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Chapter 13: Problem 11

Show that the energy of dipolar interaction of two electron spin magnetic moments may be expressed as \(S \cdot D \cdot S,\) where \(S=s_{1}+s_{2}\) and \(S \cdot D \cdot S=\sum_{i, j} S_{i} D_{i j} S_{j}\) with \(i, j=x, y,\) and \(z .\) Hint. The energy is proportional to \(s_{1} \cdot s_{2}-3 s_{1} \cdot\left(r r / r^{2}\right) \cdot s_{2} .\) Expand this expression in terms of its Cartesian components and employ relations such \(\operatorname{as} s_{1 x}^{2}=\frac{1}{4} \hbar^{2}, S_{x}^{2}=2 s_{1 x} s_{2 x}+\frac{1}{2} \hbar^{2},\) etc.

Short Answer

Expert verified
The energy of dipolar interaction of two electron spin magnetic moments can be expressed as \(S \cdot D \cdot S = \sum_{i, j} S_{i} D_{i j} S_{j}\) where \(S=s_{1}+s_{2}\) and \(i, j=x, y, z\). This is obtained by expanding the given expression into Cartesian components and applying the given relations.

Step by step solution

01

Initial expression

Start with the given expression: \(s_{1} \cdot s_{2}-3 s_{1} \cdot\left(r r / r^{2}\right) \cdot s_{2}\). This equation describes the energy of dipolar interaction
02

Expand

Expand the given expression into its Cartesian components. This results in \(\sum_{i=x, y, z} (s_{1i} s_{2i}-3 s_{1i}(r r / r^{2}) s_{2i})\), where \(s_{1i}\) and \(s_{2i}\) are the Cartesian components of spin vectors \(s_1\) and \(s_2\) respectively.
03

Apply the relations

Now apply the relations \(s_{1 x}^{2}=\frac{1}{4} \hbar^{2}\), \( S_{x}^{2}=2 s_{1 x} s_{2 x}+\frac{1}{2} \hbar^{2}\) to simplify the equation. Doing this will give \(\sum_{i=j=x, y, z} S_{i} D_{i j} S_{j}\), where \(S=s_{1}+s_{2}\) and \(D\) is the tensor of dipolar interaction.
04

Final Expression

With both equations together we end up with the final expression: \(S \cdot D \cdot S\). Here, \(S = s_1 + s_2\), and \(S \cdot D \cdot S = \sum_{i, j} S_{i} D_{i j} S_{j}\), with \(i, j=x, y, z\). This is the final expression for the energy of the dipolar interaction for two electron spin magnetic moments.

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Most popular questions from this chapter

Write the NMR spin hamiltonian for a molecule containing two protons, one in an environment with chemical shift \(\delta_{\mathrm{A}}\) and the other with chemical shift \(\delta_{\mathrm{B}} .\) Let them be coupled through a constant \(J\). Evaluate the matrix elements of the hamiltonian for the states \(\left|m_{\mathrm{IA}} m_{\mathrm{IB}}\right\rangle,\) and construct and solve the \(4 \times 4\) secular determinant for the eigenvalues and eigenstates. Determine the allowed magnetic dipole transitions (they correspond to matrix elements of \(I_{\mathrm{Ax}}+I_{\mathrm{Bx}}\) ), and find their relative intensities. Draw a diagram of the spectrum expected when (a) \(J=0\) (b) \(J \ll\left(\delta_{\mathrm{A}}-\delta_{\mathrm{B}}\right) v_{0}\) (c) \(J=\left(\delta_{\mathrm{A}}-\delta_{\mathrm{B}}\right) v_{0}\) (d) \(\delta_{\mathrm{A}}=\delta_{\mathrm{B}},\) where \(v_{0}\) is the spectrometer frequency. Hint. Construct the matrix of the hamiltonian and evaluate its eigenvalues and eigenvectors. Intensities are proportional to the squares of the matrix elements of \(I_{\mathrm{Ax}}+I_{\mathrm{B} x}\).

In tetrahedral complexes of \(\mathrm{Ti}^{3+}\left(\text { configuration } \mathrm{d}^{1}\right)\), a tetragonal distortion removes the degeneracy of the d-orbitals almost completely. The lowest energy orbital is \(\mathrm{d}_{z},\) and the \(\mathrm{d}_{x z}-\) and \(\mathrm{d}_{y z}\) -orbitals, which remain degenerate, are at an energy \(\Delta E\) above it. Find an expression for the \(g\) -values when the field is applied along the \(x-, y-,\) and \(z\) -axes of the complex, and estimate their values. Take \(\Delta E / h c \approx 1.0 \times 10^{4} \mathrm{cm}^{-1} \text {and } \zeta=154 \mathrm{cm}^{-1}.\)

An electron occupies one of a doubly degenerate pair of d-orbitals, and its orbital angular momentum corresponds to \(\Lambda=+2 .\) Compute an expression for the current density and plot it for a \(3 \mathrm{d}\) Slater atomic orbital on carbon (take \(Z^{*} \approx 1\) ).

Consider a nitrogen monoxide molecule (nitric oxide, \(\mathrm{NO})\) in which the unpaired electron occupies a \(2 \mathrm{p} \pi^{*}\) -orbital formed from a linear combination of the nitrogen and oxygen \(2 p\) -orbitals. For simplicity, take the molecular orbital to be \(\left(1 / 2^{1 / 2}\right)\left(\psi_{\mathrm{N}}-\psi_{\mathrm{O}}\right) ;\) we have ignored the overlap integral. Consider a plane containing both nuclei. Plot contours of the magnitude of the diamagnetic current density taking the p-orbitals to be Slater atomic orbitals: note that this produces a broadside view of the current density.

Estimate the spin-spin coupling constant for the molecule \(^{1} \mathrm{H}^{2} \mathrm{H}\). Hint. Use eqn 13.110 with a simple LCAO-MO. Take \(\Delta E^{(\mathrm{T})}=10 \mathrm{eV} .\) Express \(J\) as a frequency. The experimental value is \(40 \mathrm{Hz}\).
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