I'm reviewing nucleogenenesis and came across the table below. I like it because it actually labels the s and r-process elements, but I became confused as to why the neutron capture processes begin at A = 38. I thought anything with an atomic number greater than 26 has to be formed by neutron capture? What is my confusion here? Thanks in advance!
If you go back to the sourced article in your graphics by Jennifer Johnson it has a more verbose colour legend. For the grey part it explicitly states
very radioactive elements. Nothing left from stars.
and her figure caption more verbosely
Each element in this periodic table is color-coded by the relative contribution of nucleosynthesis sources, scaled to the time of Solar System formation. Only elements that occur naturally in the Solar System are shown; artificially made elements and elements produced only through radioactive decay of long-lived nuclei are shown in gray. The data plotted in this figure are available in table S1.
And that pretty much sums it up:
The colour gives the source of the elements as found on Earth.
Thus the colour does NOT indicate how the elements can be formed in general, but the origin of the isotops as found on Earth (or present on formation of Earth, thus still long after nucleosynthesis in their progenitor stars). The lifetime of the longest isotope of technetium is 4.2 million years, and of promethium of only 17 years. Thus basically all of those grey-coloured elements are decayed long ago and none of the naturally occuring specimen on Earth are anymore of primordeal origin but result of fission of higher-order elements.
EDIT to add: As to the 2nd question in your posting which is quite distinct from the title
"why the neutron capture processes begin at A = 38. I thought anything with an atomic number greater than 26 has to be formed by neutron capture?"
Those elements can actually be formed by fusion of the lighter elements in the core of the supernova due to the high density and temperature (basically similar to how we can create heavier elements in the lab, colliding heavy ions) - but that fusion will consume energy instead of releasing energy.