A Periodic Chart for Magnetic Resonance

Spin, abundance, receptivity and Larmor frequency, arranged the way the chemists arranged them and annotated for people who put things in magnets.

Mendeleev’s table is organised by the electrons. Magnetic resonance cares about the nucleus, and the nucleus keeps its own counsel: ¹²C is invisible and ¹³C is not, ¹⁶O is invisible and ¹⁷O is not, and nothing about an element’s chemistry tells you which of its isotopes will answer when you call. Nuclei with an even number of protons and an even number of neutrons have I = 0 and no magnetic moment at all. Stable nuclides are overwhelmingly even-even, so that one rule takes more than half the table away at a stroke, including the two most abundant atoms in soft tissue after hydrogen. So this chart carries a second layer over the first: for every element, the isotopes that carry spin, what they cost you in sensitivity, and where they land in frequency at whatever field you happen to have.

It reads two ways. Read down the sensitivity, and it is a table of nuclei you might one day detect, from the proton you detect without trying to the ones nobody has ever seen in a living subject. Read across the roles, and it is a table of the machine: the niobium in the wire, the helium in the cryostat, the copper in the coil, the gadolinium in the syringe, the titanium in the patient’s hip. Set the field, choose an encoding, and click anything.

What a nucleus costs you

Sensitivity in magnetic resonance is brutally non-linear. The signal from a given number of spins goes as the cube of the gyromagnetic ratio, and then you multiply by however few of the element’s atoms happen to be the right isotope. One factor of γ from the Boltzmann polarization, one from the magnetic moment each spin presents to the coil, and one from the Faraday induction that turns precession into voltage. The three of them together are why ¹³C, whose γ is a quarter of the proton’s, is already down by a factor of sixty-four before natural abundance takes its share. The ladder below is that arithmetic, drawn on a log scale spanning ten decades. The filled dot is what nature gives you; the open dot is what you would get from a fully labelled sample, and the gap between them is the entire economic case for isotopic enrichment.

The arithmetic behind the cells

What the table stores is small: the nuclear spin I, the magnetic dipole moment μ in nuclear magnetons, the natural abundance, the electric quadrupole moment Q, and, where the NMR literature has settled on one, a tabulated γ and frequency ratio. Everything else on the page is computed from those, live, at whatever field you have set, and the receptivities are recomputed even where a source quoted them, so that no displayed number can drift from the primitives it came from: Which is also why you will occasionally see a γ here that differs in its fourth figure from one you have on a card by the console. A moment measured in a real compound belongs to a shielded nucleus; correcting it for the electrons around it gives the bare-nucleus value, and the two routes propagate into two slightly different tables. The gap is a few hundredths of a per cent for light nuclei and grows steeply with the electron count: ³¹P comes out at 17.251 or 17.236 MHz/T, nine hundred parts per million apart; ⁵⁵Mn at 10.576 or 10.555; and by thallium, lead and bismuth the two routes are more than one per cent apart. The panel shows both wherever they differ by more than a hundred parts per million, so the gap is visible rather than buried.

Xenon is where this stops being academic, and it takes two steps. The table gives ¹²⁹Xe as 11.860 MHz/T; the frequency-ratio route gives 11.841; and a hyperpolarized-gas coil is tuned on 11.777, because free xenon gas sits a further 5300 ppm upfield of the IUPAC reference compound, XeOF₄. End to end that is seven parts in a thousand, a quarter of a megahertz at 3 T: a missed resonance, not a rounding error. Any nucleus whose chemical shift range runs to thousands of ppm carries the same trap, and the isotope note in the panel below says so where it matters.

gyromagnetic ratio

γ/2π = 7.6225932 μ / I  MHz per tesla

Larmor frequency

ν₀ = |γ/2π| B₀

receptivity

D = (N/100) |γ/γ_H|³ I(I+1) / ¾  relative to ¹H

thermal polarization

P = (I+1) γℏB₀ / 3kT  at 310 K

wavelength in tissue

λ = c / (ν₀ √ε_r)  ε_r ≈ 60

skin depth in copper

δ = 65.2 / √ν₀[MHz]  micrometres

The last two are there because a Larmor frequency is not only a number you type into the console. It sets the wavelength of the B₁ field inside a conductive, high-permittivity patient, and it sets how thin a skin of your copper is actually carrying current. At 3 T the lossless expression gives a proton wavelength in soft tissue of about 30 cm, which is why dielectric shading is a nuisance rather than a catastrophe; at 7 T it is around 13 cm, comparable to a head, and the standing wave becomes a dominant design problem. Read both as upper bounds. The permittivity here is one round number standing in for all tissue, where grey matter, white matter and fat differ from each other by more than a factor of two, and the expression ignores conductivity, which at 128 MHz is not a small omission: with σ near 0.5 S/m the loss tangent is above one, and the real wavelength is about a tenth shorter again, nearer 27 cm. Every stable X-nucleus sits at a longer wavelength than the proton at the same field and is correspondingly better behaved. Tritium is the one nucleus that does not, and you are not putting it in a patient. Move from ¹H to ³¹P in the same magnet and the wavelength gets two and a half times longer, while the skin depth, going as the inverse square root of frequency, grows by about sixty per cent. A coil design that ignores either will disappoint you.

One caution about receptivity: it counts atoms, not molarity in tissue. Water protons are present at about 80 molar in the proton sense; phosphocreatine is present at a few millimolar. Multiplying the two effects together, not the receptivity alone, is what tells you whether a voxel is a five-minute acquisition or an impossible one.

Reading notes

• Spin one half is a gift. Of the nuclei that resonate at all, only those with I = ½ have no electric quadrupole moment, and so only they are indifferent to the electric field gradients of their surroundings. Every other NMR-active nucleus relaxes through its quadrupole, which in a large molecule or a solid means lines broad enough to vanish into the baseline. This is why ¹H, ¹³C, ³¹P, ¹⁹F, ¹⁵N and ¹²⁹Xe carry almost all of in-vivo spectroscopy between them, and why ²³Na, the one quadrupolar workhorse, is imaged rather than resolved.
• The sign of γ is real. Negative-γ nuclei precess the other way. It does not change the frequency, but it flips the sense of every rotating frame, every phase-encoding gradient, every Overhauser enhancement, and the sign of the NOE you get from decoupling. ³He, ¹⁵N, ¹⁷O, ²⁹Si and ¹²⁹Xe are the ones you are likely to meet, and the two hyperpolarized gases are the two you are likeliest to meet.
• Quadrupolar broadening is not the same as low receptivity. ²⁷Al has 100 per cent abundance and a receptivity of 0.2, better than ³¹P, and yet nobody does aluminium spectroscopy in vivo. The reason is in the Q column, not the D column.
• Thermal polarization is the embarrassment of the field. At 3 T and body temperature a proton ensemble is about ten parts per million polarized, and an X-nucleus rather less. Hyperpolarization methods reach tens of per cent, a gain of four to five orders of magnitude, which is how a gas at atmospheric density, or a carbon at natural abundance, becomes visible at all. The cost is that the polarization is not in equilibrium and will not come back: you get one shot, decaying with T₁.
• The elements without spin still matter. Switch to the hardware encoding and the chart stops being about detection. Niobium, titanium and tin build the magnet; helium and nitrogen keep it cold; copper, silver and aluminium carry the currents; gadolinium, manganese, iron and dysprosium alter the relaxation of protons they never resonate with themselves; and the three room-temperature ferromagnets are the reason for the line on the floor.

Where this is still soft

A reference table is only as good as its weakest column, and it is more useful to say which column that is than to let a reader find out. What follows is the honest inventory. The reviewer’s notes carry the full list, entry by entry.

• Magnetic susceptibility was the softest column in the table, and is now the most recently rebuilt. Every value has been checked cell by cell against the CRC compilation named below, and forty of them moved. Two had been wrong by a factor of six: germanium and indium were carrying numbers that would have made them semimetals in bismuth’s class. Thorium was carrying a figure its own note already called inflated. What is still soft is narrower but real. Fluorine is the one value with no entry in that table at all, resting instead on a 1999 gas-phase measurement that drew a published Comment. Helium is the one cell that deliberately departs from it, because helium’s diamagnetism is calculable essentially exactly and the tabulated figure is seven per cent off. Carbon is the least representative number here: the cell carries graphite’s in-plane value, and a randomly oriented powder averages fifteen times larger. The tin cell is grey tin, which is not the tin in your solder or your Nb₃Sn. Promethium and everything past plutonium are null because the source does not list them.
• Chemical shift ranges are soft by nature and softer here. Well anchored: ¹²⁵Te, ⁹⁹Ru, ¹²⁹Xe, ²⁰⁷Pb, ²⁰⁵Tl. Constructed windows rather than tabulated ranges: ¹³³Cs, ⁹⁹Tc, ¹⁰³Rh, the silvers, the cadmiums, the indiums and the tins. Left null where no defensible endpoints existed at all: ¹⁰⁵Pd, the antimonies, ¹²⁷I and the bariums.
• Some frequency ratios do not reconcile with their own moments. Two are now withheld rather than printed. Two isotopes of one element measured in one molecule must give a ratio equal to their moment ratio, the shielding cancelling exactly; ¹⁸⁹Os and ¹⁸⁷Os are eight parts in a thousand from that, which rounding cannot explain, so the ¹⁸⁹Os ratio is gone and the ¹⁸⁷Os one kept because the osmium shift scale is quoted on it. ¹⁷⁶Lu’s ratio is gone too: no published moment reproduces it, and with I = 7 and Q = +492 fm² no solution measurement could have anchored one. ¹⁷⁵Lu’s is sound, and the seventeen hundred parts per million between it and the γ here is the gap between two successive recommended moments, not an error. ⁹⁹Tc’s Ξ implies a γ a thousandth higher than its moment gives, which means a negative absolute shielding, and for a d⁰ oxoanion that is ordinary rather than impossible. ¹⁶⁹Tm and the iridiums carry no ratio at all, for one reason: the ion is paramagnetic, no diamagnetic reference sample exists, and a quoted ratio would be an estimate wearing the clothes of a measurement.
• One value now sits on a different scale from its neighbours. The quadrupole moment of ²⁰⁹Bi is given as −42.2 fm², the 2023 determination, rather than the −51.6 of Stone 2013 and of Pyykkö’s own 2017 table: a twenty per cent revision is too large to go on printing the superseded number. Every other quadrupole moment in the chart is still on the older scale, so bismuth is the odd one out by design, and ²¹³Bi’s moment has been rescaled to match it rather than measured. Several lanthanide quadrupole moments are Stone 2013 rather than the 2018 revision. ²²⁹Th already uses the newer moment; others may deserve it.
• A few dossier numbers are inference, not citation. The nitinol susceptibility is derived from a mass susceptibility and a density. The persistent-circuit resistance is derived from a drift specification rather than quoted. Desflurane’s ¹⁹F position is inferred by analogy from isoflurane, and fluoxetine’s from its chemical family rather than from an in-vivo paper. The absolute position of 5-fluorouracil, on which every one of its metabolites is pinned, has since been confirmed to about a seventh of a ppm, but it is a pH 7.4 value for a ring that titrates with a pK_a near 8, and like everything else in that table it is anchored on a trifluoroacetic acid reference whose own position moves by about a ppm with concentration and pH.
• Stem heights in the spectra are drawing hints. They are ordered sensibly and, in the proton and xenon tables, roughly faithful. They are not amplitudes, and the single-line group’s heights are a readability ranking compressing about two orders of magnitude.

Where the numbers come from

Spins, magnetic moments, natural abundances and relative frequencies follow the IUPAC recommendations of Harris, Becker, Cabral de Menezes, Goodfellow and Granger, NMR nomenclature: nuclear spin properties and conventions for chemical shifts (2001, revised 2008). Every spin, moment and abundance was then checked by program against Stone’s IAEA compilation of nuclear magnetic dipole moments, in the machine-readable form distributed with EasySpin, so that the differences which remain are the convention splits described in the margin above rather than transcription errors. Quadrupole moments follow Pyykkö’s 2018 evaluation, which revises a number of the older values by ten to twenty per cent (⁴³Ca and ⁶⁷Zn among them), with one exception: ²⁰⁹Bi carries the 2023 redetermination instead. They are given in fm²; one fm² is 10 millibarn, or 0.01 barn, if you learned them the old way. Molar magnetic susceptibilities are the values of the CRC Handbook’s Magnetic Susceptibility of the Elements and Inorganic Compounds, which rests in turn on Landolt–Börnstein II/16 and III/19 and on Foëx’s Tables de Constantes, in CGS units of 10⁻⁶ cm³/mol at that table’s nominal room temperature of 285 to 300 K; multiply by 4π for SI. They are per mole of the element as that table names it, so the entries for hydrogen, nitrogen, oxygen and the halogens are per mole of the diatomic molecule and not per mole of atoms, while sulfur, whose standard state is S₈, is per mole of atoms. Where the table offers more than one form the element’s own note says which the cell carries, because the choice is sometimes the whole story: chlorine is the liquid, tin is grey α-tin and carbon is graphite. Which is the honest way to quote oxygen: the +3449 in its cell belongs to O₂ at about 293 K, and because oxygen is the one Curie paramagnet in the column it is nearer +3260 at body temperature. It is the reason air is paramagnetic while the tissue next to it is not. That single contrast, about 9.4 ppm of volume susceptibility in SI, is behind most of the field distortion near the sinuses and the ear canals, and behind the shimming you spend your time on at 7 T. Note also that the dossiers quote volume susceptibility in SI ppm rather than molar susceptibility in CGS; the two differ by 4π and by a factor of the molar volume, and mixing them is a classic way to be out by an order of magnitude. Concentrations, relaxation times and clinical practice in the element dossiers are drawn from the in-vivo MR literature and are typical values, not specifications: they vary with tissue, field, sequence and subject, and every one of them is a range in the real world.

Derived quantities are recomputed in the browser from the tabulated primitives, so the chart is internally consistent by construction rather than by transcription. A value shown as – is one that is not meaningful, or not carried by the source that column follows; it is never a placeholder for a number we have.

This is version 0.3, and it is a preprint in the ordinary sense: circulated to be checked, not to be cited. If you know one of these numbers better than the table does, or you can close one of the open questions in the reviewer’s notes, that is exactly the correction this document is asking for. A later version will say who supplied what.

Part of the UAB MRS resources; see also the glossary, the interactive spectrum, and the Ernst angle calculator.