A Primer on Scalar Relaxation in NMR
Stanislav Sýkora, Extra Byte, Via R.Sanzio 22C, Castano Primo, Italy 20022
in Stan's Library, Ed.S.Sykora, Vol.III. First release September 30, 2009
Permalink via DOI:  10.3247/SL3Nmr09.004
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This article responds to a query from a reader of my NMR blog who was worried about errors which might be committed when, following the common practice, one neglects scalar relaxation in NOE-based measurements of internuclear distances in proteins. It starts with a very brief, qualitative introduction to the phenomenon of relaxation in MR and follows by an order-of-magnitude assessment of the importance of scalar relaxation in various molecular systems. It concludes that in organic molecules scalar relaxation is usually negligible with respect to dipolar relaxation, except in special situations typical of small molecules with strongly hindered internal rotations. In natural proteins it is unlikely to be of any importance, while no general rule can be set up for inorganic systems with heavy nuclei.

As a primer, this article provides hardly anything new to an expert in the field except, perhaps, some slightly novel terminology leading to a relaxation pathways classification scheme which I find convenient (the relaxation-enabling interactions combined with the relaxation-driving random motions).

Relaxation phenomena in NMR

In general, relaxation processes in NMR are those responsible [1] for
   a) Non-radiative energy exchanges between the spin-system and any other degrees of freedom present in the sample (the latter constitute the so-called "lattice"), thus establishing thermodynamic equilibrium between the spin system and the rest. Most importantly, this establishes also the thermal-equilibrium nuclear polarization without which there would be no signal.
   b) Broadening of the spin-system's energy levels related to a finite permanence time of the system in any particular quantum state. This is responsible for the homogeneous broadening of spectral peaks, as opposed to heterogeneous broadening due to a superposition of many sharp peaks.

There are many different relaxation mechanisms [1-13] which concur in an approximately additive way to the observed relaxation rate(s) of a spin system or, more correctly, of each quantum transition [9,15,16,18]. The additivity of relaxation rates with respect to the various contributing mechanisms is generally quite good, with typical deviations of the order of 1% (note that a relaxation rate is the inverse of a corresponding relaxation time which implies that relaxation times are not additive).

All such mechanisms (relaxation pathways) are characterized by i) an interaction involving a spin and ii) its random temporal variation due to a stochastic motion, generally activated by molecular-level thermal effects. This picture can be used to classify the relaxation pathways by locating each one in a cell of a table whose columns are associated with various types of interactions and the rows with various types of motions. There are literally hundreds of relaxation pathways and each has its particular characteristics, above all the spin interaction which enables it, the random motion which drives it, a relaxation rate formula and its dependence on Larmor frequency, gamma ratio, motion parameters (correlation times) and their temperature dependence, etc.

On the theoretical side, the essential point is that each relaxation pathway is related to a stochastic component term in the Hamiltonian of the spin system, one which on the time-scale of NMR relaxation rates averages to zero and therefore does not lead to distinct peak splittings in NMR spectra.

In practical situations, there is often one relaxation pathway which dominates over the others, though which one it is may depend upon many factors (including temperature). The best known case, at least in proton NMR in liquid state, is the dipolar relaxation which is enabled by direct dipole-dipole interactions between nuclear spins and driven by the Brownian rotational tumbling of the whole molecule. It is to this case (and this case only) that one can apply the historic formulas of Bloembergen, Purcell and Pound (BPP) and those of Solomon [14-17].

Scalar interaction as an example of a relaxation-enabling pathway

By definition, scalar relaxation is enabled by the scalar interaction (also referred to as J-coupling) jAB between two nuclides A and B. We will use it now to illustrate some of the principles mentioned above.

A scalar interaction is due to a simultaneous Fermi contact of both nuclei A and B with each bond electron and gives rise to a spin-system Hamiltonian term of the form jAB(IA.IB), where IX is the spin vector of nuclide X and the dot denotes dot product between vectors. Actually, this is already a simplification since jAB is really a tensor, being somewhat dependent upon the orientation of the local molecular fragment with respect to the field (this stems primarily from an extra contribution of the coupling between nuclear spins and the orbital momenta of bond electrons). However, since the orientation dependence is usually mild (from quantum-theoretical standpoint it is a second order effect) it can be neglected, leaving just the orientation-independent scalar part (hence the scalar interaction name). Just keep in mind that, should we keep it, the relaxation due to the anisotropic part of jAB would be simply an additional relaxation pathway, akin to the well known and much more important relaxation pathway enabled by chemical-shift anisotropy (CSA).

However, even the isotropic (scalar) part of jAB is not necessarily constant because its value depends to some extent on the stereo-conformation of the whole molecule and is subject to random "modulations" due to internal molecular motions (vibrations, internal rotations, jumps between conformers, chemical exchange, ...). We must therefore consider jAB(t) as a time-dependent function and split it into two components:
   1) The time-averaged value JAB = <jAB(t)> which is the usual J listed in chemical reports and which manifests itself by splittings between spectral peaks, determining their multiplicity and positions.
   2) The randomly varying remainder JAB(t) = jAB(t) - JAB which averages to zero and, consequently, has no direct effect on the multiplicity and positions of spectral peaks. For those same reasons, however, this part contributes to (drives) the relaxation of nuclides A and B and, to some extent, also of any other nuclide coupled to them.

The question now is how large can be the two components of scalar interactions.

First of all, we know - both theoretically and empirically - that jAB is only appreciable when the nuclides A and B are structurally close, i.e., separated by a limited number of covalent bonds (simultaneous Fermi contact at two different nuclides requires the bonding electron wavefunction to extend all the way from one to the other). Stereo vicinity, as opposed to structural vicinity, is in this case irrelevant.

We therefore distinguish n-bond J's (n = 1, 2, ..., 9) and often write the n as a leading superscript (examples: 1JAB, 2JAB, 3JAB, ...)

As far as numeric values are concerned, a general discussion is nearly impossible since the J values range from barely measurable fractions of a Hz up to hundreds of kHz! The largest values are all related to one- and two- bond scalar couplings between heavy nuclides (statistically, the Fermi contact term grows sharply with the atomic number of the nuclide) and thus pertain to inorganic chemistry. Such cases also exhibit high quantum-based variability and must be studied case-by-case.

The situation is simpler to handle in diamagnetic organic molecules where relaxations are dominated by the proton-proton interactions, the anisotropy of the J's is quite small, and the scalar couplings are relatively easy to classify:

One-bond proton-proton J exists only in the hydrogen molecule and is therefore of little interest in this context.

Two-bond proton-proton scalar couplings ( 2JHH) are ubiquitous, primarily between the two geminal protons of methylene groups (-CH2-). In those cases where the two protons are chemically and magnetically equivalent, the geminal coupling may be unobservable in the spectrum, but it is still there and may contribute to relaxation. The 2j couplings are almost always negative and most fall in the interval between -20 Hz to -15 Hz (though values up to +40 Hz crop up occasionally [19]). What is more important in the relaxation context is that their values are little sensitive to internal conformations of the molecule which hardly ever induce variations greater than 2 Hz. This makes the 2j's rather ineffective as a scalar relaxation pathway - a feature which applies to most 1j and 2j scalar couplings (even those between heteronuclei).

Three-bond scalar couplings ( 3JHH) between vicinal protons (>CH-CH<) are also very common in organic molecules and they are those exhibiting the largest variations upon changes in molecular geometry; in particular, they depend strongly upon the dihedral angle φ defined by the three bonds. Semi-empirically, this dependence is well known (see the Mestrelab utility [20] based on the Karplus formula [21,22] and its more recent analogs) and covers the approximate range of 0 to 16 Hz. It is frequently used to study molecular stereochemistry in those cases where the dihedral bond rotamers are sufficiently persistent on NMR timescale (not very common) or where the dihedral angle is constrained to a fixed value by further-away bonds (cyclic compounds with bridges and the like). When the dihedral rotamers are inter-converting too fast to give separate spectra, one observes only an average, temperature dependent 2J coupling, while the random component 2J(t) affects just relaxation, with a correlation time given approximately by the lifetime of the individual rotamers. Given the Karplus relations, the mean amplitude of 2J(t) never exceeds 8 Hz (5 Hz is probably the most likely value).

Scalar couplings of appreciable amplitude over four- or more bonds are more rare. They fall mostly in the interval of -4 to +4 Hz, but there do exist surprising exceptions [19], particularly in rigid, extensively conjugated fragments and in sterically severely strained molecules (for example, there exists a 4JHH of 18 Hz). Because of the rigidity of such molecules, and because of the small long-range coupling values in flexible molecules, the amplitude of the variable parts nJ(t) for n ≥ 3 probably hardly ever exceeds 1 Hz (though, admittedly, this is just the Author's conjecture and there might exist a few exceptions).

Order-of-magnitude estimates of scalar relaxation rates

The above estimates of the amplitudes of the variable parts of proton scalar couplings in organic molecules enable us to compare them to the direct dipole-dipole interactions DAB. Under rapid isotropic tumbling these average to zero, but their time-variable amplitude is very large.

For geminal protons, in particular, the mean amplitude of DAB is about 30 kHz. Due to inverse third-power dependence of DAB on internuclear distance, the DAB values for vicinal proton pairs are smaller (of the order of 10 kHz) and they drop still more for protons pairs separated by three and more bonds.

Even so, if we compare these values with mean amplitudes of the scalar coupling variations, we find that the latter are typically about 3 orders of magnitude smaller. Only in extremely exceptional cases of exceptionally large long-range scalar couplings could the magnitudes of the two entities become remotely comparable. Considering that relaxation is a second-order effect and the relaxation rates correlate with the square of the variable component of the enabling interaction, it would appear that scalar relaxation should be virtually always inferior to dipolar relaxation by a factor of about 106 (give or take an order of magnitude).

This, however, is not exactly true because we must take into account also the correlation times τ of the random motions which drive the respective relaxation. For dipolar relaxations it is the characteristic time τr of the molecular rotational tumbling, while for a scalar interaction it is the mean lifetime τc of the involved conformers. The correlation time appears in relaxation rate formulas inside factors known as spectral density functions which, at least for low values of ωτ (ω is the Larmor frequency) are proportional to τ. Rather than discussing the details of the specific formulae, it is sufficient for the present purposes to understand that when τc is substantially larger than τr, it can compensate for the difference in the mean amplitudes of the interactions.

The τr values vary from tens of pico-seconds (~10-11) for small molecules to tens of micro-seconds (~10-5) for large proteins. On the other hand, τc values can vary from values just as low as τr (in which case the scalar relaxation pathway is certainly negligible) up to values comparable to the relaxation times (seconds), depending upon the internal rotational barriers and upon temperature.

This means that there indeed exist cases where the scalar relaxation handicap due to interaction amplitudes is all but offset by a very large value of the ratio τcr. In particular, this can happen in small molecules with one or more internal-rotation degrees of freedom characterized by high energy barriers. In such cases, should τc become even longer than relaxation times, the rotamers would start appearing in the NMR spectrum as independent entities. Before reaching that point, however, we are in a situation in which the rotamer spectra are still coalesced into a single, average-Hamiltonian spectrum and only the linewidths (T2) and the T1's indicate that we are not very far from the coalescence point. When this happens, scalar relaxation may dominate over dipolar relaxation.

For two different reasons, a situation of this kind is much less likely to occur in large molecules. First, the reorientation times τr are much longer so that it becomes difficult to reach a large value of τcr while still having coalesced rotamer spectra. Second, in molecules characterized by extensive steric constraints (such as non-denaturated proteins), local rotamers are all but ruled out (at most, there can be librations and vibrations about some average positions).


Analyzing the case of proton-proton scalar and dipolar relaxation pathways, we have seen how their relative importance depends upon a delicate balance of two factors:
   (1) the ratio of the amplitudes of the time-variable components of the interactions which enable the relaxations
   (2) the ratio of the correlation times of the respective random motions which drive the relaxations.

The feasible ranges for both of these factors being very large (several orders of magnitude), all kinds of situations can occur. For protons in organic molecules, factor (1) is very much biased in favor of dipolar relaxation but in small molecules with strongly expressed rotamers, and at temperatures close to the coalescence of the rotamer spectra, factor (2) can offset the bias and overturn it in favor of scalar relaxation. This is unlikely to happen in large molecules where dipolar relaxation reigns supreme.

In systems with heteronuclei, particularly inorganic ones, the situation can be very different. Given the smaller γ values and larger distances, direct dipole-dipole interactions are generally much weaker, while scalar couplings are often much larger. However, remember that it is not the value of the average J that matters, rather than the amplitude of its variable part J(t) - and that is a very unpredictable entity. The final outcome is therefore hard to predict and a case-by-case analysis is called for.

Final note: Though there was a mention of chemical exchange as a source of the random modulations of the scalar couplings, I have later always mentioned only the modulation by means of internal rotations. This was intentional, since I was trying to keep the argumentation as simple and fluid as possible. But chemical exchange is certainly a very frequent reason behind J modulation and should be kept in mind. Fortunately, as long as J is indeed subject to random variations, their exact cause does not really matter and referring to one or to the other does not alter the above discussion.


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