What are the Bak4 and Bk-7 prisms? Is there a difference when they are used in scopes and in binoculars?

Both Bak4 and Bk-7 are common types of brown glass used in prisms that fold the light path inside the scopes and binoculars. “Bak” represents “Barium light Crown” glass, while “Bk” represents “Borosilicate Crown” glass.

In general terms, Bak4 prisms are more expensive and are considered better because they give a smooth, rounded “exit pupil”, due to a slightly higher level of refraction than Bk-7. Bk-7 prisms are also good quality, but brightness falls off slightly at the edge of the field compared to Bak4, so they create a slightly square exit pupil. Sharpness and clarity aren't affected by using BK-7 and only the outer edges of the exit pupil are shaded (blurred). If your eye pupil is closed to a size that fits within the unshadowed diamond shape inside the exit pupil there is no loss of light at all with BK-7 prisms compared to Bak4.

However, BaK4 is preferable to BK-7 as a prism glass only when the focal ratio of the objective lens falls below about f/5. Virtually all scopes have focal ratios above f/5, so there's no disadvantage in using BK-7 prisms. It achieves total internal reflection just as well as BaK4. In some telescopes it's actually a slightly better choice because a prism made with BK-7 has a little less spherical overcorrection and chromatic aberration at blue/violet wavelengths than BaK4.

For binoculars, their objective lenses are mostly around f/4 or less, so BaK4 is preferred for their prisms to achieve total internal reflection at the edges of the fast f/4 light cone. Whist the binoculars with the Bk-7 prism would look no difference to the Bak4 in daylight condition, as the light levels drop, and the eye pupil expends, you start to observe the effects of the shaded regions as the image quality drops and becomes prone to chromatic aberration around the periphery of the image.