In general, observed signal depends on both the linewidth of the signal as well as the spectral resolution of the spectrometer. Therefore, you may consider the observed resolution to be convolution of these two factors. Moreover, you may be working in two different regimes; (I) when the spectral resolution is significantly smaller than the signal linewidth, and (II) when the spectral resolution is significantly Greater than the signal linewidth. This convolution cannot be ignored in high resolution Raman spectroscopy. Therefore, measuring the spectral resolution of a spectrometer, it is important to make sure that the measured signal is significantly narrow to assure that the measurement is resolution limited (e.g. analyzing the FWHM of a linewidth that are narrower than the spectral resolution such as a single mode laser or a discharge lamp to determine spectral resolution). Elements impacting the spectral resolution include; Slit width, Spectral range, Pixel width, and number of pixels, where spectral resolution is linearly depending on the first two elements mentioned and inversely related to the last two. However, the accuracy of the peak also depends on the reliability on reproducible signals. That is why sometimes one may need to use a discharge lamp or signals collected form a sample with very narrow linewidth (e.g., CCl4)to caliber the spectrometer first and to ensure there are no drafts due to effects such as temperature, etc.

When talking about spectral resolution of your spectrometer, you are usually talking about resolving power. Meaning practically, what is the closest (spectral) distance of two line, that can still be resolved as two lines. This depends, as Bahram Roughani said, on various parameters of your spectrometer, such as slit width, pixel size, numbers of grids on your grating, etc. Based on these factors your measured spectrum is a convolution of the "real" spectrum and the spectrometer function.

How is the resolving power determined? Usually one uses arbitrary criteria, e.g. the Raleigh criterium. Here, you say you can still resolve two lines, if the peak of one line is in the diffraction minimum of the other line. And this is what the 1 cm^-1 of resolution is usually referring too. However, you may will be able to resolve two lines separated by a smaller amount, as the Rayleigh criterium suggests. And there are other criteria out there, e.g. the Sparrow criterium, which determines two lines as resolvable, until the point the derivation in the center of the sum function of the two lines disappears. It will give you a number, usually smaller than the Rayleigh criterium, however it is still an arbitrary number invented to compare your spectrometer to others. Also, in the "real world" you may have to take into account different intensities in the lines, e.g. a small should in a big and broad line, you will more likely miss and not be able to tell apart, than two nice equally broad and intense lines.

The interesting thing is, you are not asking about resolving power, you are actually asking about, what error or confidence interval you should give on one single line. Which is somewhat a different thing, but they are related. If you just click quickly through your spectrum and look for the (spectrally calibrated) pixel with the highest intensity and call this the "peak" frequency, it is a reasonable choice to use the resolution (your 1 cm^-1) to as a measure for the confidence interval, where the should be (679.4 cm-1 +-1 cm-1). Here, now fitting comes into account. Fitting can improve this value significantly. Why? Because, you are assuming something else. That is, you assume that all the neighboring points of your highest intensity value are related. For example, they are all part of the same Lorentzian, or Gauss or Voigt (or whatever). Fitting one function through all this curves will statistically equalize all the random errors, e.g. one pixel got a few more photon, one some less and so on. Based on this fit you will get a much better value for your peak frequency, i.e. 0.5, 0.1, 0.01 cm^-1, depending on the quality of your fit. It is then reasonable to use the statistical error, your fit (whatever program you use) usually provides you, to use as the confidence interval of your FWHM, peak frequency or intensity. if your line shape matches very well with the fit, than this might very well be much smaller than your resolving power or resolution, simply because the statistical errors on the left and right of your will neutralize each other. A good habit is also to give the coefficient of determination (or R^2), which determines the quality of the fit.

Mentioning quality of the fit, brings us back to the assumption of a certain line shape. Here, you are putting some external (physically justified) educated guesses in, e.g. a thermally system such as a gas, might have Gaussian lines, the spectrum of cold liquid molecules might be Lorentzian, however here the spectral response curve of your spectrometer will play a role, ie. the convolution thingy. Further, you might consider to subtract backgrounds or something else, which might cause systematic errors. Bottom line is, be aware of all this issues and try to physically or mathematically justify anything you do.

EDIT: You may now ask, where in fitting your resolving power, your resolution (your 1 cm^-1), comes into play. It gives you a measure, if it is justified to fit one or two lines to a certain spectral range. That means, if you having two lines in your spectrum overlapping each other well below resolution limit, you will not be able to fit two lines to this structure and be able to tell them apart. In mathematical sense, you have over-parameterized this problem. Also, the resolution or better the spectral function of your spectrometer comes into account, when choosing the right spectral function. If you, for example analyzing a line much smaller than your response curve of your spectrometer, the peak position will still be correct, however the line shape and width will be determined by the spectrometer and not by the "real" line shape.

Hope this helps,

Michael

Thank you both Bahram Roughani and Michael Rüsing for your answers, they were very helpful.

Michael Rüsing, your explanation on the peak fitting was particularly useful and described excellently!

I have two things I forgot to add:

- The thoughts on fitting do also apply for all other, may even nonsymmetric peak (or spectral) shapes you want to fit. Best example from Raman spectroscopy could be spectra of very small nanoparticles, where you have an assymetric broadening of the peaks due to phonon confinement (see phonon confiment model), or the fitting of the LOPC (Longitudinal optical phonon plasmon coupling) to determine charge carrier densities in semiconductors. In both cases, you may fit an asymmetric line shape (determined by a given analytical expression), which will tell you not only something about the peak value, but your nanoparticle size or charge carrier density, as well. Here, just choosing a Gaussian or Lorentzian line can be a systematic error.

- Secondly, you may should not forget other systematic errors. The most prominent that comes into my mind is the calibration. Calibration can induce a systematic error, which can potentially be larger than your "resolution" error. To calibrate, you will likely couple light from a known source (spectrum of a well known chemical, or spectrum of a gas discharge lamp) to determine a pixelnumber vs wavenumber calibration. You will usually than determine a linear or quadratic fit (or other) to calculate the wavenumber for all you other pixels. After fitting and calculating, you will also get confidence intervals for your fitting parameter. Depending on the quality of your fit, you may be inducing a systematic error on all your wavenumbers. This is particular true, if you have made a bad choice for your fitting function. In most cases a linear function is good enough, as you are usually covering a very small portion of a spectrum. However, your most common dispersive element, a grating, is governed by diffraction, which is described by trigonometry, i.e. cosine functions. So, if you are covering a larger spectral area, you might will see some curvature in you calibration curve. Therefore, a linear fit will induce an increasing systematic error over your pixel-array. Therefore, it is a good habit to cover your whole interesting spectral range with an appropriate calibration source.

As a closing thought, be aware of these issues and try to figure out, which error is influencing you the most and if it is relevant to your current research question. If you want to publish your measured peak values as absolute values for others to compare, you should be very very certain, what all your errors are. In most cases, however you are only interested in relative peak changes, e.g. before annealing a given line was 674 cm^-1 and a width of 10 cm^-1, after annealing it shifted to 669 cm^-1 and has a width of 7 cm^-1. Here, most errors play much less a role.

Regards,

Michael