| Symbol | Meaning | Symbol | Meaning |
|---|---|---|---|
| Roman alphabet | |||
| a or A | Hyperfine (electron-nucleus) coupling constant | Aql,m | The mth component of an irreducible tensor of order l representing the nuclear spin operator for an interaction of type q |
| B | Magnetic Field (strictly the magnetic flux density or magnetic induction) | B0 | Static magnetic field of an NMR spectrometer |
| B1,B2 | Radiofrequency magnetic fields associated with frequencies ν1, ν2 | BL | Local magnetic field (components BxL. ByL, BzL) of random field or dipolar origin |
| C | Spin-rotation interaction tensor | Cx | Spin-rotation coupling constant of nuclide X |
| D | Dipolar interaction tensor | D | Dipolar coupling constant between two nuclei (say 1 and 2), (μ0/4π)γ1γ2(h/2π)r12-3in frequency units (footnote 1). |
| DC | Nuclear receptivity relative to that of the carbon-13 nucleus | DP | Nuclear receptivity relative to that of the proton (hydrogen-1 nucleus) |
| E | Electric field strength | F | Spectral width |
| F1, F2 (or f1,f2 | The two frequency dimensions of a two-dimensional spectrum (use F3 etc., for higher orders) |  |  |
| FG | Magnetic quantum number associated with | g | Nuclear or electronic g factor (Landé splitting factor) |
| G | Magnetic field gradient amplitude | Hij | Element of matrix representation of Hamiltonian operator (in energy units) ; superscripts indicate the nature of the operator |
| Îj | Nuclear spin operator for nucleus j (components Îjx, Îjy, Îjz) | Îj+, Îj- | 'Raising' and 'lowering' spin operators for nucleus j |
| Ij | Magnetic quantum number associated with Îj | J | Indirect coupling tensor |
| nJ | Nuclear spin-spin coupling constant through n bonds (usually given in frequency units). Parentheses may be used (for example) to indicate the species of nuclei coupled, e.g. J(13C, 1H) or, additionally, the coupling path, e.g. J(POCF). Where no ambiguity arises, the elements involved can be, alternatively, given as subscripts, e.g. JCH. The nucleus of higher mass should be given first | J(ω) | Spectral density of fluctuations at angular frequency ω. Subscripts and superscripts to J may be used to indicate the relevant quantum number change (0, 1 or 2) or the order and component of the relevant tensor quantities. |
| nK | Reduced nuclear spin-spin coupling constant (see the notes concerning nJ), Kjk= 4π2Jjk/hγjγk | L | Angular momentum |
| mj | Eigenvalue of Îjz (magnetic component quantum number) (footnote 2) | mtot | Total magnetic component quantum number for a spin system (eigenvalue of ΣjÎjz)(footnote 2) |
| mtot(X) | Total magnetic component quantum number for X-type nuclei (footnote 2) | M0 | Equilibrium macroscopic magnetization per volume of a spin system in the presence of B0 |
| MX, MY, MZ | Components of macroscopic magnetization per volume. | Mn | nth moment of spectrum (M2 = second moment, etc.) |
| nα, nβ | Populations of the α and β spin states | N | Total number of nuclei of a given type per volume in the sample |
| q | Electric field gradient tensor in units of the elementary charge (principal components qxx, qyy, qzz) (see also V) | Q | eQ is the nuclear quadrupole moment, where e is the elementary charge |
| RX1 | Spin-lattice (longitudinal) relaxation rate constant for nucleus X | RX2 | Spin-spin (transverse) relaxation rate constant for nucleus X |
| RX1ρ | Spin-lattice relaxation rate constant in the rotating frame for nucleus X | S | Signal intensity |
 | Electron (or, occasionally, nuclear) spin operator; cf. Î | t1, t2 | Time dimensions for two-dimensional NMR |
| TC | Coalescence temperature for signals in an NMR spectrum | TX1 | Spin-lattice (longitudinal) relaxation time of the X nucleus (further subscripts refer to the relaxation mechanism) |
| TX2 | Spin-spin (transverse) relaxation time of the X nucleus (further subscripts refer to the relaxation mechanism) | T*2 | Net dephasing time for MX or My (including contribution from magnetic field inhomogeneity) |
| TX1ρ | Spin-lattice relaxation time of the X nucleus in the frame of reference rotating with B1 | Td | Pulse (recycle) delay |
| Tac | Acquisition time | T(l,m)q | The mth component of an irreducible tensor of order l representing the strength of an interaction of type q |
| V | Electric field gradient tensor. V = eq, where e is the elementary charge | Vα,β | Elements of Cartesian electric field gradient tensor |
| W0, W1,W2 | Relaxation rate constants (transition probabilities per time) between energy levels differing by 0, 1, and 2 (respectively) in mtot especially, but not uniquely, for systems of two spin 1/2 nuclei | Wr | Transition probability between spin states r and s |
| Greek alphabet | |||
| α | Nuclear spin wavefunction (eigenfunction of Îjz) for the mI=+1/2 state of a spin-1/2 nucleus | αE | The Ernst angle (for optimum sensitivity) |
| β | Nuclear spin wavefunction (eigenfunction of Îjz) for the mI=-1/2 state of a spin-1/2 nucleus | γX | Magnetogyric ratio of nucleus X |
| δX | Chemical shift (for the resonance) of nucleus of element X (positive when the sample resonates to high frequency of the reference). Usually in ppm (footnote 3). Further information regarding solvent, references or nucleus of interest may be given by superscripts or subscripts or in parentheses. | Δn | Population difference between nuclear states (Δn0 Boltzmann equilibrium) |
| Δδ | Change or difference in δ | Δν1/2 | Full width in frequency units of a resonance line at half-height |
| Δσ | Anisotropy in σ [Δσ = σzz - 1/2(σxx + σyy] (footnote 4). (see also ζ | Δχ | (i) Susceptibility anisotropy (Δχ = χ|| - χ⊥ (II) difference in electronegativities |
| ε0 | Permittivity of a vacuum | ζ | Anisotropy in shielding (footnote 4), expressed as σzz - σiso. (see also Δσ) |
| η | (i) Nuclear Overhauser enhancement (so that the nuclear Overhauser effect is 1 + η); (ii) tensor asymmetry factor (e.g. in σ); (iii) viscosity | κ | Skew of a tensor. (See also footnote 7) |
| &theta: | Angle, especially for that between a given vector and B0 | μ | (i) Magnetic dipole moment (component μz along B0); (ii) electric dipole moment |
| σ0 | Permeability of a vacuum | σB | Bohr magneton |
| σN | Nuclear magneton | νj | Larmor precession frequency of nucleus j (usually given in MHz) |
| ν0 | (i) Spectrometer operating frequency; (ii) Larmor precession frequency (general, or of bare nucleus) | ν1 | Frequency of 'observing' RF magnetic Field B1 (to be distinguished from its strength, γB1, for which the symbol Ω1 is recommended) |
| ν2 | Frequency of 'irradiating' RF magnetic Field B2 (to be distinguished from its strength, γB2, for which the symbol Ω2 is recommended) | Ξx | Resonance frequency for the nucleus of element X in a magnetic Field such that the protons in tetramethylsilane (TMS) resonate at exactly 100 MHz |
| ρ | Density matrix |  | Density operator |
| ρij | Element of matrix representation of  | σ | Shielding tensor (footnotes 5 and 6) |
| σj | Isotropic) shielding constant of nucleus j. Usually given in ppm. Subscripts may alternatively indicate contributions to σ | σ||, σ⊥ | Components of shielding tensor σ parallel and perpendicular to the symmetry axis (axially-symmetric case) (footnote 5) |
 | Reduced density operator | τ | (i) Time between RF pulses (general symbol) (ii) lifetime in dynamic NMR usage |
| τc | Correlation time for molecular-level motion, especially for isotropic molecular tumbling | τd | Dwell time |
| τnull | Recovery time sufficing to give zero signal after a 180° pulse | τp | Pulse duration |
| τsc | Correlation time for relaxation by the scalar mechanism | τsr | Correlation time for spin-rotation relaxation |
| τ||, τperp | Correlation times for molecular tumbling parallel and perpendicular to the symmetry axis (axially symmetric case) | Χ | (i) Magnetic susceptibility (footnote 7) ; (ii) nuclear quadrupole coupling constant (Χ = e2qzzQ/h) |
| ωj, ω0, ω1, ω2 | As for νj, ν0, ν1, ν2 but for angular frequencies | Ω | Span of a Tensor |
| Ω1, Ω2 | R.f. magnetic Fields, expressed in angular frequency units for a nucleus of magnetogyric ratio γ (Ω1 = -γB1, Ω2 = -γB2 | ||
Parameters and Symbols for Use in Nuclear Magnetic Resonance (IUPAC Recommendations 1997)
1Note that confusion might arise when the so-called alphabet expansion is used for D, since this includes a term D which is not the dipolar coupling constant.
2M rather than m is frequently recommended, but most NMR practitioners use m so as to avoid confusion with magnetization.
3Whereas earlier IUPAC recommendations give a definition of δ which requires that the “unit" ppm is not stated when values are quoted, this is largely ignored and a change of recommendation is under consideration.
4ζ = 2Δδ/3
5The symbols σ (and related terms for components), σj, σ||, σ⊥ should refer to shielding on absolute scale (for theoretical work). For shielding relative to a reference, symbols such as σ|| - &sigmaref should be used.
6 For tensors, doubled subscript capital letters X, Y and Z should generally be used for principal components, e.g. σXX, σYY and σ ZZ for shielding. Alternatively, numerical subscripts may be used (e.g. σ11, σ22, σ33).
7 The symbol κ may also be used for magnetic susceptibility, some authors reserving χ for unrationalized units.
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