Nuclear Spin and the Magnetic Field

Introduction

Nuclear magnetic resonance (NMR) spectroscopy is the single most powerful technique for working out the structure of an organic molecule. It can reveal how many different hydrogen environments there are, how many hydrogens are in each and which groups are next to each other. All of that information starts from a surprising fact: certain atomic nuclei behave like tiny bar magnets. This page explains where that magnetism comes from, which nuclei possess it, and what happens to those nuclear magnets when they are placed in a very strong magnetic field.

Core explanation

Nuclear spin. Protons and neutrons each possess an intrinsic property called spin. In a nucleus these spins pair up where possible. If a nucleus has an even number of protons and an even number of neutrons, the spins cancel completely and the nucleus has zero spin (I = 0). Such nuclei, including ¹²C and ¹⁶O, are NMR-inactive and give no signal. If the numbers do not both pair, the nucleus has an overall spin and, because it is charged and spinning, a magnetic moment. It behaves like a tiny magnet.

The important NMR nuclei. For organic chemistry the two key nuclei are ¹H (a single proton) and ¹³C (six protons and seven neutrons). Both have spin quantum number I = ½. Other spin-½ nuclei used in research include ¹⁹F and ³¹P. Deuterium, ²H, has I = 1 and resonates at a very different frequency, which is why deuterated solvents do not interfere with ¹H spectra.

Spin states without a field. For a nucleus with spin quantum number I, there are 2I + 1 possible spin states. For I = ½ there are two. With no external magnetic field, the two states have exactly the same energy, and the nuclear magnets in a sample point in random directions.

Spin states in a field. When the sample is placed in a strong, uniform external field B₀, the two states have different energies:

- the α state (spin aligned with the field) is lower in energy; - the β state (spin opposed to the field) is higher in energy.

This is a quantised arrangement: nuclei cannot adopt intermediate orientations. The energy difference ΔE between the states is directly proportional to the field strength. Doubling B₀ doubles ΔE.

How big is the gap? Very small. In a typical 9.4 tesla magnet, the gap for ¹H corresponds to electromagnetic radiation of about 400 MHz, which lies in the radio-frequency region. That is why an instrument with this magnet is called a "400 MHz spectrometer". The gap for ¹³C in the same magnet is about a quarter as large, around 100 MHz, because the ¹³C nucleus is a weaker magnet.

Tiny population difference. Because ΔE is so small compared with the thermal energy available at room temperature, the two states are almost equally populated. In a 400 MHz field only about 3 extra nuclei in every 100 000 sit in the lower α state. NMR detects only this small excess, which is why NMR is a relatively insensitive technique and why stronger magnets, which widen the gap and enlarge the excess, give better spectra.

Natural abundance. Almost all hydrogen is ¹H (over 99.98%), so ¹H spectra are strong. Only about 1.1% of carbon is ¹³C, and its magnetic moment is weaker, so ¹³C spectra need much more acquisition time.

Formulae

Number of spin states = 2I + 1. Energy gap: ΔE = hν, where h = 6.63 × 10⁻³⁴ J s and ν is the resonance frequency; ΔE is proportional to B₀.

Step-by-step reasoning

To decide whether a nucleus is NMR-active:

1. Count its protons and neutrons. 2. If both are even, I = 0 and the nucleus gives no NMR signal. 3. Otherwise the nucleus has spin; for ¹H and ¹³C, I = ½. 4. Calculate the number of spin states as 2I + 1; two states for spin-½ nuclei.

Visual explanation

Draw a horizontal line for the single energy level with no field. To its right, as the field increases, draw two lines diverging like an opening pair of scissors: α dropping, β rising. The vertical gap between them grows steadily with B₀. The simulation lets you slide B₀ and watch the gap and frequency change.

Real-world analogy

A compass needle in the Earth's field can point north (comfortable, low energy) or be forced to point south (strained, higher energy). A nuclear magnet is similar, except that quantum rules allow only those two orientations and nothing in between.

Real-world example

Hospital MRI scanners rely on exactly this physics. They use magnets of about 1.5 or 3 tesla to split the spin states of hydrogen nuclei in water and fat in the body, then build images from the signals those protons give out.

Why?

Why does a stronger magnet give a better spectrum? A larger B₀ increases ΔE, which increases the small excess of nuclei in the lower state. More excess nuclei means more net absorption of energy, so the signal becomes stronger and signals are also spread further apart.

Common misconception

"Every atom in an organic molecule gives an NMR signal." The common isotopes ¹²C and ¹⁶O have zero spin and are invisible. Only nuclei with spin, such as ¹H and the rare ¹³C, are detected in routine organic NMR.

Worked example

Question: Calculate the energy gap for one proton in a 400 MHz spectrometer, and state whether ¹⁹F (9 protons, 10 neutrons) is NMR-active.

Reasoning: ΔE = hν = 6.63 × 10⁻³⁴ × 4.00 × 10⁸ = 2.65 × 10⁻²⁵ J. Fluorine-19 has an odd number of protons, so the spins do not all pair and the nucleus has spin.

Answer: ΔE ≈ 2.7 × 10⁻²⁵ J per nucleus; ¹⁹F is NMR-active (it has I = ½).

Quick check

1. How many spin states does a ¹³C nucleus have in a magnetic field, and which is lower in energy? Answer: Two states; the α state, aligned with the applied field, is lower in energy.

Exam focus

Know that nuclei with an odd number of protons or neutrons have spin, that ¹H and ¹³C have two spin states in a field, that the gap is proportional to field strength and that it corresponds to radio-frequency radiation. Explain why ¹²C gives no signal.

Advanced insight

The resonance frequency is ν = γB₀/2π, where γ, the gyromagnetic ratio, is a constant for each nucleus. For ¹H, γ/2π is about 42.6 MHz per tesla; for ¹³C about 10.7 MHz per tesla. This ratio of roughly four explains why a 400 MHz proton instrument observes carbon at about 100 MHz.

Summary

Nuclei with unpaired proton or neutron spins, such as ¹H and ¹³C (I = ½), act as tiny magnets. In an applied field B₀ they occupy two quantised spin states, α (lower, aligned) and β (higher, opposed). The gap is proportional to B₀ and lies in the radio-frequency region. Only a tiny excess of nuclei populates the lower state, so NMR is intrinsically insensitive and benefits from stronger magnets.

Practice questions

1. Explain why ¹²C gives no NMR signal but ¹³C does. Answer: ¹²C has six protons and six neutrons, both even, so its spins cancel and I = 0; ¹³C has seven neutrons, giving an overall spin of ½. 2. What happens to the energy gap between spin states if the field strength is doubled? Answer: It doubles, because ΔE is directly proportional to B₀. 3. In which region of the electromagnetic spectrum does NMR radiation lie, and why? Answer: The radio-frequency region, because the energy gap between nuclear spin states is extremely small. 4. Suggest two reasons why ¹³C spectra take longer to record than ¹H spectra. Answer: Only about 1.1% of carbon atoms are ¹³C, and the ¹³C nucleus is a weaker magnet, giving a smaller energy gap and population excess.