40-3 Magnetic Resonance

40-3 Magnetic Resonance#

Prompts

  • A proton has spin and a magnetic dipole moment. How does the proton differ from an electron in terms of the direction of \(\vec{\mu}\) relative to \(\vec{S}\), and which spin orientation (up or down) has lower energy?

  • For a proton in a magnetic field \(B\), what is the energy difference between the spin-up and spin-down states? What photon energy is needed to flip the spin?

  • What is magnetic resonance (NMR)? How is the RF field applied, and what happens when its frequency matches the spin-flip energy?

  • The effective field at a proton is the sum of an external field and an internal field from nearby atoms. Why does this allow NMR to identify different chemical environments?

  • For a given magnetic field, estimate the frequency of photons needed for proton NMR. Why is it in the radio-frequency range?

Lecture Notes#

Overview#

  • A proton has intrinsic spin \(\vec{S}\) and a magnetic dipole moment \(\vec{\mu}\) that are in the same direction (unlike the electron, because the proton is positively charged).

  • In a magnetic field \(\vec{B}\), the proton has two energy levels; spin up (aligned with \(\vec{B}\)) is the lower-energy state.

  • Magnetic resonance (NMR) occurs when a proton absorbs a photon whose energy equals the spin-flip energy \(\Delta E = 2\mu_z B\); the required frequency is in the radio-frequency range.

  • The effective field at a proton includes contributions from nearby atoms; this chemical shift allows NMR to identify different substances and is the basis of MRI.

Proton Spin and Magnetic Moment#

Like the electron, a proton has an intrinsic spin angular momentum \(\vec{S}\) and an associated spin magnetic dipole moment \(\vec{\mu}\). The key difference:

Electron

Proton

Charge

Negative

Positive

\(\vec{\mu}\) vs. \(\vec{S}\)

Opposite directions

Same direction

Lower energy in \(\vec{B}\)

Spin down (\(\vec{\mu}\) opposite \(\vec{B}\))

Spin up (\(\vec{\mu}\) aligned with \(\vec{B}\))

Proton vs. electron

For an electron, \(U = -\vec{\mu}\cdot\vec{B}\) with \(\vec{\mu}\) opposite \(\vec{S}\): spin down (\(\mu_z > 0\) when \(S_z < 0\)) gives lower \(U\). For a proton, \(\vec{\mu}\) and \(\vec{S}\) point the same way: spin up (aligned with \(\vec{B}\)) gives lower \(U\).

Energy Levels and Spin-Flip#

In a magnetic field \(\vec{B}\) along the \(z\) axis, the proton has two quantized orientations:

  • Spin up: \(\mu_z\) in the direction of \(\vec{B}\); energy \(U = -\mu_z B\) (lower).

  • Spin down: \(\mu_z\) opposite \(\vec{B}\); energy \(U = +\mu_z B\) (higher).

The energy difference is

(418)#\[ \Delta E = 2\mu_z B \]

A proton can jump from the lower to the higher state by absorbing a photon with energy

(419)#\[ hf = 2\mu_z B \]

This absorption is called magnetic resonance or nuclear magnetic resonance (NMR); the reversal of \(S_z\) is called spin-flipping.

Resonance Condition#

For resonance to occur, the photon frequency must satisfy

(420)#\[ f = \frac{2\mu_z B}{h} \]

  • The proton’s magnetic moment \(\mu_z\) is much smaller than the Bohr magneton (nuclear magneton scale).

  • Typical fields \(B \sim 1\)–3 T give frequencies in the radio-frequency (RF) range (tens of MHz).

In practice, an RF source drives a coil wrapped around the sample. The oscillating EM field in the coil provides photons at frequency \(f\). When \(f\) matches the resonance condition, protons absorb energy and flip spin; the absorption is detected.

NMR Spectra and Chemical Shifts#

The field \(\vec{B}\) at a proton is not purely the external field. It is the vector sum of:

  1. The external field from the magnet.

  2. The internal field from nearby atoms and nuclei.

Different chemical environments (e.g., protons in water vs. in organic molecules) experience slightly different effective fields. The resonance frequency shifts accordingly — a chemical shift. By scanning the RF frequency (or the field) and recording absorption, one obtains an NMR spectrum that identifies specific substances and molecular structure.

MRI

Magnetic resonance imaging (MRI) uses the same physics. A gradient in the magnetic field encodes spatial position into the resonance frequency, allowing construction of 3D images of proton density (e.g., in soft tissue). See Module 32-5 for a brief discussion.

Summary#

  • A proton’s spin and magnetic moment are aligned (positive charge); spin up in \(\vec{B}\) is the lower-energy state.

  • The spin-flip energy is \(\Delta E = 2\mu_z B\); resonance occurs when \(hf = 2\mu_z B\).

  • NMR uses RF photons to induce spin-flips; detection of absorption yields spectra.

  • Chemical shifts from the internal field allow identification of different substances and support applications such as MRI.