target="_blank" rel="nofollow" href="#ulink_7b21b27c-f501-5acf-9858-667d2ab13979">Figure 2.12 Comparison between a Lorentzian and a Gaussian with the same FWHM.
Table 2.3 Some features of Lorentzian and Gaussian (A, B, a are constants).
| Lorentzian | Gaussian | |
|---|---|---|
| Line-shape equation | f(x)=A(1+B2(x−x0)2 | g(x)=e−(x−x0)2a2 |
| Line width (FWHM) | 2/B | 2aIn(2)= 1.66511a |
| Normalized expression | f1(x)=(B/πA)f(x) | g1(x)=1xa2g(x) |
| Fourier transform | Exponential | Gaussian |
2.9 CW NMR
The earliest NMR experiments ran in a continuous-wave (CW) mode, where the spectrometer is tuned to observe the component of M, which is 90˚ out of phase to the rotating field B1, the so-called absorption mode signal. (Earlier in Section 2.7, we set both B1(t) and u in the direction of x′, and v in the direction of y′ in the rotating frame.) During an experiment, the magnetic field B0 is swept slowly through the resonance frequency. As each chemically identical spin group comes into resonance, it undergoes nuclear induction and a voltage is induced in the pick-up coil (cf. the three peaks of ethanol in Figure 1.4). This approach is called the CW method, where the signal of the specimen is recorded continuously on an oscilloscope. Provided that this field sweep is done sufficiently slow, the absorption mode signal at each frequency corresponds to the steady state value of v when M has come to rest in the rotating coordinate system. Hence it is also called the slow passage experiment. Since neither the resonance frequency nor the number of the equivalent groups in a specimen is known, doing an NMR experiment using the CW method could take a long time.
By examining the Bloch equation in the rotating frame [Eq. (2.23)], the following observations can be made:
1 When we are far from the resonance (i.e., |ω0 – ω| is large), we have u = v = 0 and Mz = M0. The non-zero values of u and v appear only in a small interval around ω0, that is, when there is a resonance.
2 Where T1T2(γB1)2 ≪ 1 (i.e., the rf power applied is sufficiently low so that the saturation does not occur), v can be simplified as (2.24)By comparing Eq. (2.24) with the line-shape functions in Table 2.3, we see that v is a Lorentzian centered at ω0 with a line width at half maximum of 1/(πT2). Hence, in principle, the FWHM of the resonant peak can be used to determine the T2 relaxation time.
3 When T1T2(γB1)2 is not sufficiently smaller than 1, we can have these situations:when T1T2(γB1)2 < 1, the spins are below saturation, and the signal ∝ γB1T2when T1T2(γB1)2 = 1, spins are saturated, where both signal and SNR reach the maximumwhen T1T2(γB1)2 > 1, the Signal starts to drop.
4 Only the transverse component of the precessing M induces an observable signal in the receiver coil. The transverse component can be written as (2.25a) (2.25b)
where the complex i indicates a 90˚ phase shift (i = −1); v (which is in the direction of y′) is called the absorption spectrum, where the signal is proportional to the power absorbed from the EM field; and u (which is in the direction of x′) is called the dispersion spectrum (which is a common term in optics). Note that the signal is proportional to B1 not B12 [10].
2.10 RADIO-FREQUENCY PULSES IN NMR
A much more efficient method in modern NMR experiments is to apply a short but powerful B1 pulse, which has a duration of several to tens of µs. This rf pulse will be able to cover all possible resonance frequencies simultaneously in the specimen. The reason that a short pulse can excite all possible resonant groups in the specimen is because the frequency range of a 10 µs pulse is about 105 Hz (since f = 1/T), which is sufficient to cover the range of all resonant peaks due to the differences in their chemical shifts (e.g., the three peaks in the ethanol spectrum in Figure 1.4). Since the Larmor frequency in common NMR magnets is in tens or hundreds of megahertz (e.g., at B0 = 1 Tesla, f = γB0/2π = 42.6 MHz), this short B1 pulse is commonly called a radio-frequency (rf) pulse. It is customary to label a B1 pulse with the amplitude and duration to tip M by ϕ degrees as a ϕ rf pulse. More precisely, a 90˚| x′ pulse implies the B1(t) is stationary at the x′ axis in the rotating frame and is capable of tipping the magnetization by 90˚. By setting the central carrier frequency close to the nominal Larmor frequency ω0, one single pulse can excite all possible resonance frequencies.
There are two descriptive terms for an rf pulse, whether the pulse is soft or hard (Figure 2.13), and whether the pulse has a constant or modulated amplitude. A soft rf pulse refers to its narrow band in frequency, which would have a long duration in time (since f = 1/T); in contrast, a hard rf pulse would have a narrow time duration and hence a wide band in frequency, which is commonly used in NMR spectroscopy to excite all possible resonance frequencies. An rf pulse with a constant amplitude in time is commonly used in NMR spectroscopy, while an rf pulse with a modulated amplitude in time would result in a particular waveform in frequency, which is commonly used in MRI.
Figure 2.13 Fourier transform of (a) a hard rf pulse that is short in time duration and (b) a soft rf pulse that is long in time duration.
For a square or rectangular pulse where the B1 field has a constant amplitude during the pulse duration, the tipping angle is given by the area of the pulse, as
where tp is the duration of the pulse and B1 is its magnitude. When B1(t) does not have a constant amplitude during tp, the amount of rotation ϕ is then given by the time integral of the amplitude of the rf field, as
For a pulse whose frequency response is not uniform, the central portion of the frequency response (i.e., frequency range close to ω0) should have a constant amplitude so that all resonant groups centered around ω0 can experience similar tipping angles.
2.11 FT NMR
In contrast to the situation in CW NMR (Section 2.9) where each particular
