The 5th and 6th order bows are very wide compared to the primary and
secondary. They are made by rays that
are closer to the edges of the droplets. These glancing rays are more
widely colour separated when they enter and leave the drops.
The outer red rim of the 5th order bow, produced by rays emerging
after five internal reflections, has a radius of 52.9º, slightly
greater than that the red inner rim of the secondary bow.
The 5th order bow's might therefore have a chance of being
photographed. Its green and blue sections fall within Alexander's
dark band between the primary and secondary. With the right
combination of sky lighting and perhaps polarising
filters it might be photographed. Try it!
The above was written several years before it was finally positively identified in a photograph.
Harald Edens saw the same opportunity and captured part of the 5th order bow on August 8, 2012. See the image and read full details here.
The unfiltered Bowsin simulation at left shows the 5th order
bow inside Alexander's dark band.