Chemical Shift and the δ Scale

Introduction

Protons in different chemical environments absorb at slightly different radio frequencies. The trouble is that the actual frequencies depend on the magnet: the same proton in ethanol resonates at one frequency on a 300 MHz instrument and at a different one on a 600 MHz instrument. Chemists need a scale on which a given proton always appears at the same place, whichever machine is used. The chemical shift scale, δ, does exactly this by measuring each signal relative to a reference and dividing by the operating frequency.

Core explanation

Measuring from a reference. Absolute resonance frequencies are hundreds of megahertz and differ between instruments. What matters chemically is the small difference between a proton's frequency and that of a standard. The universally agreed reference is tetramethylsilane (TMS), whose signal is assigned δ = 0. The frequency difference is small, typically from zero to a few thousand hertz.

Dividing by the operating frequency. The frequency difference in hertz is proportional to the field strength. A proton 1480 Hz from TMS on a 400 MHz instrument would be 2220 Hz from TMS on a 600 MHz instrument. Dividing the difference by the operating frequency removes this dependence, giving a ratio that is the same on every instrument. Because the ratio is tiny, it is multiplied by 10⁶ and quoted in parts per million:

δ = (ν sample − ν TMS) ÷ ν spectrometer × 10⁶

Both frequencies must be in the same units. In the example above, 1480 Hz ÷ 400 × 10⁶ Hz × 10⁶ = 3.70 ppm, and on the 600 MHz instrument 2220 ÷ 600 × 10⁶ × 10⁶ is also 3.70 ppm.

A handy shortcut. On an instrument operating at N MHz, 1 ppm corresponds to N Hz. On a 400 MHz spectrometer, 1 ppm = 400 Hz; on a 600 MHz spectrometer, 1 ppm = 600 Hz.

The range of δ values. Almost all proton signals in organic compounds fall between δ 0 and δ 12. Alkyl protons appear roughly from 0.9 to 2, protons next to oxygen around 3.3 to 4.5, alkene protons about 4.5 to 6.5, aromatic protons about 6.5 to 8, aldehyde protons 9 to 10 and carboxylic acid protons 10 to 12. Carbon-13 shifts, also measured from TMS, cover a much wider range, about 0 to 220 ppm.

Reading the axis. By long-standing convention, δ increases from right to left. TMS sits at the far right. Signals at higher δ, to the left, are described as downfield ; signals at lower δ, to the right, are upfield . These terms come from early instruments that swept the magnetic field: signals on the left needed a lower field to come into resonance. Although the historical wording seems back-to-front, it remains in everyday use, alongside the clearer "high δ" and "low δ".

Higher field, better spread. Since δ values are fixed, a stronger magnet does not move signals on the δ scale. Instead it increases the number of hertz per ppm, so neighbouring signals are separated by more hertz and overlapping peaks become resolved.

Formulae

δ (ppm) = (ν sample − ν TMS) ÷ ν spectrometer × 10⁶. Frequency difference in Hz = δ × operating frequency in MHz.

Step-by-step reasoning

To convert a frequency offset into a chemical shift:

1. Find the frequency difference between the signal and TMS, in Hz. 2. Convert the spectrometer frequency to Hz (400 MHz = 4.00 × 10⁸ Hz). 3. Divide the difference by the spectrometer frequency. 4. Multiply by 10⁶ to express the result in ppm.

Visual explanation

Sketch a horizontal axis labelled δ/ppm running from 12 on the left to 0 on the right, with a sharp TMS line at 0. Mark arrows "downfield, deshielded, higher δ" pointing left and "upfield, shielded, lower δ" pointing right. The simulation lets you switch between 300 and 600 MHz and see Hz values change while δ values stay put.

Real-world analogy

Describing a price increase as a percentage rather than in pounds lets you compare shops of any size. A ratio relative to a baseline travels well; chemical shift is a percentage-like ratio, in millionths, relative to TMS.

Real-world example

Journals and spectral databases list compounds by δ values, for instance "δ 3.70 (s, 3H)" for a methyl ester. Because δ does not depend on the instrument, a chemist in one country can compare a new spectrum directly with data recorded decades earlier on a completely different spectrometer.

Why?

Why divide by the operating frequency? The shielding that shifts a signal is proportional to the applied field, so the frequency offset grows with the field. Dividing by the operating frequency cancels this proportionality, leaving a quantity that depends only on the chemical environment.

Common misconception

"A 600 MHz spectrometer moves every signal to a larger δ value than a 300 MHz one." The δ value of a proton is the same on both. Only the separation in hertz changes, which is why higher-field instruments resolve crowded spectra more clearly.

Worked example

Question: On a 300 MHz spectrometer, a signal appears 2190 Hz downfield of TMS. Calculate δ, and state how many hertz from TMS it would be on a 500 MHz instrument.

Reasoning: δ = 2190 ÷ (300 × 10⁶) × 10⁶ = 7.30 ppm. On a 500 MHz instrument, 1 ppm = 500 Hz, so the offset is 7.30 × 500 Hz.

Answer: δ = 7.30 ppm; 3650 Hz from TMS on the 500 MHz instrument (an aromatic proton region).

Quick check

1. On a 400 MHz spectrometer, how many hertz from TMS is a signal at δ 2.00? Answer: 2.00 × 400 = 800 Hz downfield of TMS.

Exam focus

Learn the definition of δ and its unit, ppm. Be able to convert between hertz and ppm, state that δ is independent of the field strength, and use upfield and downfield correctly. Remember that the δ axis increases from right to left, with TMS at zero on the right.

Advanced insight

Coupling constants, J, are quoted in hertz, not ppm, because the splitting caused by neighbouring nuclei does not scale with the field. On a stronger instrument, multiplets therefore occupy a smaller fraction of a ppm, which is a further reason why high-field spectra look cleaner and easier to interpret.

Summary

Chemical shift, δ, locates an NMR signal relative to TMS at δ 0. It is the frequency offset divided by the spectrometer frequency, multiplied by 10⁶, in ppm, and is therefore identical on every instrument. Proton shifts usually span 0 to 12 ppm. The axis increases to the left; higher δ is downfield and lower δ is upfield. Stronger magnets spread signals over more hertz without changing their δ values.

Practice questions

1. Define chemical shift and give its unit. Answer: The difference between a signal's resonance frequency and that of TMS, divided by the spectrometer frequency and multiplied by 10⁶; measured in ppm. 2. A signal is 840 Hz from TMS on a 400 MHz instrument. Calculate δ. Answer: 840 ÷ (400 × 10⁶) × 10⁶ = 2.10 ppm. 3. Two signals are 0.05 ppm apart. How far apart are they in hertz on 300 MHz and 600 MHz instruments? Answer: 15 Hz on the 300 MHz instrument and 30 Hz on the 600 MHz instrument. 4. Which is further downfield, a signal at δ 1.2 or one at δ 7.3? Answer: The signal at δ 7.3, because downfield means higher δ, towards the left of the spectrum.