TYPES OF INTERACTION IN MAGNETISM

Ferromagnetism is a type of magnetism characterized by an spontaneous parallel alignment of atomic magnetic moments, with long range order. Examples of ferromagnets are the elements iron, nickel, and cobalt. This order disappears above a certain temperature called the Curie temperature. From the different interactions taking place in atoms constituing a solid, the question is which of those shows responsible for macroscopic magnetic effects? A quick look at the dipolar Interaction will reveil that this type cannot account for long range order. But exchange interaction, looked at in section 3.2, presents a valid explanation: The energydifference between a singlet and a triplet state is at the very core of explaining the long range order. So in the end magnetism on the bigger scale is a purely quantum mechanical effect.

MINISTRY OF EDUCATION AND TRAINING HUE UNIVERSITY COLLEGE OF EDUCATION LE THI HOAI TYPES OF INTERACTION IN MAGNETISM INDEPENDENT STUDY Scientific Advisor DR PHAM HUONG THAO HUE, 04/2016 ACKNOWLEGMENTS I’m grateful to my supervisor, Dr PHAM HUONG THAO for helping during my study My would not have been complete without her guidance at the beginning and the fruitful discussion later on Her continual encouragement careful reading, critical comments and patient guidance made my work more enjoyable and easier I would like to thanks all teachers in Physics department of Hue University’s College of Education and Foreign teachers for teaching and helping me in my courses Lastly, I would to acknowledge to my family, my friend and my classmate Their love, support and constant encouragement gave me a great deal of strength and determination that help me during the stressful time of writing this paper It is my great pleasure to thanks these people Student Le Thi Hoai TABLE OF CONTENTS LIST OF FIGURES Figure 1: Vector model of the atom: The plane of the electron’s orbit can have only certain possible orientations, we say it is spatially quantized [Reprinted from H Lueken, Magnetochemie, Auflage; Teubner Verlag]…………….…………12 Figure 2: Magnetic moment due to a current loop [Reprinted from Stephen Blundell, Magnetism in Condensed Matter, 1st Edition; Oxford Univ Press] ……….13 Figure 3: Hydrogen atom [Reprinted from Stephen Blundell, Magnetism in Condensed Matter, 1st Edition; Oxford Univ Press]…………………………….14 Figure 4: Electron spin in a magnetic field Bz [Reprinted from H Haken, H.C Wolf; Atomund Quantenphysik, Auflage; Springer Verlag]…………………16 Figure 5: Vector model of LS-coupling [Reprinted from Wolfgang Demtröder, Experimentalphysik Band 3, Auflage; Springer Verlag]………………………20 Figure 6: Hund’s rule assume combination to form S and L, or imply L-S (RussellSaunders) coupling……………… .…………20 Figure 7: Characteristic magnetic susceptibilities of diamagnetic and paramagnetic substance………………………………………………………………………….23 Figure 8: Fe, Co and Nickel are ferromagnetic so that they have a spontaneous magnetization with no applied field [Reprinted from Stephen Blundell, Magnetism in Condensed Matter, 1st Edition; Oxford Univ Press]……………………….….24 Figure 9: Dipol-Dipol interaction energy for two colinear dipoles with the same ur m= m moments [Reprinted from J Stöhr, H.C Siegmann; Magnetism, 1st Edition; Springer Verlag]…………………………………………………………27 Figure 10: H2 Molecule [Reprintedfrom H Haken, H.C Wolf; Atom- und Quantenphysik, Auflage; SpringerVerlag]…………………………………….29 Figure 11: Energies of H2 as a function of distance for singlet and triplet states according to Heitler-London [Reprinted from Andreas Bringer, Heisenberg Model; Institut für Festkörperforschung and Institut for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany]………………………….32 Figure 12: Antiparallel alignment for small interatomic distances…………….34 Figure 13: Parallel alignment for large interatomic distances……… ……… 34 Figure 14: The Bethe-Slater curve……………………………………… ……35 Figure 15: The coefficient of indirect (RKKY) exchange versus the interatomic spacing a…………………………………………………………………………36 Figure 16: Antiferromagnetic MnO [Reprinted from Blundell, Stephen; Magnetism in Condensed Matter, 1st Edition; Oxford Univ Press]……… ……36 Figure 17: Magnetization curve of iron [Reprinted from Heiko Lueken, Magnetochemie, Auflage; Teubner Verlag]………………………………….38 INTRODUCTION Reasons for choosing the topic Even though magnetic phenomena have been known for centuries but modern physics was able to put them on a solid basis Earlier attempts, especially explaining magnetism on the macroscopic scale, remained in some mysterious ways Only with the birth of quantum mechanics, the magnetic phenomena could be understood clearly (via exchange interaction) Today where the world market for magnetic media and recording equipment reaches billions dollars per year, the magnetic materials, which are the basis of the present technological revolution, remains a very interesting and active field of physical research The interactions take an important role in magnetism, especially exchange interaction Studying the interactions helps students fill gaps in basic knowledge of magnetism For the above reasons, I write the independent study: “types of interaction in magnetism “ Aims of study Studying theoretically some fundamental concepts in magnetism as magnetic moment of atom and some interactions in magnetism (dipolar interaction and exchange interaction) Contents of study The independent study focuses on the following problems: Magnetic moments of single atoms Many electron atoms with unpaired electrons Types of interaction in magnetism Body of independent study    Additional to the introduction and the conclusion, the graduate thesis’s content consists of four chapter: Chapter 1: Magnetic moments of single atoms Chapter 2: Many electron atoms with Unpaired electrons Chapter 3: Types of interaction in magnetism Chapter 4: Other types of interaction 111Equation Chapter Section 1CHAPTER MAGNETIC MOMENTS OF SINGLE ATOMS 1.1 Quantum Mechanics 1.1.1 The Schrodinger equation in spherical coordinates The energy levels of an atom are solutions to the Schrodinger equation: ΗΨ = En Ψ If the momentum p 212\* MERGEFORMAT (.) in the experession for the classical Hamiltonian for a one particle system is replaced by −ih∂ / ∂xi the corresponding operator is obtained: p2 h2 H= + V → H op = − ∇ +V, 2me 2me MERGEFORMAT (.) where V is the potential in which the electron moves In three dimensions the Schrodinger equation generalizes to:  h2  ∇ + V ÷Ψ = E Ψ, −  2me  313\* 414\* MERGEFORMAT (.) ∇ where is the Laplacian operator Using the Laplacian in spherical coodinates, the Schrodinger equation becomes:  h2  ∂  ∂2 ∂2  −  + + ÷+ V  Ψ = E Ψ , ∂y ∂z   2me  ∂x  MERGEFORMAT (.) 515\* h2  ∂  ∂  ∂  ∂  ∂2  →− Ψ + V ( r ) Ψ = E Ψ r ÷+  sin θ ÷+ 2me  r ∂r  ∂r  r sin θ ∂θ  ∂θ  r sin θ ∂φ  616\* MERG EFOR MAT In spherical coodinates, Ψ = Ψ (r ,θ ,φ ), (.) by assuming the wave function to be a product of a radial function and two angular functions Ψ = R(r )Θ(θ )Φ(φ ), 717\* MERGEFORMAT (.) And by making use of the Legendre polynomials Φ = C exp(imφ ), with l and m Pl m ( cosθ ) and being integers, the following expression for the energy result: −e mr4 En = , 2h2 n 818\* MERGEFORMAT (.) mm mr = e p me + m p mr n where is an integer, is the reduced mass given by 1.1.2 The Quantum Numbers The theory of quantum mechanics tells us that in an atom, the electrons are found in orbitals, and each orbitak has a characteristic energy Orbital means “small orbit” We are interested in two properties of orbitals – their energies and their shapes Their energies are important because we normally find atoms in their most stable states, which we call their ground states, in which electrons are at their lowest possible energies • The Principal Quantum Number n: The quantum number n is called the principal quantum number We already know this as shell/orbit The shell “K” has been given the value n=1, the “L” shell has been given the value n=2… the shell are denoted by letters as shown in the table below Number n Shell K L M N O P Q The principal quantum number serves to determine the size of the orbital, or how far the electron extends from the nucleus The higher the value of n the further from the nucleus we can expect to find it As n increases so does the energy required as well because the further out from the nucleus we go the more energy the electron must have to stay in orbit Bohr's work took into account only this first principle quantum number His theory worked for hydrogen because hydrogen just happens to be the one element in which all orbitals having the same value of n also have the same energy Bohr's theory failed for atoms other than hydrogen, however, because orbitals with the same value of n can have different energies when the atom has more than one electron • l The Orbital Angular Momentum Quantum number : The secondary quantum number, l, divides the shells up into smaller groups of subshells called orbitals The value of n determines the possible values for l For any given shell the number of subshells can be found by l = n -1 This means that for n = 1, the first shell, there is only l = 1-1 = subshells ie the shell and subshell are identical When n = there are two sets of subshells; l = and l = The value l = 0,1, 2, , ( n − 1) For a particular value of l, the magnitude of the total angular momentum ur L orbit of an electron due to its orbital motion is given by: u r L orbit = l (l + 1) h 919\* MERGEFORMAT (.) As with the principal quantum number, letters are used to denote specific orbital quantum numbers: Number l Electron s P d F G The principle quantum number describes size and energy, but the second quantum number describes shape The subshells in any given orbital differ slightly in energy, with the energy in the subshell increasing with increasing l This means that within a given shell, the s subshell is lowest in energy, p is the next lowest, followed by d, then f, and so on For example: 4s < 4p < 4d < 4f • The Magnetic quantum number ml → increasing energy : The third quantum number, ml, is known as the magnetic quantum number This integral number determines the component of the orbital angular momentum along the direction of an applied magnetic field For a given value of l, there are 10 colinear dipoles with the same ur m= m moments [Reprinted from J Stöhr, H.C Siegmann; 1st Magnetism, Edition; Springer Verlag] 3.2 temperature ⇒ magnetic dipolar interaction must be too weak to account for the ordering of most magnetic matrrials ⇒ magnetic dipolar interaction is causing the breaking up of a ferromagnetic into domains Exchange interaction – cause for a long range magnetic order The exchange interaction is an interaction of electrostatic origin that result from the indistinguishability of the electrons It can be understood only by a quantum mechanical point of view We will discuss the formulaation of the ( → [ A] ) exchange interaction arriving at the Heisenberg hamiltonian H2 The formation of a -Molecule from two hydrogen atoms depends on the relative orientation of the spins of the electrons The H2 -Molecule consisting of two protons and two electrons interacting via coulomb forces, can be described using a 4-body Schrodinger equation The problem can be facilitated further by separating the motion of the electrons from the motion of the protons The problem then consists in finding the energy E and the wave function two electrons with positions equation: uur r1,2 and spins s1,2 u r ur Ψ (r1; s1; r2 ; s2 ) from the stationary Schrodinger u r ur u r ur H Ψ r1; s1; r2 ; s2 = E Ψ r1 ; s1; r2 ; s2 ( ) ( 23123\* MERGEFORMAT (.) 29 of the ) The Hamiltonian H is composed of three terms, one for the kinetic energies of the two electrons protons protons HC HR HK , one for the Coulomb attraction between the electrons and and one for the Coulomb repulsion between the two electrons and two : H = H K + HC + H R 24124\* MERGEFORMAT (.) 2H → H : Energy depends on rel orientation of electron spins H2  : two protons and two electrons interacting via Coulomb force  States can be determined by 4-body Schrodinger equation  Adiabatic approximation: treat motion of H2 protons and electrons separately Figure 10: Molecule ⇒ [Reprintedfrom H Haken, H.C find energy E and wave function u r ur Wolf; Atom- und Ψ r1; s1; r2 ; s2 Quantenphysik, from stationary Schrodinger Auflage; Springer equation Verlag]2 r ur R Figure 10 defines the vector used: at positions and we have the two  ( protons, at u r r1 and ur r2 we have the two electrons ) uuur p1,2 The kinetic energy of the two electrons with momenta is given by: −2 −2 p + p2 HK = 25125\* MERGEFORMAT (.) 30 The Coulomb attraction between the electrons and the two protons at positions r and ur R is: HC = − 1 1 − − u r ur − ur ur r1 r2 r1 − R r2 − R 26126\* MERGEFORMAT (.) The Coulomb repilsion between the electrons and the protons is: 1 HR = u r ur + r1 − r2 R 27127\* MERGEFORMAT (.) According to Pauli’s principle, wave functions of identical fermions must change sign when two arguments are interchanged: u r ur ur u r Ψ r1; s1 ; r2 ; s2 = − r2 ; s1 ; r1 ; s2 ( ) ( ) 28128\* MERGEFORMAT (.) As there is no spin dependent term in H, the solution takes the form: u r ur u r ur H Ψ (r1; s1; r2 ; s2 ) = E Ψ r1; s1 ; r2 ; s2 ( With solution: ) 29129\* MERGEFORMAT (.) u r ur Ψ = Φ r1; r2 χ ( s1; s2 ) ( ) 30130\* MERGEFORMAT (.) Where Φ is the spatial part and χ the spin part of the wave function Equation 129 is satisfied when u r ur u r ur H Φ r1; r2 = EΦ r1; r2 ( ) 31131\* MERGEFORMAT (.) 31 ( ) Ψ χ So the choice of seems arbitary, but since the overall wave function Ψ must be antisymmetric in its arguments, it is not Φ ⟹ in case of a symmetric spatial state , the spin part χ of the wave function must be antisymmetric ⟹ in case of an antisymmetric spatial state Φ , the spin part χ of the wave function must be symmetric How are the states found for which χ is either symmetric or antisymmetric ? Eigenstate mS S ↑↑ 1 ↑↓ + ↓↑ ↓↓ -1 ↑↓ − ↓↑ 0 2 These Eigenstates symmetric obey ur2 ur ur S op = s1 + s2 Looking for Eigenstates of the toatal spin ur2 S op We see Eigenstates value of are and ⟹ in case of a symmetric spatial state must be antisymmetric singlet state exchange Φ , the spin part χS ( S = 0) 32 ( ) χ of the wave function ⟹ in case of an antisymmetric spatial state function must be symmetric singlet state Thus, there exist, for two spins Φ χT ( S = 1) , the spin part χ of the wave , three symmetric spin functions χT , corresponding to a total spin S=1 (‘parallel spin’), and one single antisymmetric function χS , corresponding to S=0 (‘antiparallel spins’) We therefore have two case: Φ and χ sym anti {Φ anti and χ sym giving S =0 ( sin glet ) giving S =1 ( triplet ) The equation 130 can now be written as a product of single electron states, u r Ψ a r1 ( ) first electron in state ur Ψ b r2 , second electron in state ( ) equations for which the condition 128 holds, one wave function case and one wave function ΨT We find two ΨS for the singlet for the triplet case: ΨS = u r ur ur u r  Ψ a r1 Ψ b r2 + Ψ a r2 Ψ b r1  χ S  2 ( ) ( ) ( ) ( ) 32132\* MERGEFORMAT (.) ΨT = u r ur ur u r  Ψ a r1 Ψ b r2 − Ψ a r2 Ψ b r1  χT  2 ( ) ( ) ( ) ( ) 33133\* MERGEFORMAT (.) 33 H2 Figure 11: Energies of as a function of distance for singlet and triplet states according to Heitler-London [Reprinted from Andreas Bringer, Heisenberg Model; Institut für Festkörperforschung and Institut for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany] Fig.11 shows according to Haitler-London, the difference in energy between the singlet and the triplet state for the The energy ES and ET H2 -molecule as a function of distance R for the singlet and the triplet state can be calculated as: u r ur µ Ψ drdr ES = ∫ Ψ *S H S 34134\* MERGEFORMAT (.) u r ur µ Ψ drdr ET = ∫ ΨT* H T 35135\* MERGEFORMAT (.) The difference between the two energies is given by: u r ur u r ur u r ur µ Ψ r Ψ r drdr ES − ET = 2∫ Ψ *a r1 Ψ *b r2 H a b 2 ( ) ( ) ( ) ( ) 36136\* MERGEFORMAT (.) 34 Consequently, the energy E in these two cases depends on the relative orientation of the electronic spins; thus, to represent the interaction between the electrons, it suffices to introduce a term in the hamiltonian containing a factor: ur ur s1.s2 37137\* MERGEFORMAT (.) Thus, the connection between the spin and spatial parts is indirect, although necessary, imposed by the antisymmetry of the total wavefunction Because of this connection, the effect of the electrostatic interaction between the electronic charges may be described as an interaction between spins With the so-called Exchange Integral J, that follows from 136: u r ur ur u r u r ur E − ET µ Ψ r Ψ r drdr J= S = ∫ Ψ *a r1 Ψ *b r2 H a b 1 2 ( ) ( ) ( ) ( ) 38138\* MERGEFORMAT (.) this leads to the idea that combined effects of the Coulomb interelectronic interactions and Pauli’s exclusion principle between two atoms with spins s2 s1 and will produce an effective interaction potential of the form: ur ur ⇒ V = −2 J ( R12 ) s1.s2 39139\* MERGEFORMAT (.) If If J > 0, ES > ET J < 0, ES < ET and the triplet state S=1 is favoured and the singlet state S=0 is favoured Generalizing this idea, in a solid, the hamiltonian describing the interaction is given by: 35 uu r uu r µ = −2 H J S S ∑ i> j ij i j 40140\* MERGEFORMAT (.) It is known as the Heisenberg Hamiltonian, where now the sum is performed each pair of atoms ( i; j ) Often it is possible to take for nearest neighbour spins and to be otherwise, J ij J to be equal to a constant J then becomes an effective exchange parameter For J > spins are parallel, the magnet is a ferromagnet For J < spins are antiparallel and the magnet is an anti-ferromagnet So (anti)ferromagnetism is not the result of a mysterious spin-spin interaction, but the manifestation of the Coulomb interaction between electrons that are indistinguishable Fermions! CHAPTER SOME TYPES OF EXCHANGE INTERACTIONS 4.1 Direct, indirect exchange and superexchange • Direct exchange Direct exchange operates between moments, which are close enough to have sufficient overlap of their wavefunctions It gives a strong but short range coupling which decreases rapidly as the ions are separated An initial simple way of understanding direct exchange is to look at two atoms with one electron each When the atoms are very close together the Coulomb interaction is minimal when the electrons spend most of their time in between the nuclei Since the electrons are then required to be at the same place in space at the same time, Pauli's exclusion principle requires that they possess opposite spins According to Bethe and Slater 36 the electrons spend most of their time in between neighboring atoms when the interatomic distance is small This gives rise to antiparallel alignment and therefore negative exchange (antiferromagnetic), Fig 12 Fig 12 Antiparallel alignment for small interatomic distances If the atoms are far apart the electrons spend their time away from each other in order to minimize the electron-electron repulsion This gives rise to parallel alignment or positive exchange (ferromagnetism), Fig 13 Fig 13 Parallel alignment for large interatomic distances For direct inter-atomic exchange j can be positive or negative depending on the balance between the Coulomb and kinetic energies The Bethe-Slater curve represents the magnitude of direct exchange as a function of interatomic distance Cobalt is situated near the peak of this curve, while chromium and manganese are on the side of negative exchange Iron, with its sign depending on the crystal structures probably around the zero-crossing point of the curve, Fig 14 • Fig 14 The Bethe-Slater curve Indirect exchange Indirect exchange couples moments over relatively large distances It is the dominant exchange interaction in metals, where there is little or no direct overlap 37 between neighboring electrons It therefore acts through an intermediary, which in metals are the conduction electrons (itinerant electrons) This type of exchange is better known as the RKKY interaction named after Ruderman, Kittel, Kasuya and Yoshida The RKKY exchange coefficient j oscillates from positive to negative as the separation of the ion changes and has the damped oscillatory nature shown in Fig 15 Therefore depending on the separation between a pair of ions their magnetic coupling can be ferromagnetic or antiferromagnetic A magnetic ion induces a spin polarization in the conduction electrons in its neighborhood This spin polarization in the itinerant electrons is felt by the moments of other magnetic ions within the range leading to an indirect coupling In rare-earth metals, whose magnetic electrons in the 4f shell are shielded by the 5s and 5p electrons, direct exchange is rather and indirect exchange via the conduction electrons gives rise to magnetic order in these materials Fig 15: The coefficient of indirect (RKKY) exchange versus the interatomic spacing a MnO • Superexchange in antiferromagnetic 38 Figure 16: Antiferromagnetic MnO [Reprinted from Blundell, Stephen; 1st Magnetism in Condensed Matter, Edition; Oxford Univ Press] • electrons interact with neighbouring electrons ⇒ direct exchange, no intermediary • BUT: direct exchange often not possible ⇒ insufficient direct overlap between orbitals ⇒ for example, 4f electrons in rare earths strongly localized, close to the nucleus, so direct exchange interaction uneffective The exchange interaction described in the previous section stated a direct exchange, so electrons would interact only with their next neighbours But if the electrons are strongly localised there is only a small probability for them to interact with electrons an neighbouring atoms This is for example the case for the 4f electrons in the rare earths but MnO still exhibits the characteristics of an antiferromagnet The reason is that in the MnO-lattice (see Fig 16) the Oxygen O2 takes up the rôle of an intermediary transmitting the exchange forces This is then called superexchange 4.2 Anisotropic exchange: Crystalline anisotropy By looking at the Heisenberg Hamiltonian: 39 ur uu r µ = −2 H J S S ∑ i> j ij i j , It is obvious that the asscociated exchange energy and therefore the magnetization should be totally isotropic, as µ H is invariant with respect to choice of coordinate systems So the experimental fact, that the magnetization curve of iron depends on the positioning of the lattice in respect to the external magnetic field, is very surprising How can the Heisenberg Hamiltonian, which is only spin dependent, be affected by the structure of the lattice? The answer is simple: Even though the spin part χ Φ and the spatial part of the wave function not depend on the same variables they are connected by the spin-orbit coupling! So transmitts the structure of the environment (symmetry of the lattice) onto χ Φ via LS-coupling (→ [Bl])! exchangeuu depends on scalar renergy uu r S1.S product ⇒ invariant with respect to choice of coord sys ⇒ magnetization of ferromagnets considered isotropic  BUT: experimentally was found: ⇒ magnetization lies along certain axes  Anisotropy is caused by spin-orbit coupling  Figure 17: Magnetization curve of iron [Reprinted from Heiko Lueken, Magnetochemie, Auflage; Teubner Verlag] 40 • spatial wave functions reflect symmetry of the lattice because of the interactions between the lattice atoms that arise from the crystalline electric • fields and the overlap of the wave functions via spin-orbit coupling, spins are made aware of this anisotropy 41 CONCLUSIONS We have seen that magnetic properties of matter originate essentially from the magnetic moment of electrons in incomplete shells in the atoms and from unpaired electrons The incomplete shell may be, for example, the 3d shell in the case of the elements of the iron group, or the 4f shell in the rare earths (Sections and 2) The cause for some solids, called (anti)ferromagnets, to exhibit macroscopic magnetic characteristics was given by the exchange interaction, which is quantum mechanical in nature Dipolar interaction, which has a tendency to align microscopic magnetic moments as well, does not present a cause for (anti)ferromagnetism due to energetical reasons (Section 3) This text was then concluded by qualitatively explaining why the exchange interaction, though short range in nature, is able to cause ferromagnetic characteristics in compounds like MnO where there is no direct overlap of the Mn-orbitals and how anisotropy can occure in crystals even though the Heisenberg hamiltonian, describing (anti)ferromagnetism, is totally isotropic (Section 4) 42 REFERENCES [1] [2] Lueken, Heiko; Magnetochemie, Auflage; Teubner Verlag Blundell, Stephen; Magnetism in Condensed Matter, 1st Edition; Oxford Univ Press [3] H Haken, H.C Wolf; Atom- und Quantenphysik, Auflage; Springer Verlag [4] Demtröder, Wolfgang; Experimentalphysik Band 3, Auflage; Springer Verlag 1st [5] J Stöhr, H.C Siegmann; Magnetism, Edition; Springer Verlag [6] Bringer, Andreas; Heisenberg Model; Institut für Festkörperforschung and Institut for Advanced Simulation, Forschungszentrum Jülich, D52425 Jülich, Germany [7] Aharoni, Amikam; Introduction to the Theory of Ferromagnetism, 2nd Edition; Oxford Univ Press [8] Guimarães, A P.; Magnetism and Magnetic Resonance in Solids, 1st John Wiley & Sons, Inc [9] Simonds, J L (1995), “Magnetoelectronics Today and Tomorrow”, Physics Today 48, 26-32 43 Edition; [...]... interaction is causing the breaking up of a ferromagnetic into domains Exchange interaction – cause for a long range magnetic order The exchange interaction is an interaction of electrostatic origin that result from the indistinguishability of the electrons It can be understood only by a quantum mechanical point of view We will discuss the formulaation of the ( → [ A] ) exchange interaction arriving at the... formation of a -Molecule from two hydrogen atoms depends on the relative orientation of the spins of the electrons The H2 -Molecule consisting of two protons and two electrons interacting via coulomb forces, can be described using a 4-body Schrodinger equation The problem can be facilitated further by separating the motion of the electrons from the motion of the protons The problem then consists in finding... the spin part χS ( S = 0) 32 ( ) χ of the wave function ⟹ in case of an antisymmetric spatial state function must be symmetric singlet state Thus, there exist, for two spins 1 2 Φ χT ( S = 1) , the spin part χ of the wave , three symmetric spin functions χT , corresponding to a total spin S=1 (‘parallel spin’), and one single antisymmetric function χS , corresponding to S=0 (‘antiparallel spins’)... orientation of the electronic spins; thus, to represent the interaction between the electrons, it suffices to introduce a term in the hamiltonian containing a factor: ur ur s1.s2 37137\* MERGEFORMAT (.) Thus, the connection between the spin and spatial parts is indirect, although necessary, imposed by the antisymmetry of the total wavefunction Because of this connection, the effect of the electrostatic interaction. .. occupied (in blue) If, in the second row, you put in the values for l starting with the maximum (in this case 2) and counting down to the negative maximum (in this case -2) including the zero and repeating this scheme as many times as possible till the spaces given by the shell you are in, are full (in this case 10 spaces), it is very easy to find S and L: By adding up the (blue) numbers of the first... singlet and a triplet state is at the very core of explaining the long range order So in the end magnetism on the bigger scale is a purely quantum mechanical effect 3.1 Magnetic dipolar interaction The electric charges present in the nucleus interact with the electrons that surround it; in an analogous way, the magnetic moments associated with the electrons and the nuclei also interact The magnetic interaction. .. an interaction between spins With the so-called Exchange Integral J, that follows from 136: u r ur ur u r u r ur E − ET µ Ψ r Ψ r drdr J= S = ∫ Ψ *a r1 Ψ *b r2 H a 2 b 1 1 2 2 ( ) ( ) ( ) ( ) 38138\* MERGEFORMAT (.) this leads to the idea that combined effects of the Coulomb interelectronic interactions and Pauli’s exclusion principle between two atoms with spins s2 s1 and will produce an effective interaction. .. degree of mutual alignment 0 ur   : 10−23 J  for m ≈ µ B , rij ≈ 1 A ÷   Magnitude this is energy is equivalent to about 1K in 28 colinear dipoles with the same ur m= m moments [Reprinted from J Stöhr, H.C Siegmann; 1st Magnetism, Edition; Springer Verlag] 3.2 temperature ⇒ magnetic dipolar interaction must be too weak to account for the ordering of most magnetic matrrials ⇒ magnetic dipolar interaction. .. order disappears above a certain temperature TC called the Curie temperature From the different interactions taking place in atoms constituing a solid, the question is which of those shows responsible for macroscopic magnetic effects? A quick look at the dipolar Interaction will reveil that this type cannot account for long range order But exchange interaction, looked at in section 3.2, presents a valid... where µ H is the Hamiltonian for an atom in an external magnetic field with: 2 Z ¶H =  pi + V , ∑ 0  i÷ i =1  2m  25 ur r uu r hL = ∑ ri × pi i ( → [ Bl ] ) 26 CHAPTER 3 TYPES OF INTERACTION IN MAGNETISM Ferromagnetism is a type of magnetism characterized by an spontaneous parallel alignment of atomic magnetic moments, with long range order Examples of ferromagnets are the elements iron, nickel,