Chapter 2: Problem 16
Paramagnetism can occur when some atoms in a crystalline solid possess a magnetic dipole moment owing to an unpaired electron with its associated orbital angular momentum. For simplicity, assume that (1) each paramagnetic atom has a magnetic dipole moment \(\mu\) and (2) magnetic interactions between unpaired electrons can be neglected. When a magnetic field is applied, the magnetic dipoles will align themselves either parallel or antiparallel to the direction of the magnetic field. If the magnetic moment is parallel to a magnetic field of induction \(\vec{B}\), the potential energy is \(-\mu B\), when the magnetic moment is antiparallel to \(\vec{B}\), the potential energy is \(+\mu B\). a. Prove that the probability for an atomic magnetic dipole moment to point parallel to the magnetic field is given at temperature \(T\) by $$ P_{\mathrm{a}}=\left(1+e^{-2 t}\right)^{-1} $$ where \(x=\mu B / k T\). Give a physical explanation for the value of \(P\) as \(T \rightarrow 0\) and as \(T \rightarrow \infty\) Hint: Determine the partition function for the system. b. Show that, for \(N\) independent magnetic dipoles, the mean effective magnetic moment parallel to the magnetic field is $$ m=N \mu \tanh (x) . $$ c. Demonstrate that the mean magnetic moment at high temperatures and/or weak magnetic fields ( \(x \& 1\) ) is proportional to \(1 / T\). This is Curie's law. d. Show that the contribution from paramagnetism to the internal energy of a crystalline solid is \(U=-m B\). Determine this paramagnetic contribution at \(T=\) \(\infty\). Why should this result have been expected? e. Develop an expression for the entropy of this paramagnetic system.
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