MRI Physics Made Easy

Chapter 1: Nuclear Spins and Larmor FrequencyKey ConceptsThis chapter covers how hydrogen nuclei behave in a magnetic field and why they precess at a specific frequency. You need to know the Larmor relationship because it explains how MRI selects and excites hydrogen nuclei.

Hydrogen nuclei contain one proton with spin and angular momentum.

In the main magnetic field, called B0, proton magnetic moments partly align:

Parallel: lower energy

Antiparallel: higher energy

The proton does not simply point straight along B0. It precesses, like a spinning top wobbling around an axis.

Larmor frequency is the frequency of this precession.

Formula: f = γB0

For hydrogen:

γ/2π ≈ 42.58 MHz/T

At 1.5 T: about 63.9 MHz

At 3 T: about 127.7 MHz

Increasing B0 increases Larmor frequency in direct proportion.

Before you continue: Can you explain why the Larmor frequency is higher at 3 T than at 1.5 T?

Key TermsNuclear spin - Intrinsic angular momentum of a nucleus.

Magnetic moment - The magnetic property produced by a spinning charged nucleus.

Precession - Circular wobbling of a proton’s magnetic moment around B0.

B0 - The main static magnetic field of the MRI scanner.

Larmor frequency - The precession frequency of a nucleus in a magnetic field.

Gyromagnetic ratio (γ) - A constant linking a nucleus’s Larmor frequency to magnetic-field strength.

Active RecallNuclear spin__________

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Magnetic moment__________

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Precession__________

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B0__________

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Larmor frequency__________

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Gyromagnetic ratio (γ)__________

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Worked Examples1. Calculate frequency at 1.5 TUse the hydrogen value: 42.58 MHz/T.

Multiply by field strength: 42.58 × 1.5.

Result: 63.87 MHz, approximately 63.9 MHz.

Now you try: Calculate the hydrogen Larmor frequency at 2 T.

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2. Compare 1.5 T and 3 TThe field strength doubles: 3 ÷ 1.5 = 2.

The Larmor frequency also doubles.

63.9 MHz × 2 = 127.8 MHz.

Therefore, hydrogen resonates at approximately 127.7-127.8 MHz at 3 T.

Now you try: If the frequency is 42.58 MHz at 1 T, what is it at 4 T?

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Practice Questions(Easy) Define precession.

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(Easy) State the formula linking Larmor frequency and magnetic-field strength.

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(Medium) Calculate the hydrogen frequency at 1.5 T.

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(Medium) Explain what happens to Larmor frequency when B0 increases.

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(Hard) Explain why an MRI scanner must match its radiofrequency pulse to the Larmor frequency.

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Answer KeyCircular wobbling of a proton’s magnetic moment around B0.

f = γB0.

42.58 × 1.5 = 63.87 MHz, approximately 63.9 MHz.

It increases directly and proportionally.

Matching the frequency allows efficient energy transfer and excites hydrogen nuclei into resonance.

Exam Tips & Common MistakesDo not confuse spin with the proton physically rotating like a tiny ball. Spin is an intrinsic quantum property.

Do not state that all nuclei point in exactly the same direction. Alignment is slightly biased toward the lower-energy parallel state.

Do not confuse frequency with field strength. Frequency changes because B0 changes.

Always include units: MHz/T, MHz, or tesla (T).

Markers expect the phrase “directly proportional” when describing the Larmor relationship.

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