ABSTRACT
On a more absolute level, a blackbody cavity surrounded by gold at its melting point (1064.43°C) can be focused upon for one calibration datum. Such a cavity is depicted in Figure 8.8. By placing a rotating sectored disk, or varying thicknesses of
an absorptive glass (grey filter), in the optical path to the goldpoint blackbody source, lower calibration temperatures can be "synthesized". Using the absorptive glass filter as example, the mathematical justification follows:
A partially transmitting glass of known absorption coefficient at a specific wavelength, k>.., absorbs increasing levels of radiant energy with increasing thickness. The decay of radiation at any given cross-sectional area within the glass would be
proportional to how much radiation was left:
dWr(>.) = -k>.Wr(>.) dx
where Wr(>.) = ¢Rr(>.), the geometric constant ¢ converting the spectral power released per unit area (in all directions) from the body to the power incident on the glass plate. Integrating from the front surface (x = 0) where the intensity is the incident intensity Wr(>.) 0 to some position x within the glass:
[WT(>.), dWr(>.) = fx -k>.dX lwT(>.)o Wr(>.) lo
which integrates to:
or:
( Wr(>.)x) In Wr(>.)o = -k>.x
Wr(>.)x exp( -k>.x) = Wr(>.)o
Since v = cf>. (hence dv = -(c/>. 2)d>.), Planck's law can be rewritten in terms of wavelength:
A exp >.kT Planck's law becomes Wein's law if the "-1" term is considered insignificant; combining constants yields:
Wr(>.) = >.-5C1 exp ( ~~ 2 ) Combining with the expression for the absorptive glass:
>,-5cl exp (-c2) exp( -k>.X) = >.T,
>. -5C1 exp ("A¥;) Taking logarithms and rearranging:
k>.X = -C2 (]:_ - _!_) >. To Tx
By varying filter thickness or using filters of different known absorption coefficients, the synthesized blackbody temperature after the filter can be calculated (note C2 is simply hc/k). By correlating these temperatures against the disappearing filament current, a calibration curve for the pyrometer can be established for the temperature range of the melting point of gold and below.
