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<h2 class="hd hd-2 unit-title">Problem 1.1</h2>
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Problem 1.1
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><b>Part A</b>
<p>From the magnetic moment of 1.7 \(\mu_{B}\) per Co in metallic Co, predict what \(M_{s}\) will be. Note that a unit cell of Co contains \(6\) atoms, and has a volume of \(6.68 \times 10^{-2} nm^3\).</p>
<p>\(M_s\) (in kA/m):</p>
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problem
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<b>Part B</b>
<p>Now consider the material MnFe<sub>2</sub>O<sub>4</sub>. In this cubic material, all the Mn ions (valence state +2) ions have their magnetic moments pointing along the +z direction, and all the Fe ions (valence state +3) have their magnetic moments pointing along the -z direction.</p>
<p>The following data correspond to 2+ ions:</p>
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<p>What is the magnetic moment of the material per formula unit? <br/><br/>To answer this question, you will need to type a formula into the box below. Please express your answer in terms of \(\mu_B\), which you can enter by typing "mu_B" into the box below. Also, you will need to express multiplication explicitly using the '*' sign. For example, if your answer is \(3 \mu_B\), you would need to type "3*mu_B" into the box below.</p>
<p>Magneteic moment per formula unit:</p>
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<p>If the cubic lattice parameter is \(0.8 nm\), and one unit cell contains \(8\) formula units, what is \(M_s\) for the material?</p>
<p>\(M_s\) (in \(kA/m\)):</p>
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<h2 class="hd hd-2 unit-title">Problem 1.2</h2>
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Problem 1.2
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<p> Consider two magnetic materials: amorphous FeCoB and crystalline NdFeB. Indicate which of the following statements apply to FeCoB and which apply to NdFeB. </p>
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<p> Indicate which hysteresis loop corresponds to amorphous FeCoB and which corresponds to crystalline NdFeB:</p>
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<h2 class="hd hd-2 unit-title">Question 1.3</h2>
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<p>Suppose you have a sphere of single-crystal cobalt with diameter \(L\). For what values of \(L\) would you expect this to be a single domain particle? Assume the magnetostatic energy of a uniformly magnetized sphere of volume \(V = 4/3 \pi (L/2)^3\) is given by \(\mu_0 M_s^2 V\).</p>
<p>For Co, \(M_s = 1400 kA/m\), \(K = 4 \times 10^5 J/m^3\), \(a = 0.25 nm\) and \(A = 10^{-18} J\).</p>
<p>Particles remain single domain when \(L\) is less than what value?</p>
<p>Hint: Remember that the expression provided in class and in the notes for domain wall energy \(E_w\) is given in \(J/m^2\).</p>
<p>\(L\) (in \(nm\)):</p>
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<p>For what values of \(L\) would you expect to observe superparamagnetism at room temperature?</p>
<p>We expect to observe superparamagnetism when \(L\) is less than what value?</p>
<p>\(L\) (in \(nm\)):</p>
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<div><b>Part 3</b><br/>
Indicate which properties would increase and which properties would decrease if the particle described in Part 1 were changed from a sphere to an ellipsoid (a needle-shaped particle). Assume that the long axis is along the c-axis.
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