Spin echoes
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص539-541
2025-12-14
70
Spin echoes
The measurement of T2 (as distinct from T2*) depends on being able to eliminate the effects of inhomogeneous broadening. The cunning required is at the root of some of the most important advances that have been made in NMR since its introduction. Aspin echo is the magnetic analogue of an audible echo: transverse magnetization is created by a radiofrequency pulse, decays away, is reflected by a second pulse, and grows back to form an echo. The sequence of events is shown in Fig. 15.39. We can consider the overall magnetization as being made up of a number of different magnetizations, each of which arises from a spin packet of nuclei with very similar precession frequencies. The spread in these frequencies arises because the applied field B0 is inhomogeneous, so different parts of the sample experience different fields. The pre cession frequencies also differ if there is more than one chemical shift present. As will be seen, the importance of a spin echo is that it can suppress the effects of both field inhomogeneities and chemical shifts. First, a 90° pulse is applied to the sample. We follow events by using the rotating frame, in which B1 is stationary along the x-axis and causes the magnetization to be into the xy-plane. The spin packets now begin to fan out because they have different Larmor frequencies, with some above the radiofrequency and some below. The detected signal depends on the resultant of the spin-packet magnetization vectors, and decays with a time-constant T2 * because of the combined effects of field inhomogeneity and spin–spin relaxation. After an evolution period τ, a 180° pulse is applied to the sample; this time, about the y-axis of the rotating frame (the axis of the pulse is changed from x to y by a 90° phase shift of the radiofrequency radiation). The pulse rotates the magnetization vectors of the faster spin packets into the positions previously occupied by the slower spin packets, and vice versa. Thus, as the vectors continue to precess, the fast vectors are now behind the slow; the fan begins to close up again, and the resultant signal begins to grow back into an echo. At time 2τ, all the vectors will once more be aligned along the y-axis, and the fanning out caused by the field inhomogeneity is said to have been refocused: the spin echo has reached its maximum. Because the effects of field inhomogeneities have been suppressed by the refocusing, the echo signal will have been attenuated by the factor e−2τ/T2 caused by spin–spin relaxation alone. After the time 2τ, the magnetization will continue to precess, fanning out once again, giving a resultant that decays with time constant T2 *. The important feature of the technique is that the size of the echo is independent of any local fields that remain constant during the two τ intervals. If a spin packet is ‘fast’ because it happens to be composed of spins in a region of the sample that experiences higher than average fields, then it remains fast throughout both intervals, and what it gains on the first interval it loses on the second interval. Hence, the size of the echo is independent of inhomogeneities in the magnetic field, for these remain constant. The true transverse relaxation arises from fields that vary on a molecular distance scale, and there is no guarantee that an individual ‘fast’ spin will remain ‘fast’ in the refocusing phase: the spins within the packets therefore spread with a time constant T2. Hence, the effects of the true relaxation are not refocused, and the size of the echo decays with the time constant T2 (Fig. 15.40).
One of the most striking applications of nuclear magnetic resonance is in medicine. Magnetic resonance imaging (MRI) is a portrayal of the concentrations of protons in a solid object. The technique relies on the application of specific pulse sequences to an object in an inhomogeneous magnetic field. If an object containing hydrogen nuclei (a tube of water or a human body) is placed in an NMR spectrometer and exposed to a homogeneous magnetic field, then a single resonance signal will be detected. Now consider a flask of water in a magnetic field that varies linearly in the z-direction according to B0 + Gzz, where Gz is the field gradient along the z-direction (Fig. 15.41). Then the water protons will be resonant at the frequencies
(Similar equations may be written for gradients along the x- and y-directions.) Application of a 90° radiofrequency pulse with ν = νL(z) will result in a signal with an intensity that is proportional to the numbers of protons at the position z. This is an example of slice selection, the application of a selective 90° pulse that excites nuclei in a specific region, or slice, of the sample. It follows that the intensity of the NMR signal will be a projection of the numbers of protons on a line parallel to the field gradient. The image of a three-dimensional object such as a flask of water can be obtained if the slice selection technique is applied at different orientations (see Fig. 15.41). In projec tion reconstruction, the projections can be analysed on a computer to reconstruct the three-dimensional distribution of protons in the object. In practice, the NMR signal is not obtained by direct analysis of the FID curve after application of a single 90° pulse. Instead, spin echoes are often detected with several variations of the 90°–τ–180° pulse sequence (Section 15.9c). In phase encoding, field gradients are applied during the evolution period and the detection period of a spin-echo pulse sequence. The first step consists of a 90° pulse that results in slice selection along the z-direction. The second step consists of application of a phase gradient, a field gradient along the y-direction, during the evolution period. At each position along the gradient, a spin packet will precess at a different Larmor frequency due to chemical shift effects and the field inhomogeneity, so each packet will dephase to a different extent by the end of the evolution period. We can control the extent of dephasing by changing the duration of the evolution period, so Fourier transformation on τ gives information about the location of a proton along the y-direction.1 For each value of τ, the next steps are application of the 180° pulse and then of a read gradient, a field gradient along the x-direction, during detection of the echo. Protons at different positions along x experience different fields and will resonate at different frequencies. There-fore Fourier transformation of the FID gives different signals for protons at different positions along x. A common problem with the techniques described above is image contrast, which must be optimized in order to show spatial variations in water content in the sample. One strategy for solving this problem takes advantage of the fact that the relaxation times of water protons are shorter for water in biological tissues than for the pure liquid. Furthermore, relaxation times from water protons are also different in healthy and diseased tissues. A T1-weighted image is obtained by repeating the spin echo sequence before spin–lattice relaxation can return the spins in the sample to equilibrium. Under these conditions, differences in signal intensities are directly related to differences in T1. A T2-weighted image is obtained by using an evolution period τ that is relatively long. Each point on the image is an echo signal that behaves in the manner shown in Fig. 15.40, so signal intensities are strongly dependent on variations in T2. However, allowing so much of the decay to occur leads to weak signals even for those protons with long spin–spin relaxation times. Another strategy involves the use of contrast agents, paramagnetic compounds that shorten the relaxation times of nearby protons. The technique is particularly useful in enhancing image contrast and in diagnosing disease if the contrast agent is distributed differently in healthy and diseased tissues. The MRI technique is used widely to detect physiological abnormalities and to observe metabolic processes. With functional MRI, blood flow in different regions of the brain can be studied and related to the mental activities of the subject. The technique is based on differences in the magnetic properties of deoxygenated and oxygenated haemoglobin, the iron-containing protein that transports O2 in red blood cells. The more paramagnetic deoxygenated haemoglobin affects the proton resonances of tissue differently from the oxygenated protein. Because there is greater blood flow in active regions of the brain than in inactive regions, changes in the intensities of pro ton resonances due to changes in levels of oxygenated haemoglobin can be related to brain activity. The special advantage of MRI is that it can image soft tissues (Fig. 15.42), whereas X-rays are largely used for imaging hard, bony structures and abnormally dense regions, such as tumours. In fact, the invisibility of hard structures in MRI is an advantage, as it allows the imaging of structures encased by bone, such as the brain and the spinal cord. X-rays are known to be dangerous on account of the ionization they cause; the high magnetic fields used in MRI may also be dangerous but, apart from anecdotes about the extraction of loose fillings from teeth, there is no convincing evidence of their harmfulness, and the technique is considered safe.
Fig. 15.39 The sequence of pulses leading to the observation of a spin echo.
Fig. 15.40 The exponential decay of spin echoes can be used to determine the transverse relaxation time.
Fig. 15.41 In a magnetic field that varies linearly over a sample, all the protons within a given slice (that is, at a given field value) come into resonance and give a signal of the corresponding intensity. The resulting intensity pattern is a map of the numbers in all the slices, and portrays the shape of the sample. Changing the orientation of the field shows the shape along the corresponding direction, and computer manipulation can be used to build up the three-dimensional shape of the sample.
Fig. 15.42 The great advantage of MRI is that it can display soft tissue, such as in this cross-section through a patient’s head. (Courtesy of the University of Manitoba.)
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