Determine the precessional frequency of an atom having magnetic moment μ in an external magnetic field B. This precession is known as the Larmor precession. Strategy We have already seen that the torque τ is equal to μ × B, but we also know from classical mechanics that the torque is dL/dt. The

Question

Determine the precessional frequency of an atom having magnetic moment [latex]\vec{\mu}[/latex] in an external magnetic field [latex]\vec{لآ}[/latex]. This precession is known as the Larmor precession.

Strategy We have already seen that the torque [latex]\vec{ au}[/latex] is equal to [latex]\vec{\mu} imes \vec{B}[/latex], but we also know from classical mechanics that the torque is [latex]d \vec{L} / d t[/latex]. The torque in Figure 7.5 is perpendicular [latex]\vec{\mu}, \vec{L} ext {, and } \vec{B}[/latex] and is out of the page. This must also be the direction of the change in momentum [latex]d \vec{L}[/latex] as seen in Figure 7.5. Thus [latex]\vec{L} ext { and } \vec{\mu}[/latex] precess about the magnetic field. The Larmor frequency [latex]\omega_{L} ext { is given by } d \phi / d t[/latex].

Accepted Answer

The magnitude of [latex]d \vec{L} ext { is given by } L \sin heta d \phi[/latex] (see Figure 7.5), so [latex]\omega_{L}[/latex] is given by

[latex]\omega_{L}=\frac{d \phi}{d t}=\frac{1}{L \sin heta} \frac{d L}{d t}[/latex] (7.29)

We now insert the magnitude of [latex]L=2 m \mu / e[/latex] from Equation (7.26). The value of dL/dt, the magnitude of [latex]\vec{\mu} imes \vec{B}[/latex], can be determined from Figure 7.5 to be [latex]\mu B \sin heta[/latex]. Equation (7.29) becomes

[latex]\vec{\mu}=-\frac{e}{2 m} \vec{L}[/latex] (7.26)

[latex]\omega_{L}=\left(\frac{e}{2 m \mu \sin heta} ight) \mu B \sin heta=\frac{e B}{2 m}[/latex] (7.30)

Question 7.5: Determine the precessional frequency of an atom having magne...

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