Here - PMOD/WRC
Here - PMOD/WRC
Here - PMOD/WRC
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∂L<br />
L(<br />
λ,<br />
T ) = L(<br />
λ0,<br />
T ) +<br />
∂λ<br />
L<br />
M<br />
λ=<br />
λ0<br />
1 ∂L<br />
Λ +<br />
2 ∂λ<br />
λ=<br />
λ0<br />
( Λ)<br />
( λ0,<br />
T ) = ( 1+<br />
δ ( λ0,<br />
T )) L(<br />
λ0,<br />
T ) ∫ S( λ0,<br />
λ0<br />
+ Λ) dΛ<br />
(4)<br />
The δ(λ 0 , Τ) in above equation is expressed as follows.<br />
δ<br />
( λ , T )<br />
0<br />
=<br />
∫<br />
⎛<br />
⎜<br />
∂L<br />
⎝<br />
∂λ<br />
λ=<br />
λ0<br />
1 ∂L<br />
Λ +<br />
2 ∂λ<br />
From the equations (3) and (4), the following equation<br />
( 1+<br />
δ ( λ0,<br />
T )) L(<br />
λ0,<br />
T ) ∫ S( λ0,<br />
λ0<br />
+ Λ)<br />
dΛ<br />
CM<br />
, λ =<br />
0<br />
( 1+<br />
δ ( λ0,<br />
T0<br />
)) L(<br />
λ0,<br />
T0<br />
) ∫ S( λ0,<br />
λ0<br />
+ Λ)<br />
dΛ<br />
( 1+<br />
δ ( λ0,<br />
T ))<br />
= Cλ0<br />
( 1+<br />
δ ( λ0,<br />
T0<br />
))<br />
Cλ<br />
≅ C<br />
,<br />
( 1 (<br />
0,<br />
T0<br />
) (<br />
0,<br />
T )) (6)<br />
0 M λ<br />
+ δ λ −δ<br />
λ<br />
0<br />
is derived.<br />
The equation (5) shows that C λ0 is expressed by C M,λ0 and<br />
two modification factors, which are δ(λ 0 , Τ) and δ(λ 0 , Τ 0 ).<br />
Determination of HTBB Temperature<br />
0<br />
λ=<br />
λ0<br />
∫<br />
L(<br />
λ , T ) S<br />
At first, we have determined the temperature of the<br />
HTBB, substituting the C M,λ for the C λ . The temperature of<br />
the HTBB was determined from the equation (2) at each<br />
wavelength in which the multichannel spectroradiometer<br />
offers the spectral data. Figure 1 shows a result of<br />
temperature determination.<br />
Figure 1. The temperature of the HTBB determined from<br />
spectral data of the multichannel spectroradiometer.<br />
The determined temperature is almost constant in<br />
longer wavelength region but depends on wavelength in<br />
shorter wavelength region. The modification factors<br />
mentioned above are one of the causes of this dependence.<br />
The following equation (7) is derived from<br />
differentiating equation (2) and shows the other causes of<br />
the dependence.<br />
dT<br />
T<br />
⎛<br />
⎜<br />
⎛ T ⎞ e<br />
=<br />
⎜<br />
⎝<br />
T<br />
⎟<br />
⎝ 0 ⎠<br />
⎛ hc ⎞ e<br />
+<br />
⎜<br />
λ0kT<br />
⎟<br />
⎝ ⎠<br />
The dλ 0 , dT 0 and dC λ0 show the accuracy of reading<br />
wavelength λ 0 of the multichannel spectroradiometer, the<br />
uncertainty of the CUBB temperature T 0 and the<br />
uncertainty of the ratio C λ0 , respectivily. The dC λ0 has been<br />
derived from the modification factors mentioned above,<br />
2<br />
Λ<br />
2<br />
( Λ) Λ ⎟S( λ , λ + Λ)<br />
( λ , λ + Λ)<br />
0<br />
0<br />
⎞<br />
⎟<br />
⎠<br />
0<br />
dΛ<br />
0<br />
dΛ<br />
( hc / λ0kT0<br />
−hc<br />
/ λ0kT<br />
) ( hc / λ0kT0<br />
−hc<br />
/ λ0kT<br />
)<br />
dλ<br />
⎛<br />
0 T ⎞ e<br />
C<br />
λ0<br />
( hc / λ kT −hc<br />
/ λ kT )<br />
0<br />
0<br />
C<br />
λ0<br />
⎞<br />
−1⎟<br />
⎠ λ0<br />
dC<br />
0<br />
C<br />
λ0<br />
λ0<br />
+<br />
⎜<br />
T<br />
⎟<br />
⎝ 0 ⎠<br />
C<br />
λ0<br />
(5)<br />
dT<br />
T<br />
(7)<br />
0<br />
0<br />
and signal noise (due to stray light and the measurement<br />
repeatability) of the multichannel spectroradiometer.<br />
The evaluation of the three terms in the equation (7)<br />
showed the third term was much larger than the first and<br />
second terms in shorter wavelength region; and the first<br />
and second terms had weak dependence on wavelength<br />
compared to the third term.<br />
Figure 2 shows the third term in the equation (7). The<br />
modification factors due to the bandpass property and the<br />
signal noise of the multichannel spectroradiometer is<br />
shown in this figure.<br />
Figure 2. The relation between dT and dC λ0 , and a<br />
measured C λ0 and a fitting curve by Planck’s low. The<br />
dC λ0 has been evaluated from the bandpass property and the<br />
signal noise of the multichannel spectroradiometer.<br />
To evaluate the modification factors, we have measured<br />
some spectral lines by the multichannel spectroradiometer<br />
and estimated the spectral radiance responsivity S(λ 0 , λ).<br />
In this estimation, we assumed the S(λ 0 , λ) was close to a<br />
symmetric function with respect to λ 0 and we neglected the<br />
term that has Λ m whose m is odd number in the equation<br />
(5).<br />
Figure 2 shows that the uncertainty of the ratio C λ0<br />
increases at a wavelength below 600 nm, and hence we<br />
have determined HTBB temperature from the measured<br />
C λ0 , whose wavelength is above 600 nm.<br />
Figure 2 also shows a measured ratio C λ0 and a fitting<br />
curve by Planck’s low. The C λ0 data, whose wavelength is<br />
from 600 nm to 780 nm, is used for this fitting. The fitting<br />
curve agreed well with the measured C λ0 . The HTBB<br />
temperature has determined from this fitting is 2965.7 K<br />
and the uncertainty of this fitting is under 0.1 K.<br />
Conclusions<br />
We have determined the temperature of a blackbody<br />
based on a spectral comparison with a<br />
copper-freezing-point blackbody. A multichannel<br />
spectroradiometer was used for this comparison. From the<br />
evaluation of the signal noise and the bandpass property of<br />
the multichannel spectroradiometer, we determined an<br />
appropriate wavelength range for the spectral comparison.<br />
The results of the spectral comparison agreed well with<br />
Planck’s low.<br />
References<br />
van der Ham, E. W. M., Bos, H. C. D., Schrama, C. A., Primary<br />
realization of a spectral irradiance scale employing<br />
monochromator-based cryogenic radiometry between 200nm<br />
and 20µm, metrologia, 40, S177-S180, 2003.<br />
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