The Magic Angle

I recently found another error in the 4th edition of Intermediate Physics for Medicine and Biology. In Chapter 18 about magnetic resonance imaging, Homework Problem 18 reads

Problem 18  Suppose the two dipoles of the water molecule shown below point in the z direction while the line between them makes an angle θ with the x axis. Determine the angle θ for which the magnetic field of one dipole is perpendicular to the dipole moment of the other. For this angle the interaction energy is zero. This θ is called the “magic angle” and is used when studying anisotropic tissue such as cartilage [Xia (1998)].

Technically there is nothing wrong with this problem. However, if I were doing it over I would have the angle θ measured from the z axis, not the x axis. One reason is that this is the way θ is defined most often in the literature. Another is that in the solution manual we solve the problem as if θ were relative to the z axis, so the book and the solution manual are not consistent on the definition of θ. I should add, this problem was not present in the 3rd edition of Intermediate Physics for Medicine and Biology. It is a new problem I wrote for the 4th edition, so I can’t blame Russ Hobbie for this one (rats).

The citation in the homework problem is to the paper

Xia, Y. (1998) “Relaxation Anisotropy in Cartilage by NMR Microscopy (μMRI) at 14-μm Resolution,” Magnetic Resonance in Medicine, Volume 39, Pages 941–949.

The author, Yang Xia, is a good friend of mine, and a colleague here in the Department of Physics at Oakland University. He is well-known around OU because over the last decade he had the most grant money from the National Institutes of Health of anyone on campus. He uses a variety of techniques, including micro-magnetic resonance imaging (μMRI), to study cartilage and osteoarthritis. The abstract of his highly-cited paper reads

To study the structural anisotropy and the magic-angle effect in articular cartilage, T₁ and T₂ images were constructed at a series of orientations of cartilage specimens in the magnetic field by using NMR microscopy (μMRI). An isotropic T₁, and a strong anisotropic T₂ were observed across the cartilage tissue thickness. Three distinct regions in the microscopic MR images corresponded approximately to the superficial, transitional, and radial histological zones in the cartilage. The percentage decrease of T₂ follows the pattern of the curve of (3cos²θ - 1)² at the radial zone, where the collagen fibrils are perpendicular to the articular surface. In contrast, little orientational dependence of T₂ was observed at the transitional zone, where the collagen fibrils are more randomly oriented. The result suggests that the interactions between water molecules and proteoglycans have a directional nature, which is somehow influenced by collagen fibril orientation. Hence, T₂ anisotropy could serve as a sensitive and noninvasive marker for molecular-level orientations in articular cartilage.

Perhaps a better reference for our homework problem is another paper of Xia’s.

Xia, Y. (2000) “Magic Angle Effect in MRI of Articular Cartilage: A Review,” Investigative Radiology, Volume 35, Pages 602–621.

There in Fig. 3 of Xia’s review is a picture almost identical to the figure that immediately follows Homework Problem 18 in our book, except the angle θ is measured from the direction of the static magnetic field rather than perpendicular to it. Xia writes

T2 corresponds to the decay in phase coherence (dephasing) between the individual nuclear spins in a sample (protons in our case). Because each proton has a magnetic moment, it generates a small local dipolar magnetic field that impinges on its neighbor’s space (Fig. 3).⁴³ This local field fluctuates constantly because the molecule is tumbling randomly. The T2 process can occur under the influence of this fluctuating magnetic field. At the end of signal excitation during an MRI experiment, the net magnetization (which produces the MRI signal) is coherent and points along a certain direction in space in the rotating frame of reference. This coherent magnetization vector soon becomes dephased because the local magnetic fields associated with the magnetic properties of neighboring nuclei cause the precessing nuclei to acquire a range of slightly different precessional frequencies. The time scale of this signal dephasing is reported as T2 and is characteristic of the molecular environment in the sample.^43,44

For simple liquids or samples containing simple liquidlike molecules, the molecules tumble rapidly. The dipolar spin Hamiltonian (H_D) that describes the dipolar interaction is averaged to zero, and its contribution to the spin relaxation vanishes. Relaxation characteristics exhibit a simple exponential decay that is well described by the Bloch equations.⁴⁵ For samples containing molecules that are less mobile, H_D is no longer averaged to zero and makes a significant contribution to the relaxation, resulting in a shorter T2. When H_D is not zero, it is dominated by a geometric factor, (3cos²θ - 1), where θ is the angle between the position vector joining the two spins and the external magnetic field (see Fig. 3). A useful feature of this geometric factor is that it approaches zero as θ approaches 54.74° (Fig. 4). Therefore, even when H_D is not zero, the contribution of H_D to spin relaxation can be minimized if θ is set to 54.74°. This angle is called the magic angle in NMR.⁴⁶

So, in the errata you will now find this:

Page 539: In Chapter 18, Homework Problem 18, “while the line between them makes an angle θ with the x axis” should be “while the line between them makes an angle θ with the z axis”. Also, in the accompanying figure following the homework problem, the angle θ should be measured from the z (vertical) axis, not the x (horizontal) axis. Corrected 1-18-13.

Is this the last error that we’ll find in our book? I doubt it; there are sure to be more we haven’t found yet. If you find any, please let us know.