## Sunday, November 19, 2006

### Theory

NMR means nuclear magnetic resonance. It' s an analytical technique by which atomic nuclei and, indirectly, their surroundings, are revealed. It's called magnetic because a magnet is used to enhance the (otherwise negligible) energy of the nuclei. It's a resonance phenomenon because the nuclei, when excited by a wave of the right frequency, respond with an analogue wave. The NMR experiment is conceptualized in a rotating coordinate system. For the moment being you just have to pretend it is a plain Cartesian system of coordinates xyz. A radio-frequency pulse tilts the magnetization of the nuclei, initially in the z direction, in the xy plane for detection. Here each nucleus rotates at its own resonance frequency while two detectors sample the total magnetization at regular intervals along the x and y axes. The regular interval is called dwell time. The measured intensities along the x axis are called the real part of the spectrum and intensities along y form the imaginary part. Real and imaginary are just names. They could have been called right and left or red and white and it would have been the same (or better). This is the main difference indeed between an NMR instrument and a hi-fi tuner. It probably serves to justify the price difference. You have to realize that the real and imaginary parts are both true experimental values of the same importance. These intensities are stored on a hard disk in the same time order in which they are sampled (i.e. chronologically). A couple of a real and an imaginary values, collected at the same time, constitute a complex point. iNMR normally displays only the real part of the spectrum. A complex spectrum is like a vector in physics. It can be characterized by its x and y components or by its magnitude (amplitude) and direction. iNMR lets you display the magnitude of the spectrum if you want. The direction of a complex point is called phase. The so called "phase correction" is a process which mixes the real (x) and imaginary (y) components. A radio-frequency wave has frequency, amplitude and phase. Thus complex numbers are the natural choice to describe a RF signal. In an older experimental scheme a single detector measures the magnetization along both axes. In this case the sampling cannot be simultaneous. It is in fact sequential. In this case the spectrum is known as real only. Actually it can (and is) manipulated just as a normal simultaneous, complex, spectrum. A simple ad hoc correction is needed when transforming the spectrum in the "frequency domain".
Let's explain what this last term means. What we have been speaking of up to know is a not very meaningful function of time called FID (free induction decay) (induction is another way to refer to magnetization). It is not very meaningful because all the nuclei resonates at the same time. If there were me and Pavarotti singing together it would be easy for you to discriminate our single contribution to the choir. But you are not trained to discriminate among a collection of atomic nuclei resonating together. The Fourier transformation (FT) is a mathematical tool that separates the contribution of each nucleus by its resonance frequency. The FID is a function of time, while the transformed spectrum is a function of frequency. The FT requires a computer and this is why you need a computer and a software application if you want to do some NMR. At first sight it may seem that the function of frequency should extend between - and +. Actually the sampling theorem states it only has to be calculated in the interval from 0 (included) to Ny (excluded). Ny = Nyquist frequency = sampling rate = reciprocal of the dwell time. Let's take a pause. What's the angle whose sine is 1? My pocket calculator says: 90°. I say: 90°±n360°. Who is right? Both! Coming back to our NMR experiment, suppose a signal is so fast that it rotates by exactly 360° during the dwell time. The two detectors will see it always in the same position, so they will believe it simply doesn't move (it has zero frequency). The same happens with four different signals which rotate by -350°, 10°, 370° and 730° during the dwell time. There is absolutely no way to tell which is which, unless you shorten the dwell time (a common experimental practice). A final case: you have two signals A and B and A moves of 361° respect to B each sampling interval. You will get the impression that A is moving only of 1° each time. In conclusion, the maximum difference in frequency that can be detected is = number of cycles / time interval = 1 / dwell time = Nyquist. q.e.d. In NMR this quantity is called spectral width. All the books report different expressions for the Nyquist frequency and the spectral width. One day we should open a discussion on the subject.
On a purely mathematical basis it doesn't matter how large the actual frequency range you have to record. You can pretend the range starts at zero and extend up to comprise the maximum signal separation. In practice detectors work in the low-frequency range. So you have technical limitations, and this is only the first one. The resonance frequencies today are approaching the GHz. The frame of reference also rotates at a similar frequency, so the apparent frequency is in the range of KHz. With this reduced frequency the dwell time needs to be in the order of milliseconds. The problem is that you need to filter out all other frequencies because they contain nasty noise (didn't I say it was an hi-fi matter?). So we need the rotating frame to move from the ideal world of theory and to become a practical reality. How is the rotating frame accomplished experimentally? A detector receives two signals, one coming from the sample under study and another which is a duplicate of the exciting frequency. The detector actually detects the difference between the two frequencies. To fully exploit the power of the pulse, the transmitter is put at the centre of the spectrum. Signals falling at the left of it appear as negative frequencies. (well, here it is not important if you use the delta or the tau scale and if they are positve or negative; only the concept matters). We have said that the spectrum begins at zero. In fact, if you perform a plain FT, the whole left side would appear shifted by the Nyquist frequency, then to the right of the right side!
The FT and its inverse show a number of interesting properties. The first one predicts that, if you complex-conjugate the FID, the transformed spectrum will be the mirror image of the original spectrum. In fact, if you invert the y component of a vector, you obtain its image across a mirror put along the x axis. Anything rotating counter-clockwise (positive frequency) will appear as rotating clockwise (negative frequency) and vice versa. This mirror image is mathematically called "complex conjugate". Some spectrometers already perform this operation when acquiring. This is another reason (together with sequential acquisition) why spectra coming from different instruments require different processing.