1. 10.5 Fourier Transform NMR Instrumentation
the free-induction decay (FID)
chemical shifts and digitization rates
heterodyned detection
example FIDs
multiplexed excitation
single-nucleus spectrometer
ensemble averaging and the multiplex advantage
multi-nuclear spectrometer
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2. The Free-Induction Decay (FID)
Suppose the rf magnetic field, B1 , is abruptly cut off after the excess spins have been rotated into the x,y-plane. The coherent magnetic moment would continue to precess about the z-axis. The size of the moment will decrease due to dephasing and spin-lattice relaxation. These two processes are first order making the signal decay exponentially with a time constant, T. Although T varies from nucleus to nucleus and with chemical shift, a typical value is 1 s.
For a single proton with no chemical shift, the result is an exponentially decaying cosine with a frequency of 300 MHz and a decay constant of 1 s. This is called the FID, or free-induction decay.
The Fourier transform of this signal yields a spectrum consisting of a single Lorentzian-shaped band centered at 300 MHz with a FWHM = 1/π = 0.32 Hz.
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3. Chemical Shifts and Digitization
For a fixed magnetic field (no sweep coils), chemical shifts will change the frequency of the FID. For example, a chemical shift of 17 ppm would change the Larmor frequency to 300,000, ,005,100 = 299,994,900 Hz. Thus, an NMR spectrum will produce an FID composed of exponentially decaying cosines over the range of 299,994,900 to 300,000,000 MHz.
To digitize the FID, the data rate would have to be a minimum of 600 MHz. To obtain the theoretical resolution one would have to digitize for at least 3-5 time constants, i.e. 3-5 seconds. The signal would then consist of Gsamples! Even today a transform of this size would require a computer with an unusually large memory and speed.
Note that chemical shifts are actually measured versus the reference compound, tetramethylsilane (TMS), and not versus 300 MHz. Also, the proton frequency of TMS is never adjusted to be exactly 300 MHz.
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4. Heterodyned Detection
To circumvent the data rate and size problem, the FID signal is heterodyned with the 300 MHz source.
The mixer produces two sets of signals: 600 MHz ± fsignal and 0 ± fsignal. A low pass filter will remove the 600 MHz portion and pass the 0 ± fsignal portion.
With heterodyning, a resonance at 300 MHz will produce an exponential decay without an underlying cosine. That is, the signal is a 0 Hz cosine. The Fourier transform of this signal will be a Lorentzian peak at 0 Hz.
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5. Chemical Shifts and Heterodyning
A 17 ppm chemical shift produces a resonance at 299,994,900 Hz. After heterodyning the FID will have a frequency of 5.1 kHz. The Fourier transform will be a Lorentzian peak centered at 5.1 kHz. This will be labeled as 17 ppm on the spectrum.
Heterodyne detection drastically reduces the required performance of the data system. If the highest FID frequency is 5.1 kHz, the ADC only needs to take data at a rate of 10.2 kHz. For a 3 s observation time the number of data values collected is down to 30,600. This is a lot more manageable than directly digitizing a signal near 300 MHz!
In the following example FIDs it is assumed that TMS is exactly 300 MHz, and that all protons have the same dephasing time constant. Although both of these assumptions are erroneous, the observed FIDs will be very close to the figures. A decay of 0.1 s was used to help visualize the data.
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6. Single Chemical Shifts
0 Hz, 0 ppm
"zero beat"
300 Hz, 1 ppm
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7. Multiple Chemical Shifts
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8. Multiplexed Excitation
In order to multiplex the NMR signal, all of the nuclei have to be excited at the same time. For a chemical shift of 17 ppm this means that frequencies over the range 300,000,000 ± 5,100 Hz have to be generated.
A band of frequencies is generated by pulsing the 300 MHz "carrier." If the carrier is turned on and off in 10 μs, the result is a convolution in the frequency domain by a sinc function with its first nodes at ±100 kHz.
At first a 10 μs pulse seems to be too small (generates too large a range of frequencies). However, all the frequencies generated should have amplitudes within a percent of each other. To determine the amplitude of 299,994,900 Hz evaluate the following sinc function.
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9. Single-Nucleus FT Spectrometer
The timing circuit turns on the pulse unit. After a small delay the timing circuit tells the ADC when to start digitizing. It also tells the ADC the rate at which the FID will be digitized.
The rf pulse width is determined by the range of frequencies necessary to excite simultaneously all the chemical shifts. The pulse rate is determined by the decay constant of the FID (3-5 decay constants).
For a one-second data collection time and a spectral width of 17 ppm (5.1 kHz), the multiplex advantage would be
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10. Ensemble Averaging
Because the excitation is pulsed the experiment is amenable to ensemble averaging. This is shown diagrammatically below for a heterodyned frequency of 20 Hz and a decay time of 0.1 s. In this figure excitation is repeated every 3 decay times. A shorter time between repetitions will decrease the size of (Na - Nb), thus the sensitivity. It will also impose a rectangle on the decay, causing the frequency domain Lorentzian peaks to be convolved with a sinc function.
A typical data collection time for a run-of-the-mill organic compound would be 5-15 min. If three seconds are used to collect each FID, a total of 300 data ensembles can be averaged. The overall signal-to-noise enhancement (averaging plus the multiplex advantage) is given by
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11. Multi-Nucleus FT Spectrometer
The diagram at the right is the general concept of how the rf frequency is varied at the sample, while keeping all the intermediate amplifier frequencies the same.
This is an extremely clever use of heterodyning! Go through the diagram with the rf synthesizer set to 65.4 MHz (13C) instead of 290 MHz (1H) and see how the frequencies after the signal mixer stay the same.
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