# Emissivity of transparent window

I've been reading this article where the IR radiance of an IR-window ($MgF_2$, 1.75mm thick, passband 3-5$\mu m$) is calculated. At some point during the calculation they mention in a footnote:

The spectral emittance has been approximated by means of a step function: $\epsilon_\lambda$ = 0.08 and 0.06 from 3$\mu$ to 4$\mu$ and 4$\mu$ to 5$\mu$, respectively (cf. Ref. 9)

Ref. 9 is

S. S. Ballard, K. A. McCarthy, and W. L. Wolfe, IRIA Stateof- the-Art Report: Optical Materials for Infrared Instrumentation: Supplement (The University of Michigan, Ann Arbor, 1961), p. 17.

although I have no further access to it so I can't check.

Now, to my own knowledge, the emissivity of a transparent material is given by

$$\epsilon = 1-e^{-\alpha d}$$

where d is the material's thickness (1.75mm, or 0.175 cm) and $\alpha$ the absorption coefficient. The absorption coefficient for $MgF_2$ is somewhere around $40*10^{-3}$ cm$^{-1}$ at 2.7 $\mu m$(Source) or 5.5-6* $10^{-3}$ cm$^{-1}$ at 2.8 and 5.1 $\mu m$ (Source)

Question 1: Is there an explanation for the order of magnitude difference between both sources, other than "one must be wrong"?

Question 2: Calculating the emissivity using those two sources yields me numbers of 0.007 (first source) or 0.001 (second source). These are both a factor 10 and 70 different from what's used in the paper. The resulting graphs & data from the paper don't appear to be wrong, does that mean I used a wrong equation / did something else wrong?

EDIT: In addition, this thesis shows on page 52 the emissivity of Germanium to be ~35% (0.35) at 10µm for 1.14mm thickness. Using an absorption coefficient for Germanium I found here (0.035 cm$^{-1}$) and the earlier equation yields an emissivity of 0.004 - which is about TWO orders of magnitude different. I'd say since I found two sources that yield wrong answers now, it's more sensible that I'm doing something wrong. Please enlighten me :)

• I agree, after looking at your two sources, that "one of them must be wrong" seems like a reasonable conclusion. If the units of the second source were intended to be $\rm{mm^{-1}}$, that would explain the discrepancy. – Floris Sep 29 '16 at 12:44
• Incidentally, I want to congratulate you for this question. You show clear research, find discrepancies, and conclude that you are probably wrong, given the mismatch between your approach, and several published sources. I can't immediately find where you went wrong, but applaud your attitude. Sometimes, you are right and everyone else is wrong. But sometimes, it's the other way around... – Floris Sep 29 '16 at 12:46
• If you are at an institution, go find your local reference librarian - they love this kind of thing! Given that it has been referenced multiple times, the library at the University of Michigan will know where a copy is. Note that the IRIA here seems to be the InfraRed Information and Analysis center at U. Mich at the 'Willow Run Laboratories', run under an Office of Naval Research contract. – Jon Custer Sep 29 '16 at 13:10
• The University of Michigan on-line catalog lists (and has on-line) the original 1959 report, but does not seem to have the supplement of 1961 available. In the 1959 report they reference A Duncanson and RW Stevenson, Proc. Phys. Soc. 72 1001-1006 (1958). Ballard's report is also listed in various reference works which may be easier to find. – Jon Custer Sep 29 '16 at 14:14