The Geometry of Magnetic Resonance
David B Pengra (dbpengra@uw.edu)
Department of Physics, Box 351560, University of Washington, Seattle, WA 98195
1. Introduction
In a 1957 paper, "Geometrical Representation of the Schrödinger Equation for Solving Maser Problems," Feynman, Vernon and Hellwarth presented a simple, rigorous mapping of a two-level quantum system onto three-dimensional real space. The point of this approach, they argued, is that "the simplicity of the pictorial representation enables one to gain physical insight and to obtain results quickly which display the main features of interest." Readers familiar with the justification of the Bloch equation of nuclear magnetic resonance (NMR) via Ehrenfest's theorem applied to angular momentum operators [Slichter, 1996] or the textbook example of time-dependent perturbation theory applied to a two-level atomic system to derive Rabi oscillations [Foote, 200x] will recognize this mapping. From group theory one learns that the trick is nothing more than the SU(2)-O$^+_3$ homomorphism [Arfken, 1985, pp. X], wherein a two-dimensional complex "spinor" representation of the rotation group is connected to its three-dimensional vector representation via the "Cayley-Klein" parameters [Jaynes, 1955]. But Feynman, Vernon and Hellwarth note that insights gained through the 3D version apply to a wider class of problems than NMR, and serve to answer the question, What is the geometry of resonance? The answer: it is often the geometry of NMR.
This geometry is inherent in the three-dimensional real vector description of magnetic resonance given by the Bloch equation,
\begin{equation} \label{eq:Bloch} \frac{d\bvec{M}}{dt} = -\gamma\bvec{H}\times\bvec{M} - \frac{\bvec{M}-\chi\bvec{H}}{T} \;, \end{equation}
which is the standard starting point for learning NMR theory. In the above $\bvec{M}$ is the magnetization density of the sample, $\bvec{H}$ is an applied magnetic field typically defined in terms of a large static field $\bvec{H}_0$ polarized along the $z$ axis plus a time dependent field $\bvec{H}_1(t)$ usually polarized perpendicular to $\bvec{H}_0$, $\gamma$ is the gyromagnetic ratio connecting the magnetic moment of the system to its angular momentum, $\chi$ is the magnetic susceptibility of $\bvec{M}$, and $T$ is a time constant associated with the relaxation of $\bvec{M}$ to steady state. When expressed in component form, $T\equiv T_1$ for the longitudinal component $M_z$ (defined by the direction of a large constant field $\bvec{H}_0$, conventionally taken to be along the $z$ axis), and $T\equiv T_2$ for the transverse components $M_{x,y}$. The theory of relaxation [Slichter, 1996, Ch. 2] implies that $T_2\le T_1$, as one cannot have a nonzero value of $M_{x,y}$ when $M_z=\chi H_0$ at equilibrium. In liquids $T_1\approx T_2$ holds in cases of low-viscosity liquids used in common teaching apparatus. However, due to even small variations in $\bvec{H}_0$ across the sample, the transverse magnetization may disappear quickly, characterized by $T_2^{\ast} \ll T_2$ which includes these instrumental effects.
In spite of the real-space representation of resonance given by Eq. ($\ref{eq:Bloch}$), grasping its geometry remains challenging,especially for beginning students, when the evolution of $\bvec{M}$ is beyond a simple precession about the fixed $\bvec{H}_0$ field. Once one adds $\bvec{H}_1(t)$, needed to excite resonance and change the energy of the system, it becomes difficult to visualize the dynamics without transforming the problem into a rotating frame of reference.
This, then, is the purpose of the work presented here: A simple, Python-based animated simulation of the Bloch equation and its extension (according to the model used by Erwin Hahn [Hahn, 1950]) to an ensemble of non-interactong subsystems are shown to display nearly all observable phenomena in typical magnetic resonance apparatus found in teaching laboratories. The goal of the simulations is not just to reproduce experimental results, but to provide a clear visual link between the underlying geometrical evolution of the system and what can be seen on an oscilloscope screen.
A wide range of concepts and phenomena are made visible. From the simulation of a single copy of Eq. ($\ref{eq:Bloch}$) one can illustrate the shift between lab and rotating frames of reference; the response to linearly versus rotationally oscillating applied fields; the relaxation of $\bvec{M}$ according to constants $T_1$ and $T_2^*$; steady-state and transient evolution over a range of applied fields; a slow sweep through resonance, which displays the real and imaginary parts of the magnetic susceptibility $\chi(\omega)$, with the phase shift being visible by the direction change in the steady-state magnetization vector [Slichter, 1996, pp. X]; a faster sweep that produces the phenomenon of asymmetric "wiggles" in which the signal reaches a maximum when approching the resonance, then oscillates after moving through it [Boembergen, et al., 1948; Jacobsohn and Wangsness, 1948]; the application of short bursts of the alternating field—pulses—to create a "free induction" signal, and their use in measuring the longitudinal relaxation time constant $T_1$.
By expanding the simulation to include an ensemble of Bloch equations, with each member of the ensemble subject to a different static field magnitude $H_{0,i}$ distributed about a central $H_0$, the user can see how each "isochrome" (in Hahn's terminology) contributes to the ensemble average, and the full range of phenomena associated with spin echoes can be visualized. Nearly all the observations reported by Hahn in his groundbreaking paper of 1950 are illustrated—for example, that the FID gets narrower but the peak amplitude remains unchanged when the distribution of $H_{0,i}$ is broadened; the origin of $T_2^*$ and its clear distinction from $T_2$; how the shape of the FID signal is proportional to the Fourier transform of the distribution function of $H_{0,i}$ by observing the phases of the isochromes as they evolve in time; the production of a spin echo from two successive $\pi/2$ pulses, and the configuration of the isochrome ensemble into a figure-eight on the surface of the Bloch sphere at the moment of the echo's peak; and, the formation of two more echoes following a third $\pi/2$ pulse. Multipulse sequences, such as the short-recovery repetition of $\pi/2$ pulses to measure $T_1$ commonly used in magnetic resonance imaging, or the famous Carr-Purcell $\pi/2, N\pi$ sequence to measure $T_2$ from the evelope of successive spin echoes can be visualized easily.
A reader familiar with NMR may not find anything new in the items just described, but the simulations allow exploration of many hard-to-calculate and hard-to-visualize aspects and reveal some surprises. Some examples: (1) How the asymmetry seen in the fast-passage case of the "wiggles" arises: the system responds weakly until the frequency of the oscillating field gets close to the resonance, at which point the magnetization is pitched into a nonequilibrium state that in the rotating reference frame precesses around the slowly changing effective field at a rate equal to the difference between the resonant frequency and the frequency of the oscillator. (2) How the "rotating-wave approximation" commonly invoked to simplify the theory of the applied field is justified in practice—the motion of the magnetization under the influence of a linearly oscillating applied field is seen in the rotating frame to adhere to the in-phase component ever more closely as the ratio of the applied field to the static field amplitudes $H_1/H_0$ becomes smaller. (3) How Hahn's "eight-ball" configuration of isochromes at the moment of maximum spin-echo amplitude is a universal topology following any type of second pulse sufficient to create an echo, and that the optimal $\pi$ second pulse produces a configuration with a figure-eight whose loops are shrunk to zero area. (4) How the configuration of isochromes continues to evolve following the echo from a second $\pi/2$ pulse to display patterns with increasing numbers of loops—"internal echoes" that produce no signal because these subsequent configurations are all symmetric about the origin of the Bloch sphere. (5) How in multipulse sequences, the configuration of isochromes that produce echoes display an astonishing (and mesmerising) range of patterns: rosettes, multiple figure-eights, and a variety of Bloch-sphere enrobing loops, depending on the sequence parameters.
Next: Basics of the Simulations