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<h2 class="hd hd-2 unit-title">Magnetic Quantities and Units</h2>
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<p>When talking about magnetism, it is necessary to keep track of a number of terms, quantities, and units. A summary is given below:</p>
<ul>
<li>\(H\) is the magnetic field. It has units of \(A/m\). The magnetic field represents an energy gradient, and is proportional to the force experienced by a magnetic monopole in the field.</li>
<li>\(B\) is the magnetic flux density. It has units of \(T\) or \(Wb/m^2\). The magnetic flux density describes the number of magnetic field lines per unit area.</li>
<li>\(M\) is the magnetization, which is measured in \(A/m\). The magnetization describes the magnetic moment, and tells us the response of a material to a magnetic field. </li>
</ul>
<p>\(M\) and \(B\) both tell us how a material responds to a magnetic field \(H\). We can relate \(H\) to \(B\) and \(M\) in a number of different ways:</p>
<p>In free space, the magnetic flux density is given as:</p>
<ul>
<ul>
<ul>
<li>\(B = \mu _0 H\)</li>
</ul>
</ul>
</ul>
<p>where \(\mu _0 = 4 \pi \times\ 10^{-7} H/m\) is the permeability of free space.</p>
<p>In a material, the magnetic flux density becomes:</p>
<ul>
<ul>
<ul>
<li>\(B = \mu_0(H+M)\) or</li>
<li>\(B = \mu_0 \mu_r H\)</li>
</ul>
</ul>
</ul>
<p>where \(\mu_r\) is the relative permeability. </p>
<p>Alternately, we can write an expressions for the magnetization:</p>
<ul>
<ul>
<ul>
<li>\(M = H(\mu_r -1)\) </li>
<li>\(M = \chi H\)</li>
</ul>
</ul>
</ul>
<p>where \(\chi\) is the susceptibility and equals \((\mu _r -1)\).</p>
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Example 1.1A
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<p>A piece of iron has a relative permeability of 5000. When placed in a magnetic field with a strength of 100 \(A/m\), what is the magnetic flux density within the material?</p>
<p>\(B\) (in \(T\)):</p>
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Problem 1.1B
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<p>What is the magnetization of the material?</p>
<p>\(M\) (in \(kA/m\)):</p>
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<h2 class="hd hd-2 unit-title">Magnetism: Origins and Materials</h2>
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<h2>Magnetism: Origins and Materials</h2>
<h4>Origins of Magnetic Behavior:</h4>
<ul>
<li>The angular momentum of electrons produces magnetization due to moving charges.</li>
<li>Both electron spin and the orbital motion of electrons contribute to magnetization.</li>
<li>The contributions from paired electrons cancel each other out, so strong magnetic effects are only observed in materials with a large number of unpaired electrons.</li>
<li>One electron has a moment of \(1 \mu_B = 9.27 \times 10^{-24} A \cdot m^2\). This is known as the Bohr magneton.</li>
</ul>
<p></p>
<p></p>
<h4>Types of Magnetic Materials:</h4>
<ul>
<li><strong>Paramagnet:</strong> The existing magnetic moments in the material are thermally disordered. These moments align when a magnetic field is applied. At zero field, these materials have no net magnetization.</li>
<li><strong>Diamagnet: </strong>These materials do not have any innate magnetic moment. A small magnetic moment is induced when a magnetic field is applied. This magnetic moment opposes the applied field.</li>
<li><strong>Ferromagnet: </strong>In these materials, all of the magnetic moments are aligned, giving ferromagnets a net magnetization, even at zero field.</li>
<li><strong>Antiferromagnet: </strong>The magnetic moments of the atoms of a antiferromagnet are aligned, but the neighboring spins are of opposite sign, meaning that these materials exhibit no net magnetization at zero field.</li>
<li><strong>Ferrimagnet: </strong>Like an antiferromagnet, the neighboring spins of a ferrimagnet point in opposite directions. However, since the neighboring spins are of different magnitudes (since they are on different types of atoms), the spins do not completely cancel. Ferrimagnets have a net magnetism at zero field, although it is often weaker than the magnetization of ferromagnets. </li>
</ul>
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<p>Iron has a magnetic moment of \(2.6 \mu_B\) per Fe atom in metallic Fe. Use this information to predict the saturation magnetism \(M_s\) of iron. Note that a unit cell of Fe contains 2 atoms, and has a volume of \(2.36 \times 10^{-2} \; nm^{3}\).</p>
<p>\(M_s\) (in \(kA/m\)):</p>
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<h2 class="hd hd-2 unit-title">Magnetic Anisotropy and Domain Walls</h2>
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<h2>Magnetic Anisotropy and Domains</h2>
<h4>Magnetic Anisotropy</h4>
<p>In a material with magnetic anisotropy, the magnetization preferentially aligns along certain directions and energy is required (the anisotropy energy) to tilt the magnetization away from these directions. There are several sources of magnetic anisotropy, including:</p>
<ul>
<li>Magnetocrystalline anisotropy: The underlying structure of the magnetic material means that certain crystalline directions are easier to magnetize than others. The preferred direction of magnetization is known as the easy axis.</li>
<li>Shape anisotropy: When a magnetic particle is elongated, the direction of magnetization tends to occur along the long axis in order to minimize the energy of the stray magnetic field.</li>
</ul>
<h4>Domains</h4>
<ul>
<li><span style="font-size: 1em; line-height: 1.6em;">Domains form in magnetic materials in order to minimize energy.</span></li>
<li><span style="font-size: 1em; line-height: 1.6em;">Domain wall width is determined by the balance of two competing factors:</span></li>
<ul>
<ul>
<li><span style="font-size: 1em; line-height: 1.6em;">The minimization of exchange energy favors the formation of wide domain walls.</span></li>
<li><span style="font-size: 1em; line-height: 1.6em;">The minimization of magnetocrystalline anisotropy favors the formation of narrow domain walls.</span></li>
</ul>
</ul>
<li><span style="font-size: 1em; line-height: 1.6em;">Domain wall width: \(\delta = \pi \sqrt{A/2Ka}\) where \(a\) is the lattice parameter, \(A\) is the exchange constant, and \(K\) is the magnetocrystalline anisotropy.</span></li>
<li><span style="font-size: 1em; line-height: 1.6em;">Domain wall energy (per unit area):\(E_w = \pi \sqrt{2AK/a}\)</span></li>
<li><span style="font-size: 1em; line-height: 1.6em;">If a particle is small enough, it forms a single domain particle because it is not energetically worthwhile to form domain walls to lower the energy of the stray field.</span></li>
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Example 1.3A
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<p>Estimate the domain wall energy of a domain wall in iron. The anisotropy energy \(K\) is \(4.8 \times 10^{4} J/m^3\) and the exchange constant \(A\) is \(4.4 \times 10^{-21} J\). The lattice spacing in iron is \(0.29 \; nm\).</p>
<p>\(E_w\) (in \(J/m^2\)):</p>
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Exercise 1.3B
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<p>What is the width of the domain wall described above?</p>
<p>\(\delta\) (in \(nm\)):</p>
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