Heaps of Precessing Protons

Spin Dynamics,
by Malcolm Levitt.

Last week’s post quoted from Spin Dynamics: Basics of Nuclear Magnetic Resonance, by Malcolm Levitt. This week I’ll talk more about this excellent textbook. Russ Hobbie and I cite Spin Dynamics in Intermediate Physics for Medicine and Biology when relating the proton relaxation time constants T₁ and T₂ to the correlation time τ_c. Our Fig. 18.12 shows this relationship in a log-log plot.

Fig. 18.12  Plot of T₁ and T₂ vs correlation time of the fluctuating magnetic field at the nucleus. The dashed lines are for a Larmor frequency of 29 MHz; the solid lines are for 10 MHz. Experimental points are shown for water (open dot) and ice (solid dots).

What do we mean by the “correlation time”? Levitt explains.

The parameter τ_c is called the correlation time of the fluctuations. Rapid fluctuations have a small value of τ_c, while slow fluctuations have a large value of τ_c. For rotating molecules in a liquid, τ_c is in the range of tens of picoseconds to several nanoseconds.

Qualitatively, the correlation time indicates how long it takes before the random field changes sign.

In practice, the correlation time depends on the physical parameters of the system, such as the temperature. Generally, correlation times are decreased by warming the sample, since an increase in temperature corresponds to more rapid molecular motion. Conversely, correlation times are increased by cooling the sample.

Levitt presents a plot similar to Fig. 18.12 in IPMB, except on linear-linear rather than log-log axes. 

Adapted from Fig. 16.16 of Spin Dynamics. The T₁ relaxation time as a function of the correlation time for random field fluctuations.

His curve is calculated for a static magnetic field of 11.74 T, which corresponds to a Larmor frequency, f_Larmor, of 500 MHz (a considerably stronger magnetic field than in our Fig. 18.12). The minimum of the curve is when τ_c equals the reciprocal of 2πf_Larmor, or about 0.32 ns. Levitt writes

It is a fortuitous circumstance that the most common experimental situation in solution NMR, namely medium-size molecules in non-viscous solutions near room temperature, falls close to the T₁ minimum. The small values of T₁ permit more rapid averaging of NMR signals, and hence a relatively high signal-to-noise ratio within a given experimental time. 

Think of the correlation time as a measure of the molecule’s rotation or tumbling time, characteristic of the molecular environment. One reason magnetic resonance imaging provides such excellent soft tissue contrast is because the relaxation times T₁ and T₂ are so sensitive to their surroundings. Relaxation happens most quickly when the tumbling time is similar to the period of precession, just as spin flipping is most effective when the radiofrequency field is in resonance with the precessing protons.

I like Spin Dynamics, in part because it has its own sound track. Russ and I have a lot of auxiliary stuff associated with Intermediate Physics for Medicine and Biology, but we don’t have a sound track. I’ll have to work on that.

To close, I quote from Levitt’s lyrical introduction to Spin Dynamics. Enjoy!

“Commonplace as such experiments have become in our laboratories, I have not yet lost that sense of wonder, and of delight, that this delicate motion should reside in all ordinary things around us, revealing itself only to him who looks for it.”

E. M. Purcell, Nobel Lecture, 1952

In December 1945, Purcell, Torrey and Pound detected weak radiofrequency signals generated by the nuclei of atoms in ordinary matter (in fact, about 1 kg of paraffin wax). Almost simultaneously, Bloch, Hansen and Packard independently performed a different experiment in which they observed radio signals from the atomic nuclei in water. There two experiments were the birth of the field we now know as Nuclear Magnetic Resonance (NMR).

Before then, physicists knew a lot about atomic nuclei, but only through experiments on exotic states of matter, such as those found in particle beams, or through energetic collisions in accelerators. How amazing to detect atomic nuclei using nothing more sophisticated than a few army surplus electronic components, a rather strong magnet, and a block of wax!

In his Nobel prize address, Purcell was moved to the poetic description of his feeling of wonder, cited above. He went on to describe how

“in the winter of our first experiments… looking on snow with new eyes. There the snow lay around my doorstep—great heaps of protons quietly precessing in the Earth’s magnetic field. To see the world for a moment as something rich and strange is the private reward for many a discovery…”

In this book, I want to provide the basic theoretical and conceptual equipment for understanding these amazing experiments. At the same time, I want to reinforce Purcell’s beautiful vision—the heaps of snow, concealing innumerable nuclear magnets, in constant precessional motion. The years since 1945 have shown us that Purcell was right. Matter really is like that. My aim in this book is to communicate the rigorous theory of NMR, which is necessary for really understanding NMR expeirments, but without losing sight of Purcell’s heaps of precessing protons.