Crystallography requires the collection of many measurements (could be a few thousand to even millions depending on the size of the molecule and complexity of the crystal (technically speaking, the size of the crystal's unit cell is a major determining factor for the size of the data set). I'm not going to assume this is a small molecule crystal like a salt or a large molecule like a protein. its not necessary here.
I'll call this a set of intensity measurements, even though what is actually used is the square root of the intensities measured, called the structure factor.
The only way we really know that the molecular structure model is correct is that it generates as accurately as possible the intensities that were experimentally measured.
R, which I believe stands for residual, is a fractional difference between the measured intensities and what any proposed molecular model gives.
The residual is calculated as the absolute value of F(model)-F(measured). This means that 0.0 is a perfect match while 1.0 is a perfectly awful fit that shows the model is perfectly awful.
In practice 0.60 is usually as bad as random model will give you (there's a statistical argument as to why it doesn't go higher). Also the measurements are usually not perfect - they contain errors in measurements or artifacts from imperfect crystals or the detector, so an R value of < 0.20 (20%) is typically what you see in a reasonable paper. I think its commonly less than 0.15 for most structures now in fact.
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