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Spin and Nuclear Magnetic Resonance

Dipole moments, quantum spin, and fine structure splitting

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Magnetic Dipole Moments

The current through a closed loop for an electron is approximated

by a single charge moving at speed v in a circular path

The angular momentum of a particle moving in a circular path is given by

Therefore, the magnetic dipole moment μ is given in terms of angular momentum by

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Magnitude of a Dipole Moment

The previous equation is a close approximation, however for more complicated charge distributions we multiply by a dimensionless scalar known as the Gyromagnetic Ratio, or the ‘g-factor’.

For an electron, angular momentum has values given by

So the magnitude of the magnetic dipole moment for an electron is

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Spin as an Analog for Angular Momentum

At the scale of atoms, particles aren’t seen to be traveling in circular paths; rather, they have intrinsic angular momentum! Spin in quantum mechanics is a direct analog of classical angular momentum. Another name for spin is ‘total angular momentum’.

So the magnitude of the magnetic dipole moment for an electron due to its spin is

Solutions to the Dirac equation for energy levels accurately predicts all four quantum numbers correctly, including s

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Energy of a Dipole in an External B Field

An external magnetic field exerts a torque on the dipole moment,

Because of this torque, work is done on the moment.

Integrating to find the potential energy of the system,

For an external B field in the +z direction

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Energy States: Proton in an External B Field

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Nuclear Magnetic Resonance

Using our knowledge of fine structure splitting, we can build an apparatus that can measure the resonance response of a sample as shown (right):