTER 2 MAGNETISM. 2.1 Magnetic materials CHAPTER 2 MAGNETISM Magnetism plays a crucial role in the development of memories for mass storage, and in sensors to name a few. Spintronics is an integration of the magnetic material with semiconductor

More information

Basic Magnetism (I. Fundamentals)

Basic Magnetism (I. Fundamentals) Paolo Allia DISAT Politecnico di Torino Associate, INRiM - Torino Basic Magnetism (I. Fundamentals) P. Allia - Italian School of Magnetism 018 1 A journey through Magnetism or, From atoms to macroscopic

More information

复习题. 2 Calculate the intensity of magnetic field in the air gap of the magnetic circuit shown in the figure. Use the values N=200,

复习题. 2 Calculate the intensity of magnetic field in the air gap of the magnetic circuit shown in the figure. Use the values N=200, 复习题 1 Calculate the magnetic moment of a sphere of radius R made from a magnetic material with magnetic susceptibility, when it is magnetized by an external magnetic field H. How is the value of the moment

More information

Lecture 5. Chapters 3 & 4. Induced magnetization: that which is induced in the presence of an applied magnetic field. diamagnetic.

Lecture 5. Chapters 3 & 4. Induced magnetization: that which is induced in the presence of an applied magnetic field. diamagnetic. Lecture 5 Induced magnetization: that which is induced in the presence of an applied magnetic field diamagnetic paramagnetic Remanent magnetization: that which remains in the absence of an external field

More information

Colossal magnetoresistance:

Colossal magnetoresistance: Colossal magnetoresistance: Ram Seshadri (seshadri@mrl.ucsb.edu) The simplest example of magnetoresistance is transverse magnetoresistance associated with the Hall effect: H + + + + + + + + + + E y - -

More information

J 12 J 23 J 34. Driving forces in the nano-magnetism world. Intra-atomic exchange, electron correlation effects: Inter-atomic exchange: MAGNETIC ORDER

J 12 J 23 J 34. Driving forces in the nano-magnetism world. Intra-atomic exchange, electron correlation effects: Inter-atomic exchange: MAGNETIC ORDER Driving forces in the nano-magnetism world Intra-atomic exchange, electron correlation effects: LOCAL (ATOMIC) MAGNETIC MOMENTS m d or f electrons Inter-atomic exchange: MAGNETIC ORDER H exc J S S i j

More information

7.2 Dipolar Interactions and Single Ion Anisotropy in Metal Ions

7.2 Dipolar Interactions and Single Ion Anisotropy in Metal Ions 7.2 Dipolar Interactions and Single Ion Anisotropy in Metal Ions Up to this point, we have been making two assumptions about the spin carriers in our molecules: 1. There is no coupling between the 2S+1

More information

Cover Page. The handle holds various files of this Leiden University dissertation.

Cover Page. The handle   holds various files of this Leiden University dissertation. Cover Page The handle http://hdl.handle.net/1887/49403 holds various files of this Leiden University dissertation. Author: Keesman, R. Title: Topological phases and phase transitions in magnets and ice

More information

Solid state physics. Lecture 9: Magnetism. Prof. Dr. U. Pietsch

Solid state physics. Lecture 9: Magnetism. Prof. Dr. U. Pietsch Solid state physics Lecture 9: Magnetism Prof. Dr. U. Pietsch Diamagnetism and Paramagnetsim Materie in magnetic field m 0 0 H M H(1 H 0 0M m M magnetiszation magnetic susceptibility - magnetic permeability

More information

Ferromagnetism and antiferromagnetism ferromagnetism (FM) antiferromagnetism (AFM) ferromagnetic domains nanomagnetic particles

Ferromagnetism and antiferromagnetism ferromagnetism (FM) antiferromagnetism (AFM) ferromagnetic domains nanomagnetic particles Dept of Phys Ferromagnetism and antiferromagnetism ferromagnetism (FM) exchange interaction, Heisenberg model spin wave, magnon antiferromagnetism (AFM) ferromagnetic domains nanomagnetic particles M.C.

More information

Magnetism in Condensed Matter

Magnetism in Condensed Matter Magnetism in Condensed Matter STEPHEN BLUNDELL Department of Physics University of Oxford OXFORD 'UNIVERSITY PRESS Contents 1 Introduction 1.1 Magnetic moments 1 1 1.1.1 Magnetic moments and angular momentum

More information

MSE 7025 Magnetic Materials (and Spintronics)

MSE 7025 Magnetic Materials (and Spintronics) MSE 7025 Magnetic Materials (and Spintronics) Lecture 4: Category of Magnetism Chi-Feng Pai cfpai@ntu.edu.tw Course Outline Time Table Week Date Lecture 1 Feb 24 Introduction 2 March 2 Magnetic units and

More information

Spin Superfluidity and Graphene in a Strong Magnetic Field

Spin Superfluidity and Graphene in a Strong Magnetic Field Spin Superfluidity and Graphene in a Strong Magnetic Field by B. I. Halperin Nano-QT 2016 Kyiv October 11, 2016 Based on work with So Takei (CUNY), Yaroslav Tserkovnyak (UCLA), and Amir Yacoby (Harvard)

More information

Phase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality. Hans-Henning Klauss. Institut für Festkörperphysik TU Dresden

Phase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality. Hans-Henning Klauss. Institut für Festkörperphysik TU Dresden Phase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality Hans-Henning Klauss Institut für Festkörperphysik TU Dresden 1 References [1] Stephen Blundell, Magnetism in Condensed

More information

Magnetism. Ram Seshadri MRL 2031, x6129, Some basics:

Magnetism. Ram Seshadri MRL 2031, x6129, Some basics: Magnetism Ram Seshadri MRL 2031, x6129, seshadri@mrl.ucsb.edu Some basics: A magnet is associated with magnetic lines of force, and a north pole and a south pole. he lines of force come out of the north

More information

PHY331 Magnetism. Lecture 4

PHY331 Magnetism. Lecture 4 PHY331 Magnetism Lecture 4 Last week Discussed Langevin s theory of diamagnetism. Use angular momentum of precessing electron in magnetic field to derive the magnetization of a sample and thus diamagnetic

More information

S j H o = gµ o H o. j=1

S j H o = gµ o H o. j=1 LECTURE 17 Ferromagnetism (Refs.: Sections 10.6-10.7 of Reif; Book by J. S. Smart, Effective Field Theories of Magnetism) Consider a solid consisting of N identical atoms arranged in a regular lattice.

More information

Introduction to magnetism of confined systems

Introduction to magnetism of confined systems Introduction to magnetism of confined systems P. Vavassori CIC nanogune Consolider, San Sebastian, Spain; nano@nanogune.eu Basics: diamagnetism and paramagnetism Every material which is put in a magnetic

More information

Mean-field theory. Alessandro Vindigni. ETH October 29, Laboratorium für Festkörperphysik, ETH Zürich

Mean-field theory. Alessandro Vindigni. ETH October 29, Laboratorium für Festkörperphysik, ETH Zürich Alessandro Vindigni Laboratorium für Festkörperphysik, ETH Zürich ETH October 29, 2012 Lecture plan N-body problem Lecture plan 1. Atomic magnetism (Pescia) 2. Magnetism in solids (Pescia) 3. Magnetic

More information

PHYSICS 4750 Physics of Modern Materials Chapter 8: Magnetic Materials

PHYSICS 4750 Physics of Modern Materials Chapter 8: Magnetic Materials PHYSICS 475 Physics of Modern Materials Chapter 8: Magnetic Materials 1. Atomic Magnetic Dipole Moments A magnetic solid is one in which at least some of the atoms have a permanent magnetic dipole moment

More information

Magnetic Materials. The inductor Φ B = LI (Q = CV) = L I = N Φ. Power = VI = LI. Energy = Power dt = LIdI = 1 LI 2 = 1 NΦ B capacitor CV 2

Magnetic Materials. The inductor Φ B = LI (Q = CV) = L I = N Φ. Power = VI = LI. Energy = Power dt = LIdI = 1 LI 2 = 1 NΦ B capacitor CV 2 Magnetic Materials The inductor Φ B = LI (Q = CV) Φ B 1 B = L I E = (CGS) t t c t EdS = 1 ( BdS )= 1 Φ V EMF = N Φ B = L I t t c t B c t I V Φ B magnetic flux density V = L (recall I = C for the capacitor)

More information

Physics of Magnetism. Chapter references are to Essentials of Paleomagnetism, UC Press, 2010

Physics of Magnetism. Chapter references are to Essentials of Paleomagnetism, UC Press, 2010 Physics of Magnetism Chapter references are to Essentials of Paleomagnetism, UC Press, 2010 http://magician.ucsd.edu/essentials 1 Magnetic units (sorry!) SI cgs Magnetic fields as the gradient of a scalar

More information

Paramagnetism and Diamagnetism. Paramagnets (How do paramagnets differ fundamentally from ferromagnets?)

Paramagnetism and Diamagnetism. Paramagnets (How do paramagnets differ fundamentally from ferromagnets?) Paramagnetism and Diamagnetism Paramagnets (How do paramagnets differ fundamentally from ferromagnets?) The study of paramagnetism allows us to investigate the atomic magnetic moments of atoms almost in

More information

Competing Ferroic Orders The magnetoelectric effect

Competing Ferroic Orders The magnetoelectric effect Competing Ferroic Orders The magnetoelectric effect Cornell University I would found an institution where any person can find instruction in any study. Ezra Cornell, 1868 Craig J. Fennie School of Applied

More information

MOLECULAR MAGNETISM. Leigh Jones Room 133 School of Chemistry NUI Galway. Introduction to Molecular Magnetism

MOLECULAR MAGNETISM. Leigh Jones Room 133 School of Chemistry NUI Galway. Introduction to Molecular Magnetism 4 th year undergraduate course ecture 5 MOECUAR MAGNETISM eigh Jones Room 133 School of Chemistry NUI Galway ecture 6: 5: Outcomes Introduction to Molecular Magnetism To understand the difference between

More information

Antiferromagnetic Textures

Antiferromagnetic Textures Antiferromagnetic Textures This image cannot currently be displayed. ULRICH K. RÖSSLE IFW DRESDEN SPICE Workshop Antiferromagnetic Spintronics 26.-30/09/2016 Schloss Waldthausen u.roessler@ifw-dresden.de

More information

Chapter 8 Magnetic Resonance

Chapter 8 Magnetic Resonance Chapter 8 Magnetic Resonance 9.1 Electron paramagnetic resonance 9.2 Ferromagnetic resonance 9.3 Nuclear magnetic resonance 9.4 Other resonance methods TCD March 2007 1 A resonance experiment involves

More information

The Quantum Theory of Magnetism

The Quantum Theory of Magnetism The Quantum Theory of Magnetism Norberto Mains McGill University, Canada I: 0 World Scientific Singapore NewJersey London Hong Kong Contents 1 Paramagnetism 1.1 Introduction 1.2 Quantum mechanics of atoms

More information

Magnetism and Magnetic Switching

Magnetism and Magnetic Switching Magnetism and Magnetic Switching Robert Stamps SUPA-School of Physics and Astronomy University of Glasgow A story from modern magnetism: The Incredible Shrinking Disk Instead of this: (1980) A story from

More information

l μ M Right hand Screw rule

l μ M Right hand Screw rule Magnetic materials Magnetic property The response of the materials to external magnetic field All the materials are magnetic, only the degree of response varies, which is measured in terms of their magnetization

More information

Crystalline and Magnetic Anisotropy of the 3d Transition-Metal Monoxides

Crystalline and Magnetic Anisotropy of the 3d Transition-Metal Monoxides Crystalline and of the 3d Transition-Metal Monoxides Institut für Festkörpertheorie und -optik Friedrich-Schiller-Universität Max-Wien-Platz 1 07743 Jena 2012-03-23 Introduction Crystalline Anisotropy

More information

μ (vector) = magnetic dipole moment (not to be confused with the permeability μ). Magnetism Electromagnetic Fields in a Solid

μ (vector) = magnetic dipole moment (not to be confused with the permeability μ). Magnetism Electromagnetic Fields in a Solid Magnetism Electromagnetic Fields in a Solid SI units cgs (Gaussian) units Total magnetic field: B = μ 0 (H + M) = μ μ 0 H B = H + 4π M = μ H Total electric field: E = 1/ε 0 (D P) = 1/εε 0 D E = D 4π P

More information

Examination paper for TFY4245 Faststoff-fysikk, videregående kurs

Examination paper for TFY4245 Faststoff-fysikk, videregående kurs Side 1 av 6 Department of Physics xamination paper for TFY445 Faststoff-fysikk, videregående kurs Academic contact during examination: Ragnvald Mathiesen Phone: 976913 xamination date: 0.06.015 xamination

More information

Magnetism of Atoms and Ions. Wulf Wulfhekel Physikalisches Institut, Karlsruhe Institute of Technology (KIT) Wolfgang Gaede Str. 1, D Karlsruhe

Magnetism of Atoms and Ions. Wulf Wulfhekel Physikalisches Institut, Karlsruhe Institute of Technology (KIT) Wolfgang Gaede Str. 1, D Karlsruhe Magnetism of Atoms and Ions Wulf Wulfhekel Physikalisches Institut, Karlsruhe Institute of Technology (KIT) Wolfgang Gaede Str. 1, D-76131 Karlsruhe 1 0. Overview Literature J.M.D. Coey, Magnetism and

More information

The Oxford Solid State Basics

The Oxford Solid State Basics The Oxford Solid State Basics Steven H. Simon University of Oxford OXFORD UNIVERSITY PRESS Contents 1 About Condensed Matter Physics 1 1.1 What Is Condensed Matter Physics 1 1.2 Why Do We Study Condensed

More information

Exchange bias in core/shell magnetic nanoparticles: experimental results and numerical simulations

Exchange bias in core/shell magnetic nanoparticles: experimental results and numerical simulations Exchange bias in core/shell magnetic nanoparticles: experimental results and numerical simulations Xavier Batlle, A. Labarta, Ò. Iglesias, M. García del Muro and M. Kovylina Goup of Magnetic Nanomaterials

More information

7. Basics of Magnetization Switching

7. Basics of Magnetization Switching Beyond CMOS computing 7. Basics of Magnetization Switching Dmitri Nikonov Dmitri.e.nikonov@intel.com 1 Outline Energies in a nanomagnet Precession in a magnetic field Anisotropies in a nanomagnet Hysteresis

More information

Chapter 3. Magnetic Model. 3.1 Magnetic interactions

Chapter 3. Magnetic Model. 3.1 Magnetic interactions Chapter 3 Magnetic Model In this chapter, the micromagnetic model for the description of the magnetic properties of a laterally nanostructured film during growth is presented. The main physical idea of

More information

EFFECTIVE MAGNETIC HAMILTONIANS: ab initio determination

EFFECTIVE MAGNETIC HAMILTONIANS: ab initio determination ICSM212, Istanbul, May 3, 212, Theoretical Magnetism I, 17:2 p. 1 EFFECTIVE MAGNETIC HAMILTONIANS: ab initio determination Václav Drchal Institute of Physics ASCR, Praha, Czech Republic in collaboration

More information

2 B B D (E) Paramagnetic Susceptibility. m s probability. A) Bound Electrons in Atoms

2 B B D (E) Paramagnetic Susceptibility. m s probability. A) Bound Electrons in Atoms Paramagnetic Susceptibility A) Bound Electrons in Atoms m s probability B +½ p ½e x Curie Law: 1/T s=½ + B ½ p + ½e +x With increasing temperature T the alignment of the magnetic moments in a B field is

More information

First-Principles Calculation of Exchange Interactions

First-Principles Calculation of Exchange Interactions Chapter 2 First-Principles Calculation of Exchange Interactions Before introducing the first-principles methods for the calculation of exchange interactions in magnetic systems we will briefly review two

More information

PHY331 Magnetism. Lecture 3

PHY331 Magnetism. Lecture 3 PHY331 Magnetism Lecture 3 Last week Derived magnetic dipole moment of a circulating electron. Discussed motion of a magnetic dipole in a constant magnetic field. Showed that it precesses with a frequency

More information

Macroscopic properties II

Macroscopic properties II Paolo Allia DISAT Politecnico di Torino acroscopic properties II acroscopic properties II Crucial aspects of macroscopic ferromagnetism Crystalline magnetic anisotropy Shape anisotropy Ferromagnetic domains

More information

Luigi Paolasini

Luigi Paolasini Luigi Paolasini paolasini@esrf.fr LECTURE 4: MAGNETIC INTERACTIONS - Dipole vs exchange magnetic interactions. - Direct and indirect exchange interactions. - Anisotropic exchange interactions. - Interplay

More information

Chapter 2 Magnetic Properties

Chapter 2 Magnetic Properties Chapter 2 Magnetic Properties Abstract The magnetic properties of a material are the basis of their applications. Specifically, the contrast agents that will be developed in Chaps. 4 and 5 use their magnetic

More information

The Physics of Ferromagnetism

The Physics of Ferromagnetism Terunobu Miyazaki Hanmin Jin The Physics of Ferromagnetism Springer Contents Part I Foundation of Magnetism 1 Basis of Magnetism 3 1.1 Basic Magnetic Laws and Magnetic Quantities 3 1.1.1 Basic Laws of

More information

Lecture contents. Magnetic properties Diamagnetism Band paramagnetism Atomic paramagnetism Ferromagnetism. Molecular field theory Exchange interaction

Lecture contents. Magnetic properties Diamagnetism Band paramagnetism Atomic paramagnetism Ferromagnetism. Molecular field theory Exchange interaction 1 Lecture contents Magnetic properties Diamagnetism and paramagnetism Atomic paramagnetism Ferromagnetism Molecular field theory Exchange interaction NNSE 58 EM Lecture #1 [SI] M magnetization or magnetic

More information

Spin excitations in magnetic structures of different dimensions

Spin excitations in magnetic structures of different dimensions Spin excitations in magnetic structures of different dimensions Wulf Wulfhekel Physikalisches Institut, Universität Karlsruhe (TH) Wolfgang Gaede Str. 1, D-76131 Karlsruhe 0. Overview Chapters of spin

More information

Contents. Acknowledgments

Contents. Acknowledgments MAGNETIC MATERIALS Fundamentals and Applications Second edition NICOLA A. SPALDIN University of California, Santa Barbara CAMBRIDGE UNIVERSITY PRESS Contents Acknowledgments page xiii I Basics 1 Review

More information

Supplementary Figure 1 Representative sample of DW spin textures in a

Supplementary Figure 1 Representative sample of DW spin textures in a Supplementary Figure 1 Representative sample of DW spin textures in a Fe/Ni/W(110) film. (a) to (d) Compound SPLEEM images of the Fe/Ni/W(110) sample. As in Fig. 2 in the main text, Fe thickness is 1.5

More information

Examples of Lifshitz topological transition in interacting fermionic systems

Examples of Lifshitz topological transition in interacting fermionic systems Examples of Lifshitz topological transition in interacting fermionic systems Joseph Betouras (Loughborough U. Work in collaboration with: Sergey Slizovskiy (Loughborough, Sam Carr (Karlsruhe/Kent and Jorge

More information

Examination paper for TFY4245 Faststoff-fysikk, videregående kurs

Examination paper for TFY4245 Faststoff-fysikk, videregående kurs Side 1 av 7 Department of Physics Examination paper for TFY445 Faststoff-fysikk, videregående kurs Academic contact during examination: Ragnvald Mathiesen Phone: 976913 Examination date: 08.06.017 Examination

More information

2.1 Experimental and theoretical studies

2.1 Experimental and theoretical studies Chapter 2 NiO As stated before, the first-row transition-metal oxides are among the most interesting series of materials, exhibiting wide variations in physical properties related to electronic structure.

More information

Magnetic Oxides. Gerald F. Dionne. Department of Materials Science and Engineering Massachusetts Institute of Technology

Magnetic Oxides. Gerald F. Dionne. Department of Materials Science and Engineering Massachusetts Institute of Technology Magnetic Oxides Gerald F. Dionne Department of Materials Science and Engineering Massachusetts Institute of Technology Spins in Solids Summer School University of Virginia Charlottesville, VA 21 June 2006

More information

Anisotropic Magnetic Structures in Iron-Based Superconductors

Anisotropic Magnetic Structures in Iron-Based Superconductors Anisotropic Magnetic Structures in Iron-Based Superconductors Chi-Cheng Lee, Weiguo Yin & Wei Ku CM-Theory, CMPMSD, Brookhaven National Lab Department of Physics, SUNY Stony Brook Another example of SC

More information

Magnetic properties of spherical fcc clusters with radial surface anisotropy

Magnetic properties of spherical fcc clusters with radial surface anisotropy Magnetic properties of spherical fcc clusters with radial surface anisotropy D. A. Dimitrov and G. M. Wysin Department of Physics Kansas State University Manhattan, KS 66506-2601 (December 6, 1994) We

More information

Electronic Higher Multipoles in Solids

Electronic Higher Multipoles in Solids Electronic Higher Multipoles in Solids Yoshio Kuramoto Department of Physics, Tohoku University Outline Elementary examples of multiple moments Role of multipole moments in solids Case studies octupole

More information

Magnetism: Spin-orbit coupling magnetic exchange and anisotropy

Magnetism: Spin-orbit coupling magnetic exchange and anisotropy VASP workshop Rennes 2016 Magnetism: Spin-orbit coupling magnetic exchange and anisotropy Xavier Rocquefelte Institut des Sciences Chimiques de Rennes (UMR 6226) Université de Rennes 1, FRANCE INTRODUCTION

More information

SOLID STATE PHYSICS. Second Edition. John Wiley & Sons. J. R. Hook H. E. Hall. Department of Physics, University of Manchester

SOLID STATE PHYSICS. Second Edition. John Wiley & Sons. J. R. Hook H. E. Hall. Department of Physics, University of Manchester SOLID STATE PHYSICS Second Edition J. R. Hook H. E. Hall Department of Physics, University of Manchester John Wiley & Sons CHICHESTER NEW YORK BRISBANE TORONTO SINGAPORE Contents Flow diagram Inside front

More information

Lecture 11: Transition metals (1) Basics and magnetism

Lecture 11: Transition metals (1) Basics and magnetism Lecture 11: Transition metals (1) Basics and magnetism Oxidation states in transition metal compounds Ligand field theory Magnetism Susceptibility Temperature dependence Magnetic moments Figure: Wikipedia

More information

The Basics of Magnetic Resonance Imaging

The Basics of Magnetic Resonance Imaging The Basics of Magnetic Resonance Imaging Nathalie JUST, PhD nathalie.just@epfl.ch CIBM-AIT, EPFL Course 2013-2014-Chemistry 1 Course 2013-2014-Chemistry 2 MRI: Many different contrasts Proton density T1

More information

Luigi Paolasini

Luigi Paolasini Luigi Paolasini paolasini@esrf.fr LECTURE 7: Magnetic excitations - Phase transitions and the Landau mean-field theory. - Heisenberg and Ising models. - Magnetic excitations. External parameter, as for

More information

PHY331 Magnetism. Lecture 6

PHY331 Magnetism. Lecture 6 PHY331 Magnetism Lecture 6 Last week Learned how to calculate the magnetic dipole moment of an atom. Introduced the Landé g-factor. Saw that it compensates for the different contributions from the orbital

More information

THE INFLUENCE OF A SURFACE ON HYSTERESIS LOOPS FOR SINGLE-DOMAIN FERROMAGNETIC NANOPARTICLES

THE INFLUENCE OF A SURFACE ON HYSTERESIS LOOPS FOR SINGLE-DOMAIN FERROMAGNETIC NANOPARTICLES THE INFLUENCE OF A SURFACE ON HYSTERESIS LOOPS FOR SINGLE-DOMAIN FERROMAGNETIC NANOPARTICLES A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science By Saad Alsari

More information

SUPPLEMENTARY NOTE 1: ANISOTROPIC MAGNETORESISTANCE PHE-

SUPPLEMENTARY NOTE 1: ANISOTROPIC MAGNETORESISTANCE PHE- SUPPLEMENTARY NOTE 1: ANISOTROPIC MAGNETORESISTANCE PHE- NOMENOLOGY In the main text we introduce anisotropic magnetoresistance (AMR) in analogy to ferromagnets where non-crystalline and crystalline contributions

More information

Spin Interactions. Giuseppe Pileio 24/10/2006

Spin Interactions. Giuseppe Pileio 24/10/2006 Spin Interactions Giuseppe Pileio 24/10/2006 Magnetic moment µ = " I ˆ µ = " h I(I +1) " = g# h Spin interactions overview Zeeman Interaction Zeeman interaction Interaction with the static magnetic field

More information

Theory of Electron Spin Resonance in Ferromagnetically Correlated Heavy Fermion Compounds

Theory of Electron Spin Resonance in Ferromagnetically Correlated Heavy Fermion Compounds magnetochemistry Article Theory of Electron Spin Resonance in Ferromagnetically Correlated Heavy Fermion Compounds Pedro Schlottmann Department of Physics, Florida State University, Tallahassee, FL 32306,

More information

SECOND PUBLIC EXAMINATION. Honour School of Physics Part C: 4 Year Course. Honour School of Physics and Philosophy Part C C3: CONDENSED MATTER PHYSICS

SECOND PUBLIC EXAMINATION. Honour School of Physics Part C: 4 Year Course. Honour School of Physics and Philosophy Part C C3: CONDENSED MATTER PHYSICS 2753 SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C3: CONDENSED MATTER PHYSICS TRINITY TERM 2011 Wednesday, 22 June, 9.30 am 12.30

More information

Lecture 24 - Magnetism

Lecture 24 - Magnetism Lecture 24: Magnetism (Kittel Ch. 1112) Quantum Mechanics Magnetism ElectronElectron Interactions Physics 460 F 2006 Lect 24 1 Outline Magnetism is a purely quantum phenomenon! Totally at variance with

More information

High T C copper oxide superconductors and CMR:

High T C copper oxide superconductors and CMR: High T C copper oxide superconductors and CMR: Ram Seshadri (seshadri@mrl.ucsb.edu) The Ruddlesden-Popper phases: Ruddlesden-Popper phases are intergrowths of perovskite slabs with rock salt slabs. First

More information

Electron Correlation

Electron Correlation Series in Modern Condensed Matter Physics Vol. 5 Lecture Notes an Electron Correlation and Magnetism Patrik Fazekas Research Institute for Solid State Physics & Optics, Budapest lb World Scientific h Singapore

More information

MAGNETIC MATERIALS. Fundamentals and device applications CAMBRIDGE UNIVERSITY PRESS NICOLA A. SPALDIN

MAGNETIC MATERIALS. Fundamentals and device applications CAMBRIDGE UNIVERSITY PRESS NICOLA A. SPALDIN MAGNETIC MATERIALS Fundamentals and device applications NICOLA A. SPALDIN CAMBRIDGE UNIVERSITY PRESS Acknowledgements 1 Review of basic magnetostatics 1.1 Magnetic field 1.1.1 Magnetic poles 1.1.2 Magnetic

More information

Metropolis Monte Carlo simulation of the Ising Model

Metropolis Monte Carlo simulation of the Ising Model Metropolis Monte Carlo simulation of the Ising Model Krishna Shrinivas (CH10B026) Swaroop Ramaswamy (CH10B068) May 10, 2013 Modelling and Simulation of Particulate Processes (CH5012) Introduction The Ising

More information

Phenomenology and Models of Exchange Bias in Core /Shell Nanoparticles

Phenomenology and Models of Exchange Bias in Core /Shell Nanoparticles Phenomenology and Models of Exchange Bias in Core /Shell Nanoparticles Xavier Batlle and Amílcar Labarta Departament de Física Fonamental and Institut de Nanociència i Nanotecnologia Universitat de Barcelona,

More information

Magnetism of the Localized Electrons on the Atom

Magnetism of the Localized Electrons on the Atom Magnetism of the Localized Electrons on the Atom 1. The hydrogenic atom and angular momentum 2. The many-electron atom. Spin-orbit coupling 4. The Zeeman interaction 5. Ions in solids 6. Paramagnetism

More information

~~r ~o~/, ' , I. l: z: n.-b -z 01. ?;Cl. 60) Pro CD'fCJ7 '; ftu-0j~

~~r ~o~/, ' , I. l: z: n.-b -z 01. ?;Cl. 60) Pro CD'fCJ7 '; ftu-0j~ i -1- ~~r ~o~/, ------', I l: z: n.-b -z 01?;Cl 60) 1---.-- Pro CD'fCJ7 '; ftu-0j~ APPLICATIONS 1/ What features of atomic structure determine whether an element is diamagnetic or paramagnetic? Explain.

More information

Magnetization reversal and ferrimagnetism in Pr 1 x Nd x MnO 3

Magnetization reversal and ferrimagnetism in Pr 1 x Nd x MnO 3 Bull. Mater. Sci., Vol. 37, No. 4, June 2014, pp. 809 813. Indian Academy of Sciences. Magnetization reversal and ferrimagnetism in Pr 1 x Nd x MnO 3 SANJAY BISWAS, MOMIN HOSSAIN KHAN and SUDIPTA PAL*

More information

Fe Co Si. Fe Co Si. Ref. p. 59] d elements and C, Si, Ge, Sn or Pb Alloys and compounds with Ge

Fe Co Si. Fe Co Si. Ref. p. 59] d elements and C, Si, Ge, Sn or Pb Alloys and compounds with Ge Ref. p. 59] 1.5. 3d elements and C, Si, Ge, Sn or Pb 7 1.75 1.50 Co Si 0.8 0. 3.50 3.5 Co Si 0.8 0. H cr Magnetic field H [koe] 1.5 1.00 0.75 0.50 0.5 C C IF "A" P Frequency ωγ / e [koe] 3.00.75.50.5.00

More information

Collective Effects. Equilibrium and Nonequilibrium Physics

Collective Effects. Equilibrium and Nonequilibrium Physics Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March 2006 Collective Effects in Equilibrium and Nonequilibrium Physics Website: http://cncs.bnu.edu.cn/mccross/course/ Caltech

More information

M. A. Gusmão IF-UFRGS

M. A. Gusmão IF-UFRGS M. A. Gusmão IF-UFRGS - 217 1 FIP164-217/2 Text 9 Mean-field approximation - II Heisenberg Hamiltonian in wave-vector space As we saw in Text 8, the uniform susceptibility does not diverge in the case

More information

Making the Invisible Visible: Probing Antiferromagnetic Order in Novel Materials

Making the Invisible Visible: Probing Antiferromagnetic Order in Novel Materials Making the Invisible Visible: Probing Antiferromagnetic Order in Novel Materials Elke Arenholz Lawrence Berkeley National Laboratory Antiferromagnetic contrast in X-ray absorption Ni in NiO Neel Temperature

More information

Crystal Field Theory

Crystal Field Theory Crystal Field Theory It is not a bonding theory Method of explaining some physical properties that occur in transition metal complexes. Involves a simple electrostatic argument which can yield reasonable

More information

compound Cs 2 Cu 2 Mo 3 O 12

compound Cs 2 Cu 2 Mo 3 O 12 133 Cs-NMR study on aligned powder of competing spin chain compound A Yagi 1, K Matsui 1 T Goto 1, M Hase 2 and T Sasaki 3 1 2 Sophia University, Physics Division, Tokyo, 102-8554, Japan National Institute

More information

Interaction of matter with magnetic fields

Interaction of matter with magnetic fields LN07-1 Interaction of matter with magnetic fields All substances have magnetic properties, which can be determined by examining their behaviour in the presence of an external magnetic field, H. N S When

More information

Static and dynamic Jahn-Teller effects and antiferromagnetic order in PrO 2 : A mean-field analysis

Static and dynamic Jahn-Teller effects and antiferromagnetic order in PrO 2 : A mean-field analysis Static and dynamic Jahn-Teller effects and antiferromagnetic order in PrO 2 : A mean-field analysis Jens Jensen Niels Bohr Institute, Universitetsparken 5, 2100 Copenhagen, Denmark Received 7 June 2007;

More information

Phase transitions and critical phenomena

Phase transitions and critical phenomena Phase transitions and critical phenomena Classification of phase transitions. Discontinous (st order) transitions Summary week -5 st derivatives of thermodynamic potentials jump discontinously, e.g. (

More information

Lecture 2: Magnetic Anisotropy Energy (MAE)

Lecture 2: Magnetic Anisotropy Energy (MAE) Lecture : Magnetic Anisotropy Energy (MAE) 1. Magnetic anisotropy energy = f(t). Anisotropic magnetic moment f(t) [111] T=3 K Characteristic energies of metallic ferromagnets M (G) 5 3 [1] 1 binding energy

More information

arxiv:cond-mat/ v3 [cond-mat.stat-mech] 14 Nov 2002

arxiv:cond-mat/ v3 [cond-mat.stat-mech] 14 Nov 2002 arxiv:cond-mat/0210674v3 [cond-mat.stat-mech] 14 Nov 2002 Why is Bloch s T 3/2 -law also observed in d=2 dimensions? U. Krey, Inst. für Physik II, Universität Regensburg, 93040 Regensburg, Germany Oct.

More information

Follow this and additional works at: Part of the Physics Commons

Follow this and additional works at:   Part of the Physics Commons Wright State University CORE Scholar Browse all Theses and Dissertations Theses and Dissertations 2016 Surface Effect Of Ferromagnetic Nanoparticles On Transition Between Single- And Multi-Domain Structure

More information

Magnetism to Spintronics

Magnetism to Spintronics Magnetism to Spintronics Introduction to Solid State Physics Kittel 8 th ed Chap. 11-13 & Condensed Matter Physics Marder 2 nd ed Chap. 24-26 Magnetism -1 Why do most broken permanent magnets repel each

More information

Material Science. Chapter 16. Magnetic properties

Material Science. Chapter 16. Magnetic properties Material Science Chapter 16. Magnetic properties Engineering materials are important in everyday life because of their versatile structural properties. Other than these properties, they do play an important

More information