<p></p><ul style="display: flex;"><li style="flex:1"><strong>Magnetism </strong></li><li style="flex:1"><strong>Magnets </strong></li></ul><p></p><p>Zero external field <br>Finite external field </p><p>• Types&nbsp;of magnetic systems • Pauli&nbsp;paramagnetism in metals • Landau diamagnetism&nbsp;in metals • Larmor&nbsp;diamagnetism in insulators • Ferromagnetism&nbsp;of electron gas • Spin&nbsp;Hamiltonian </p><p>Paramagnets Diamagnets Ferromagnets </p><p>• Mean&nbsp;field approach • Curie&nbsp;transition </p><p>Antiferromagnets Ferrimagnets </p><ul style="display: flex;"><li style="flex:1">…</li><li style="flex:1">…</li><li style="flex:1">…</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1"><strong>Pauli paramagnetism </strong></li><li style="flex:1"><strong>Pauli paramagnetism </strong></li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">ε<sub style="top: 0.235em;">↑ </sub>= <em>p</em><sup style="top: -0.315em;">2 </sup>/ 2<em>m</em>−<em>e</em>=<em>B</em>/ 2<em>mc </em></li><li style="flex:1">ε = <em>p</em><sup style="top: -0.315em;">2 </sup>/ 2<em>m</em>+<em>e</em>=<em>B</em>/ 2<em>mc </em></li></ul><p></p><p>Let us first look at magnetic properties of a free electron gas. </p><p><sup style="top: -0.405em;">↓</sup>G </p><p><em>d</em><sup style="top: -0.3em;">3 </sup><em>p </em></p><p>Electron are spin-1/2 particles <br>#of majority spins: </p><p></p><ul style="display: flex;"><li style="flex:1"><em>N</em><sub style="top: 0.2201em;">↑,↓ </sub>=<em>V </em></li><li style="flex:1"><sub style="top: 0.2201em;">3 </sub><em>f </em>(ε<sub style="top: 0.2201em;">↑,↓ </sub></li></ul><p></p><p>)</p><p>∫</p><p>In external magnetic field <strong>B </strong>– Zeeman splitting <br>#of minority spins: </p><p>(2π=) </p><p></p><ul style="display: flex;"><li style="flex:1">ε<sub style="top: 0.23em;">↑ </sub>= <em>p</em><sup style="top: -0.315em;">2 </sup>/ 2<em>m</em>−<em>e</em>=<em>B</em>/ 2<em>mc </em></li><li style="flex:1">ε<sub style="top: 0.23em;">↓ </sub>= <em>p</em><sup style="top: -0.315em;">2 </sup>/ 2<em>m</em>+<em>e</em>=<em>B</em>/ 2<em>mc </em></li></ul><p></p><p>Magnetization (magnetic moment per unit volume): <br>- minority spins <br>Fermi level </p><p><em>e</em>= </p><ul style="display: flex;"><li style="flex:1"><em>M </em>= </li><li style="flex:1">(<em>N</em><sub style="top: 0.22em;">↑ </sub>− <em>N</em><sub style="top: 0.22em;">↓ </sub>) </li></ul><p></p><p>: aligned along the field and proportional </p><p>2<em>Vmc </em></p><p>to <em>B </em>in low fields <br>- magnetic succeptibility </p><p>χ</p><p><em>M </em>= χ<em>B </em></p><p>- majority spins <br>- paramagnetism </p><p>χ &gt; 0 </p><p></p><ul style="display: flex;"><li style="flex:1"><strong>Pauli succeptibility </strong></li><li style="flex:1"><strong>Landau quantization </strong></li></ul><p></p><p>G</p><p></p><ul style="display: flex;"><li style="flex:1">ε<sub style="top: 0.235em;">↑ </sub>= <em>p</em><sup style="top: -0.315em;">2 </sup>/ 2<em>m</em>−<em>e</em>=<em>B</em>/ 2<em>mc </em></li><li style="flex:1">ε<sub style="top: 0.235em;">↓ </sub>= <em>p</em><sup style="top: -0.315em;">2 </sup>/ 2<em>m</em>+<em>e</em>=<em>B</em>/ 2<em>mc </em></li></ul><p></p><p>A free electron in magnetic field: </p><p>ˆ</p><p><em>B </em>&amp; <em>z </em></p><p>G</p><p>2<br>2</p><p>µ+<em>e</em>=<em>B</em>/2<em>mc </em></p><p>G</p><p>⎛⎜⎝<br>⎞⎟⎠</p><p>=</p><p><em>ieA </em></p><p><em>A</em><sub style="top: 0.17em;"><em>y </em></sub>= <em>Bx</em>; <em>A </em>= <em>A </em>= 0 </p><p></p><ul style="display: flex;"><li style="flex:1"><em>V</em></li><li style="flex:1"><em>g e</em>=<em>B </em></li></ul><p></p><p>Schrödinger equation: </p><p></p><ul style="display: flex;"><li style="flex:1"><em>x</em></li><li style="flex:1"><em>z</em></li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">−</li><li style="flex:1">∇+ </li><li style="flex:1">ψ = εψ </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1"><em>N</em><sub style="top: 0.22em;">↑ </sub>− <em>N</em><sub style="top: 0.22em;">↓ </sub>= </li><li style="flex:1"><em>g</em>(ε)<em>d</em>ε ≈ </li></ul><p><em>V</em></p><p>∫</p><p></p><ul style="display: flex;"><li style="flex:1">2<em>m </em></li><li style="flex:1">=<em>c </em></li></ul><p></p><p>2 <sub style="top: 0.195em;">µ−<em>e</em>=<em>B</em>/2<em>mc </em></sub></p><p>2 <em>mc </em></p><p><em>n</em>,<em>k</em><sub style="top: 0.17em;"><em>z </em></sub></p><p><em>B=1</em>T corresponds to <em>e</em>=<em>B </em>/ <em>mc </em>=1<em>K </em>×<em>k</em><sub style="top: 0.17em;"><em>B </em></sub>provided <em>m </em>is free electrons’s mass </p><p>Solutions: labeled by two indices </p><p>G</p><p>ψ</p><p><sub style="top: 0.175em;"><em>nk </em></sub>(<em>r</em>) = exp(<em>ik</em><sub style="top: 0.175em;"><em>y </em></sub><em>y </em>+<em>ik</em><sub style="top: 0.175em;"><em>z </em></sub><em>z</em>)ϕ<sub style="top: 0.175em;"><em>n </em></sub>(<em>x </em>− =<em>ck</em><sub style="top: 0.175em;"><em>y </em></sub>/ <em>eB</em>) </p><p>For any fields, </p><p><em>e</em>=<em>B</em>/ <em>mc </em>ꢀ µ </p><p>ϕ<sub style="top: 0.17em;"><em>n </em></sub></p><p>- wave functions of a harmonic oscillator <br>Magnetic succeptibility: </p><p><sup style="top: 0em;">Energies: </sup>ε<sub style="top: 0.175em;"><em>nk </em></sub>= =<sup style="top: -0.295em;">2</sup><em>k</em><sub style="top: 0.175em;"><em>z</em></sub><sup style="top: -0.295em;">2 </sup>/ 2<em>m</em>+(<em>e</em>=<em>B</em>/ <em>mc</em>)(<em>n</em>+1/ 2) </p><p>2</p><p>- strongly degenerate!! </p><p><em>e</em>= </p><p></p><ul style="display: flex;"><li style="flex:1">⎛</li><li style="flex:1">⎞</li></ul><p></p><p>χ<sub style="top: 0.17em;"><em>P </em></sub></p><p>=</p><p><em>g</em></p><p>⎜⎝<br>⎟⎠</p><p>2<em>mc </em></p><p>We “quantized” momenta transverse to the field <br>(Landau levels) </p><p>1</p><p></p><ul style="display: flex;"><li style="flex:1"><strong>Landau diamagnetism </strong></li><li style="flex:1"><strong>Electrons in metals </strong></li></ul><p></p><p>A free electron in magnetic field: moves along spiral trajectories and create magnetic field themselves. <br>We know that there are diamagnetic metals. How can we explain their existence? <br>This magnetic field is directed antiparallel to the external one <br>Different spectra of electrons in metals and free electrons. </p><p><em>m</em><sup style="top: -0.3em;">* </sup></p><p>Example: renormalized electron mass <br>Diamagnetism <br>This only affects orbital motion, not Zeeman splitting </p><p><em>dk </em></p><p>Energy: </p><p><em>E </em>= 2 </p><p>ε<sub style="top: 0.17em;"><em>nk </em></sub><em>f </em>ε </p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li></ul><p></p><p>∑</p><p></p><ul style="display: flex;"><li style="flex:1"><em>s</em></li><li style="flex:1"><em>nk </em></li></ul><p></p><p>∫</p><p>2π </p><p>= −<em>V </em><sup style="top: -0.3em;">−1 </sup></p><p><em>n</em></p><p>χ<sub style="top: 0.17em;"><em>P</em></sub><sup style="top: -0.295em;">* </sup>= χ<sub style="top: 0.17em;"><em>P</em></sub>; χ<sub style="top: 0.17em;"><em>L</em></sub><sup style="top: -0.295em;">* </sup>= χ<sub style="top: 0.17em;"><em>L </em></sub>(<em>m</em>/ <em>m</em><sup style="top: -0.295em;">*</sup>)<sup style="top: -0.295em;">2 </sup></p><p></p><ul style="display: flex;"><li style="flex:1"><em>M</em></li><li style="flex:1">∂<em>E </em>/ </li></ul><p></p><p>∂</p><p><em>B</em></p><p>Magnetization: </p><p>2</p><p>13</p><p><em>e</em>= </p><p>1</p><p>⎛⎝<br>⎞</p><p></p><ul style="display: flex;"><li style="flex:1">Result for the succeptibility: </li><li style="flex:1">We can explain paramagnetic and diamagnetic metals! </li></ul><p></p><p>χ<sub style="top: 0.17em;"><em>L </em></sub>= − </p><p><em>g </em>= −&nbsp;χ<sub style="top: 0.17em;"><em>P </em></sub></p><p></p><ul style="display: flex;"><li style="flex:1">⎜</li><li style="flex:1">⎟</li></ul><p>⎠</p><p>2<em>mc </em></p><p>3</p><p>But there is no way we can explain ferromagnetic and antiferromagnetic metals in non-interacting electron model. <br>Total succeptibility: </p><p>χ</p><p>=</p><p>χ<sub style="top: 0.17em;"><em>P </em></sub></p><p>+</p><p>χ<sub style="top: 0.17em;"><em>L </em></sub></p><p>=</p><p>2χ<sub style="top: 0.17em;"><em>P </em></sub>/3 </p><p>&gt;</p><p>0</p><p>: paramagnetic! <br>Try insulators? </p><p></p><ul style="display: flex;"><li style="flex:1"><strong>Larmor diamagnetism </strong></li><li style="flex:1"><strong>Larmor diamagnetism </strong></li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1"><em>e</em><sup style="top: -0.295em;">2 </sup></li><li style="flex:1"><em>e</em><sup style="top: -0.295em;">2</sup><em>ZN </em></li></ul><p></p><p>Consider an ionic insulator with filled shells. Electrons are localized at the ions. </p><p>∆<em>E</em><sub style="top: 0.17em;"><em>g </em></sub>= </p><p></p><ul style="display: flex;"><li style="flex:1"><em>B</em><sup style="top: -0.3em;">2 </sup></li><li style="flex:1"><em>r</em><sup style="top: -0.3em;">2 </sup></li></ul><p></p><p>=</p><p><em>B</em><sup style="top: -0.3em;">2 </sup><em>r</em><sup style="top: -0.3em;">2 </sup></p><p>G</p><p>2</p><p>∑</p><p><em>i</em></p><p>Electron Hamiltonian: </p><p></p><ul style="display: flex;"><li style="flex:1">12<em>mc</em><sup style="top: -0.295em;">2 </sup></li><li style="flex:1">12<em>mc</em><sup style="top: -0.295em;">2 </sup></li></ul><p></p><p>2</p><p><em>g</em></p><p></p><ul style="display: flex;"><li style="flex:1">G</li><li style="flex:1">G G&nbsp;G </li><li style="flex:1">G</li></ul><p></p><p>⎛⎜⎝<br>⎞⎟⎠</p><p>=</p><p></p><ul style="display: flex;"><li style="flex:1"><em>ieA </em></li><li style="flex:1"><em>e</em></li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">G</li></ul><p></p><p><em>i g</em></p><p>ˆ</p><p><em>H</em></p><p></p><ul style="display: flex;"><li style="flex:1">= − </li><li style="flex:1">∇+ </li><li style="flex:1">+</li></ul><p></p><p><em>BS</em>; <em>A </em></p><p>=</p><p><em>B</em></p><p>×</p><p><em>r</em><br>2<em>m </em></p><p>=</p><p></p><ul style="display: flex;"><li style="flex:1"><em>c</em></li><li style="flex:1"><em>mc </em></li></ul><p></p><p>1 ∂<sup style="top: -0.295em;">2</sup><em>E e</em><sup style="top: -0.295em;">2</sup><em>Z </em></p><p>Succeptibility: </p><p><em>c</em><sub style="top: 0.17em;"><em>A </em></sub><em>r</em><sup style="top: -0.3em;">2 </sup></p><p>- ionic charge </p><p><em>Z</em></p><p><strong>r </strong>small: consider terms with the field as a perturbation. </p><p></p><ul style="display: flex;"><li style="flex:1">χ = − </li><li style="flex:1">= − </li></ul><p></p><p><em>V </em>∂<em>B</em><sup style="top: -0.295em;">2 </sup><br>6<em>mc</em><sup style="top: -0.295em;">2 </sup></p><p><em>g</em></p><p><em>c</em><sub style="top: 0.175em;"><em>A</em></sub><sub style="top: 0.03em;">- #of atoms per unit volume </sub></p><p>Correction to the energy in the ground state: </p><p>ˆ</p><p>∆<em>E</em><sub style="top: 0.17em;"><em>g </em></sub>= <em>g</em>.<em>s H g</em>.<em>s </em></p><p>Diamagnetism! <br>Total spin and total momentum of electrons in a filled shell are zero; </p><p><em>r</em><sup style="top: -0.3em;">2 </sup></p><p>If the ionic shells are not filled – can get paramagnetic contribution due to other terms. </p><ul style="display: flex;"><li style="flex:1">only the term with </li><li style="flex:1">contributes. </li></ul><p></p><p><em>e</em><sup style="top: -0.295em;">2 </sup><br>8<em>mc</em><sup style="top: -0.295em;">2 </sup><em>e</em><sup style="top: -0.295em;">2 </sup><br>12<em>mc</em><sup style="top: -0.295em;">2 </sup></p><p>∆<em>E</em><sub style="top: 0.17em;"><em>g </em></sub>= </p><p><em>B</em><sup style="top: -0.3em;">2 </sup><em>x</em><sup style="top: -0.3em;">2 </sup>+ <em>y</em><sup style="top: -0.3em;">2 </sup></p><p>=</p><p></p><ul style="display: flex;"><li style="flex:1"><em>B</em><sup style="top: -0.3em;">2 </sup></li><li style="flex:1"><em>r</em><sup style="top: -0.3em;">2 </sup></li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li></ul><p></p><p>Can explain paramagnetic and diamagnetic insulators, but not ferromagnetic and antiferromagnetic ones! </p><p></p><ul style="display: flex;"><li style="flex:1">∑</li><li style="flex:1">∑</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1"><em>i</em></li><li style="flex:1"><em>i</em></li><li style="flex:1"><em>i</em></li></ul><p></p><ul style="display: flex;"><li style="flex:1"><em>i</em></li><li style="flex:1"><em>i</em></li></ul><p></p><ul style="display: flex;"><li style="flex:1"><em>g</em></li><li style="flex:1"><em>g</em></li></ul><p></p><p><strong>Ferromagnetism for localized electrons </strong></p><p><strong>Exchange interaction </strong></p><p>Try to&nbsp;visualize for localized electrons (atoms; artificial atoms – quantum dots; defects etc) <br>Two electrons: antisymmetric wave function! Take an electron level </p><p>ε</p><p>The simplest model: <br>Wave function of two electrons: </p><p>δ</p><p>Discrete electron states; spacing (each level is doubly degenerate) </p><p>δ</p><p>1</p><p><em>S </em>= 0,1 </p><p>for </p><p></p><ul style="display: flex;"><li style="flex:1">ψ (<em>r </em>,<em>r </em>) = </li><li style="flex:1">ϕ (<em>r </em>)ϕ (<em>r </em>) ±ϕ (<em>r </em>)ϕ (<em>r </em>) </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">[</li><li style="flex:1">]</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">2</li><li style="flex:1">1</li><li style="flex:1">1</li><li style="flex:1">2</li><li style="flex:1">2</li><li style="flex:1">2</li><li style="flex:1">1</li><li style="flex:1">1</li><li style="flex:1">2</li></ul><p></p><p>2</p><p>To put an electron into the system costs electrostatic energy <em>U </em>and exchange </p><p>energy <em>-J </em></p><p>Energy splitting due to interaction: spin-dependent! </p><p><sup style="top: -0.16em;">Same spin: </sup><em>U </em>+δ − <em>J </em></p><p>ε →ε +<em>C </em>± <em>J </em>/ 2 </p><p>Energy to pay: </p><p><sup style="top: -0.14em;">Opposite spin: </sup><em>U </em></p><p><em>C </em>= <em>d r d r U</em>(<em>r </em>− <em>r </em>) ϕ<sub style="top: 0.18em;">1</sub>(<em>r </em>) <sup style="top: -0.42em;">2 </sup>ϕ<sub style="top: 0.18em;">2 </sub>(<em>r </em>) <sup style="top: -0.42em;">2 </sup></p><p></p><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">2</li><li style="flex:1">1</li><li style="flex:1">2</li><li style="flex:1">1</li><li style="flex:1">2</li></ul><p></p><p>∫</p><p><em>J </em>&lt;δ <sup style="top: -0.01em;">Non-magnetic state </sup></p><p><em>J </em>&gt;δ Ferromagnetic state </p><p><em>J </em>= 2 <em>d r d r U</em>(<em>r </em>− <em>r </em>)ϕ<sub style="top: 0.18em;">1</sub>(<em>r </em>)ϕ<sub style="top: 0.18em;">1</sub><sup style="top: -0.31em;">*</sup>(<em>r </em>)ϕ<sub style="top: 0.18em;">2</sub><sup style="top: -0.31em;">*</sup>(<em>r </em>)ϕ<sub style="top: 0.18em;">2 </sub>(<em>r </em>) </p><p></p><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">2</li><li style="flex:1">1</li><li style="flex:1">2</li><li style="flex:1">1</li><li style="flex:1">2</li><li style="flex:1">1</li><li style="flex:1">2</li></ul><p></p><p>∫</p><p>G G </p><p>ˆ</p><p>Spin Hamiltonian </p><p><em>H </em>= <em>const </em>− <em>JS</em><sub style="top: 0.18em;">1</sub><em>S</em><sub style="top: 0.18em;">2 </sub></p><p>2</p><p></p><ul style="display: flex;"><li style="flex:1"><strong>Itinerant ferromagnetism </strong></li><li style="flex:1"><strong>Itinerant ferromagnetism </strong></li></ul><p></p><p>If we can not get ferromagnetism with free electrons, try interacting electrons. <br>Working the interaction terms out for free electrons (see <em>Advanced </em></p><p><em>Quantum Mechanics, lecture 4</em>) </p><p>35</p><p>3<em>N </em></p><p>=</p><p><sup style="top: -0.315em;">2</sup><em>k</em><sub style="top: 0.18em;"><em>F</em></sub><sup style="top: -0.315em;">2 </sup>2<em>m </em></p><p></p><ul style="display: flex;"><li style="flex:1">Hartree-Fock approximation: </li><li style="flex:1">Kinetic energy: </li></ul><p></p><p><em>E</em><sub style="top: 0.18em;"><em>kin </em></sub></p><p>=</p><p><em>NE</em><sub style="top: 0.18em;"><em>F </em></sub></p><p>=</p><p>5</p><p><em>E</em><sub style="top: 0.17em;">int </sub>= <em>g V g</em></p><p>int </p><p>3</p><p><em>k</em><sub style="top: 0.17em;"><em>F </em></sub></p><p>G G&nbsp;<em>e</em><sup style="top: -0.295em;">2 </sup></p><p></p><ul style="display: flex;"><li style="flex:1">G</li><li style="flex:1">G</li><li style="flex:1">G</li><li style="flex:1">G</li><li style="flex:1">G</li><li style="flex:1">G</li></ul><p></p><p>Potential energy: </p><p><sup style="top: -0.005em;">NB: </sup><em>N </em>=<em>Vk</em><sub style="top: 0.18em;"><em>F</em></sub><sup style="top: -0.31em;">3 </sup>/(3π <sup style="top: -0.31em;">2 </sup>) </p><p><em>E</em><sub style="top: 0.17em;">int </sub>= −&nbsp;<em>Ne</em><sup style="top: -0.295em;">2 </sup></p><p>2</p><p>12</p><p>≈</p><p><em>drdr </em>' </p><p>ψ<sub style="top: 0.17em;"><em>i </em></sub>(<em>r</em>) <sup style="top: -0.4em;">2 </sup>ψ <sub style="top: 0.17em;"><em>j </em></sub>(<em>r </em>') −ψ<sub style="top: 0.17em;"><em>i </em></sub>(<em>r</em>)ψ <sub style="top: 0.17em;"><em>j </em></sub>(<em>r </em>')ψ<sub style="top: 0.17em;"><em>i </em></sub>(<em>r </em>')ψ <sub style="top: 0.17em;"><em>j </em></sub>(<em>r</em>) </p><p></p><ul style="display: flex;"><li style="flex:1">*</li><li style="flex:1">*</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">⎡</li><li style="flex:1">⎤</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">G</li><li style="flex:1">G</li></ul><p></p><p>4</p><p>π</p><p>∑<br>∫</p><p></p><ul style="display: flex;"><li style="flex:1">⎢</li><li style="flex:1">⎥</li></ul><p></p><ul style="display: flex;"><li style="flex:1">⎦</li><li style="flex:1">⎣</li></ul><p></p><p><em>r </em>−<em>r </em>' </p><p><em>ij </em></p><p><sup style="top: -0.16em;">Try now a spin-polarized ground state: </sup><em>k</em><sub style="top: 0.17em;"><em>F</em></sub><sup style="top: -0.3em;">↑ </sup>≠ <em>k</em><sub style="top: 0.17em;"><em>F</em></sub><sup style="top: -0.3em;">↓ </sup></p><p>Kinetic energy loss can be compensated by the potential energy gain! <br>Fock (exchange) <br>Hartree (direct) </p><p>interaction interaction. Only exists if spin projections are the same </p><p>in the states <em>i </em>and <em>j </em></p><p></p><ul style="display: flex;"><li style="flex:1">5</li><li style="flex:1">1</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">For </li><li style="flex:1">- spin-polarized ground state (ferromagnetism!!) </li></ul><p></p><p><em>k</em><sub style="top: 0.17em;"><em>F </em></sub><em>a</em><sub style="top: 0.17em;"><em>B </em></sub>&lt; </p><p>2π 2<sup style="top: -0.295em;">1/3 </sup>+1 </p><p><em>a</em><sub style="top: 0.17em;"><em>B </em></sub>= =<sup style="top: -0.295em;">2 </sup>/ <em>me</em><sup style="top: -0.295em;">2 </sup></p><p>Never occurs in real life. <br>- Bohr radius </p><p></p><ul style="display: flex;"><li style="flex:1"><strong>Antiferromagnetic ordering </strong></li><li style="flex:1"><strong>Antiferromagnetic ordering </strong></li></ul><p></p><p>2</p><p>A different situation: a pair of magnetic atoms in an insulating matrix Consider d-electrons, 5 electrons per atom <br>2<sup style="top: -0.135em;">nd </sup>order corrections to the ground state: virtual states </p><p><em>M</em><sub style="top: 0.185em;"><em>i</em>ν </sub><br>∆<em>E </em>= − </p><p>∑</p><p><em>E </em>− <em>E</em><sub style="top: 0.185em;"><em>i </em></sub></p><p>νν</p><p></p><ul style="display: flex;"><li style="flex:1">Virtual </li><li style="flex:1">Ground </li></ul><p><em>U </em>– ionization energy; <em>t </em>– overlap between the atoms </p><p><em>t</em></p><p>ꢀ</p><p><em>U</em></p><p>∆<em>E </em>= −25<em>t</em><sup style="top: -0.31em;">2 </sup>/<em>U </em>ꢀ <em>t</em>,<em>U </em></p><p>Unperturbed ground state: either parallel or antiparallel spins. <br>Ferromagnetic state: no second-order correction (forbidden by Pauli principle) </p><p>Antiferromagnetic state preferential! </p><p></p><ul style="display: flex;"><li style="flex:1"><strong>Spin Hamiltonian </strong></li><li style="flex:1"><strong>Mean field approach </strong></li></ul><p></p><p>Let us single out one particular spin at <em>i</em>. <br>Does not work for many atoms – but still represents a good model to treat </p><p>G</p><p><em>i</em></p><p>G</p><p>ˆ</p><p>magnetism </p><p>ˆ<br>ˆ</p><p><em>H </em>= −<em>JS S &nbsp;</em>+(all other spins) </p><p>G G </p><p><em>i</em></p><p>∑</p><p><em>j</em></p><p>ˆ ˆ <br>ˆ</p><p><em>H </em>= −&nbsp;<em>J S &nbsp; S </em></p><p><em>i </em>and <em>j </em>– lattice sites </p><p><em>j</em></p><p>∑</p><p></p><ul style="display: flex;"><li style="flex:1"><em>ij </em></li><li style="flex:1"><em>j</em></li></ul><p><em>ij </em></p><p></p><ul style="display: flex;"><li style="flex:1">G</li><li style="flex:1">G</li></ul><p></p><p>ˆ</p><p>GG </p><p>Approximation: this spin sees the average field (Weiss field) </p><p>ˆ<br>ˆ</p><p><em>h </em>= −<em>J </em><br><em>S</em></p><p>Common approximation: only nearest neighbours interact; the same exchange integrals for all bonds </p><p>∑</p><p><em>H </em>= <em>hS</em><sub style="top: 0.18em;"><em>i </em></sub></p><p><em>j j</em></p><p>G G </p><p><em>i</em></p><p>Now we need to calculate the average field self-consistently. </p><p>ˆ ˆ <br>ˆ</p><p>Heisenberg model: </p><p><em>H </em>= −<em>J S &nbsp; S </em></p><p>∑</p><p><em>j</em></p><p>Each site has </p><p><em>N</em></p><p>nearest neighbours. For each neighbour, </p><p><em>ij </em></p><p><em>P </em>∝ exp(−<em>h</em>/ 2<em>k</em><sub style="top: 0.18em;"><em>B</em></sub><em>T</em>) </p><p>↓</p><p>chance to be “up” (parallel to the field) </p><p><em>P</em><sup style="top: -0.49em;">↑ </sup>∝ exp(+<em>h</em>/ 2<em>k</em><sub style="top: 0.18em;"><em>B</em></sub><em>T</em>) </p><p><em>J </em>&gt; 0 (<em>J </em>&lt; 0) </p><p>favors ferromagnetism (antiferromagnetism) chance to be “down” (antiparallel to the field) <br>Exact solution: only known for 1D chain (Bethe 1931) – no magnetism! </p><p></p><ul style="display: flex;"><li style="flex:1"><em>JN </em></li><li style="flex:1"><em>JN </em></li><li style="flex:1"><em>h</em></li></ul><p></p><p>Can also be treated for high spins (classical) Let us see what we can do with approximate solutions. <br>Equation for the field: </p><p></p><ul style="display: flex;"><li style="flex:1"><em>h </em>= − </li><li style="flex:1"><em>P </em>− <em>P </em></li></ul><p></p><p>=</p><p>tanh </p><p></p><ul style="display: flex;"><li style="flex:1">(</li><li style="flex:1">)</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">↑</li><li style="flex:1">↓</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2</li></ul><p></p><p>2<em>k</em><sub style="top: 0.18em;"><em>B</em></sub><em>T </em></p><p>3</p><p></p><ul style="display: flex;"><li style="flex:1"><strong>Mean field approach </strong></li><li style="flex:1"><strong>Curie transition </strong></li></ul><p></p><p>Magnetization close to transition temperature </p><p><em>JN </em></p><p></p><ul style="display: flex;"><li style="flex:1"><em>JN </em></li><li style="flex:1"><em>h</em></li></ul><p></p><p><em>T </em>= </p><p><em>h </em>= </p><p>tanh </p><p><em>c</em></p><p>tanh <em>x </em>≈ <em>x </em>− <em>x</em><sup style="top: -0.315em;">3 </sup>/3 </p><p>4<em>k</em><sub style="top: 0.18em;"><em>B </em></sub></p><p>2</p><p>2<em>k</em><sub style="top: 0.18em;"><em>B</em></sub><em>T </em></p><p>Solution: </p><p><em>h </em>= 12<em>k</em><sub style="top: 0.185em;"><em>B</em></sub><sup style="top: -0.31em;">2</sup><em>T </em>(<em>T </em>−<em>T</em>) </p>