Friday, 20 June 2008

Real NMR

We often forget that NMR spectra are complex entities. By complex I don’t mean complicated or difficult to understand (which is, unfortunately, a very common thing too), but by the numbers in the complex space, that is, formed by real and imaginary numbers. One reason that can justify why NMR spectra are barely considered as formed by complex numbers is that NMR software, in general, only displays the real part. However, the imaginary part is present in the background (unless discarded e.g. to save memory) and it’s actually very important. The imaginary part makes it possible to correct the phase of the spectra in such a way that we can get nice absorptive lines (e.g. Lorentzian / Gaussian lines) which provide higher resolution than out-of-phase spectra (e.g. combination of absorptive and dispersive lines. See figure below).

NMR spectra, being of complex nature, can also be displayed in magnitude mode which makes the spectra phase insensitive. This can be advantageous in those cases in which phase correction is difficult, but at the expense of poorer resolution (though in the case of 13C NMR, magnitude mode can facilitate automatic processing, as I have blogged before). In addition, phase information can be very useful in some cases (e.g. DEPT, NOESY, edited NMR experiments, etc).

Acquisition of NMR spectra as complex numbers is also important (in addition to making phase correction possible) to distinguish positive from negative frequencies (quadrature detection. See this link for more information).

Mnova does not have a built-in function to show the imaginary part of the spectrum, but in the case you really need to see it, I will offer here two solutions:

1. One obvious way is by realizing that that the Re and Im channels differ by a phase shift of 90º. Thus, if we apply a (zero order) phase correction of +/-90º, the real part displayed in the software becomes analogous or equivalent to the imaginary counterpart.

2. There is a second alternative: it’s possible to write a very simple script which swaps the real with the imaginary part of the spectrum. This is the script:

  1. function swapRI()

  2. {

  3. var w = new DocumentWindow(mainWindow.activeWindow());

  4. //get active spectrum

  5. var spectrum = nmr.activeSpectrum();

  6. if (!spectrum.isValid() || spectrum.isReal)

  7. return;

  8. if (spectrum.dimCount > 1)

  9. return; //1D only

  10. var dCount = spectrum.dimCount;

  11. var npts = spectrum.count(1);

  12. //To print the value of one of the spectral points

  13. for (var i=0; i<npts; i++)

  14. {

  15. var tmp = spectrum.real(i);

  16. spectrum.setReal(i, spectrum.imag(i));

  17. spectrum.setImag(i, tmp);

  18. }

  19. spectrum.update();

  20. w.update();

  21. }

  22. }

Having said that NMR spectra are of complex nature and why this is important, then, how is it possible, from a hardware point of view, to get real and imaginary numbers? After all, NMR detectors are just measuring ‘real’ values (i.e. induced oscillating voltage in a resonant RF coil). When we say Re and Im numbers in this context, we could have said X and Y values, or u and v values, etc. What matters here is that we are using 2 (orthogonal) detectors to measure (generally simultaneously) the magnetization along X and Y axis. We know that the Re and Im components in the complex space are related by a phase angle of 90º, analogous to the X and Y axis. Thus, the values recorded by the detector placed along the X axis can be considered as the real part of the spectrum (or the cosine component), whereas the values measured in the detector placed in the Y axis will correspond to the imaginary part (or sine component) of the spectrum.

What I would like to highlight here is that the Re and Im, in principle, could appear as 2 INDEPENDENT physical measures, even though they are (generally) recorded simultaneously. In other words, the Re and Im channels are UNCORRELATED. However, this is not completely true. I have already mentioned that the dispersive and absorptive lines (corresponding to Real and Im components of a phased spectrum) are related by a phase shift of 90º. More strictly speaking, the Re and Im components are connected by the well known Kramers–Kronig relation being the Hilbert Transform the mathematical bridge connecting both components. This transform makes it possible to obtain the imaginary part from the real one and the other way round. How about the noise? Is the noise in the imaginary and real parts uncorrelated? In that case, theoretically, it should be possible to achieve a further sqrt(2) gain in sensitivity when quadrature detection is used. This issue will be the subject of my next post.

Quadrature versus linear detection, and where do complex MR data come from
Nothing is better