BaK4 vs BK7 prisms

I was shopping for binoculars recently, and in the process of researching what to buy I saw a big fuss about BaK4 (barium crown glass) prisms. The Internet made many vague claims of "sharper" and "expensive therefore better" in comparing BaK4 to BK7 (borosilicate crown glass), but what is really happening?

As far as I can piece together, it comes down to index of refraction. Based on data from Wikipedia:

Wavelength (nm)	n (BK7)	n (BaK4)
390	1.532	1.604
550	1.518	1.587
750	1.512	1.578

BaK4 has slightly greater dispersion, which is bad, but a higher index of refraction. This decreases the critical angle for total internal reflection (at 550 nm) from 41.2° to 39.1°. It doesn't seem like much, but for 90° prisms, this increases the allowable angle of incidence from 5.8° to 9.5°, which shortens the minimum focal ratio (f/) needed to ensure all incoming light is reflected:

FOV (°)	min f/ (BK7)	min f/ (BaK4)
0	5.0	3.0
6	10.4	4.4
7	12.7	4.8
8	16.3	5.2

A shorter minimum f/ ratio helps in a couple ways:

1. We might not need a telephoto group to keep binocular length manageable. For example, a 42mm objective requires a 440mm focal length at f/10.4, but only 185mm at f/4.4.
2. As FOV and focal length increase, the real image produced at the focal plane gets larger and larger. At some point, it becomes larger than the objective. This happens at 5.8° FOV (f/9.9) for BK7 and 9.4° (f/6.1) for BaK4. 

In essence, BaK4 prisms make it easier to design compact optics with a wide FOV, making it well-suited for use in binoculars. With BK7 prisms, it's infeasible to rely entirely on total internal reflection, so you end up with mirror coatings (which don't reflect as much light) and not-fully-round exit pupils.

However, BaK4 has higher dispersion than BK7, so it's a tradeoff, and there are definitely applications that would favor the lower-dispersion BK7. Telescopes, for instance, usually have a FOV less than 2° and a focal ratio around f/10, so the smaller angle of incidence is not an issue for the 90° prism. And 45° prisms don't have a problem with angle of incidence because everything hits the reflecting surface far beyond the critical angle.