SAXS, SANS, ..., (more generally all structural measurements) evaluate the "instantaneous" spatial correlations of the scatterers present in the sample (electrons for X-rays, nuclei for neutrons, ...) For this purpose, it is assumed that the scattered intensity integrates all the possible energy exchanges and measures S(Q), which is the integral over energy exchanges of the more general dynamic structure factor S(Q, omega). Consequently, the units don't include energy or time.

What is really measured is a cross section that results from the interaction of the radiation with the sample. The units are m2 (in SI) or cm2 (in the more common CGS units).

This number, through calibration procedures, can be evaluated exactly. However, it must be normalised in some way in order to be compared with other measurements, independently of the size of the sample.

The more common normalisation in SAXS and SANS (because they measure often complex systems) is the volume of the sample (in cm3). Consequently, the "absolute units" in SAXS and SANS are cm^-1 (cm2/cm3), or m^-1 in SI.

Instead, in WAXS and WANS, where the chemical composition of the samples are currently simpler, it is frequent to normalise the raw data by the cross section of the atoms present in the sample. In that case, the "absolute units" are unitless (cm2/cm2).

Note that, for the calibration, the cross section of scatterers (electrons, nuclei) must be known.

http://dx.doi.org/10.1107/S0021889803000220

They define a weighted R-factor,  as shown in the attached image, to quantify the discrepancy between a model and an experimental scattering intensity.  They multiply one of the scattering intensities by a scale factor, ScaFac_i, which reportedly minimizes the least-squares fit.  According to their definition of the scale factor, also attached, it has units of intensity, or 1/m.  A consequence of this is that the weighted R-factor has units of [(I(Q)-1)^2]/I(Q).

http://dx.doi.org/10.1107/S0021889803000220

I understand what they are doing but I would expect the scale factor and the R-factor to be unitless.  If this is not the case, then scaling one of the datasets would change it units to I^2, or cm^{-2}.  Also, without this being the case, the resulting R-factor would vary depending on the experimental setup. 

If rather than minimizing the least-squares fit we minimize the R-factor, see attached, we do have a unitless scale factor and a unitless R-factor.

W/m^2 is a unit of power density. 1/m can be a unit of absorption length. None of this makes any sense in the way you claim the authors are using these terms.

The only help I can offer is to ignore the publication which prompted this question without looking back.

SAXS, SANS, ..., (more generally all structural measurements) evaluate the "instantaneous" spatial correlations of the scatterers present in the sample (electrons for X-rays, nuclei for neutrons, ...) For this purpose, it is assumed that the scattered intensity integrates all the possible energy exchanges and measures S(Q), which is the integral over energy exchanges of the more general dynamic structure factor S(Q, omega). Consequently, the units don't include energy or time.

What is really measured is a cross section that results from the interaction of the radiation with the sample. The units are m2 (in SI) or cm2 (in the more common CGS units).

This number, through calibration procedures, can be evaluated exactly. However, it must be normalised in some way in order to be compared with other measurements, independently of the size of the sample.

The more common normalisation in SAXS and SANS (because they measure often complex systems) is the volume of the sample (in cm3). Consequently, the "absolute units" in SAXS and SANS are cm^-1 (cm2/cm3), or m^-1 in SI.

Instead, in WAXS and WANS, where the chemical composition of the samples are currently simpler, it is frequent to normalise the raw data by the cross section of the atoms present in the sample. In that case, the "absolute units" are unitless (cm2/cm2).

Note that, for the calibration, the cross section of scatterers (electrons, nuclei) must be known.

I don't intend to review or criticize the short paper of A.V. Sokolova et al. But, in order to clarify the problem, I think that you must take the different I(s) of the paper as measured in different arbitrary units, what happens quite often (particularly in X-ray scattering). This explains the need of a factor ScaFac which, otherwise, would be ideally equal to 1. Instead, ScaFac should not be different for each point i, for a given set of points.

In other words, the suggested procedure, despite its limitations (namely, s resolution) must be seen like a numerical receipt to make, as well as possible, the matching of two I(s) of the same sample measured under different conditions.

Anyway, eq. (1) calculates a number Rf dimensionless even if I(s) were expressed in absolute units, because there is a product of intensities in both numerator and denominator.

Please note that "Weight" means a statistical unitless quantity, allowing to give more importance to some experimental points (e.g. proportional to the magnitude or its square, as suggested in the text).

So, you must look these expressions like semi-empirical procedures of data treatment and not more!

Another minor remark: A cross section (in cm2) should not be associated with any geometrical area in the sample. It comes from the interaction potential between X-rays and electrons or electronic density). It is the combination of all the interactions, all cross sections, that generates I(s) as described by scattering theory, essentially through the interference of spherical waves issued from a scattering volume in the sample.

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