Here - PMOD/WRC

∂L
L(
λ,
T ) = L(
λ0,
T ) +
∂λ
L
M
λ=
λ0
1 ∂L
Λ +
2 ∂λ
λ=
λ0
( Λ)
( λ0,
T ) = ( 1+
δ ( λ0,
T )) L(
λ0,
T ) ∫ S( λ0,
λ0
+ Λ) dΛ
(4)
The δ(λ 0 , Τ) in above equation is expressed as follows.
δ
( λ , T )
0
=
∫
⎛
⎜
∂L
⎝
∂λ
λ=
λ0
1 ∂L
Λ +
2 ∂λ
From the equations (3) and (4), the following equation
( 1+
δ ( λ0,
T )) L(
λ0,
T ) ∫ S( λ0,
λ0
+ Λ)
dΛ
CM
, λ =
0
( 1+
δ ( λ0,
T0
)) L(
λ0,
T0
) ∫ S( λ0,
λ0
+ Λ)
dΛ
( 1+
δ ( λ0,
T ))
= Cλ0
( 1+
δ ( λ0,
T0
))
Cλ
≅ C
,
( 1 (
0,
T0
) (
0,
T )) (6)
0 M λ
+ δ λ −δ
λ
0
is derived.
The equation (5) shows that C λ0 is expressed by C M,λ0 and
two modification factors, which are δ(λ 0 , Τ) and δ(λ 0 , Τ 0 ).
Determination of HTBB Temperature
0
λ=
λ0
∫
L(
λ , T ) S
At first, we have determined the temperature of the
HTBB, substituting the C M,λ for the C λ . The temperature of
the HTBB was determined from the equation (2) at each
wavelength in which the multichannel spectroradiometer
offers the spectral data. Figure 1 shows a result of
temperature determination.
Figure 1. The temperature of the HTBB determined from
spectral data of the multichannel spectroradiometer.
The determined temperature is almost constant in
longer wavelength region but depends on wavelength in
shorter wavelength region. The modification factors
mentioned above are one of the causes of this dependence.
The following equation (7) is derived from
differentiating equation (2) and shows the other causes of
the dependence.
dT
T
⎛
⎜
⎛ T ⎞ e
=
⎜
⎝
T
⎟
⎝ 0 ⎠
⎛ hc ⎞ e
+
⎜
λ0kT
⎟
⎝ ⎠
The dλ 0 , dT 0 and dC λ0 show the accuracy of reading
wavelength λ 0 of the multichannel spectroradiometer, the
uncertainty of the CUBB temperature T 0 and the
uncertainty of the ratio C λ0 , respectivily. The dC λ0 has been
derived from the modification factors mentioned above,
2
Λ
2
( Λ) Λ ⎟S( λ , λ + Λ)
( λ , λ + Λ)
0
0
⎞
⎟
⎠
0
dΛ
0
dΛ
( hc / λ0kT0
−hc
/ λ0kT
) ( hc / λ0kT0
−hc
/ λ0kT
)
dλ
⎛
0 T ⎞ e
C
λ0
( hc / λ kT −hc
/ λ kT )
0
0
C
λ0
⎞
−1⎟
⎠ λ0
dC
0
C
λ0
λ0
+
⎜
T
⎟
⎝ 0 ⎠
C
λ0
(5)
dT
T
(7)
0
0
and signal noise (due to stray light and the measurement
repeatability) of the multichannel spectroradiometer.
The evaluation of the three terms in the equation (7)
showed the third term was much larger than the first and
second terms in shorter wavelength region; and the first
and second terms had weak dependence on wavelength
compared to the third term.
Figure 2 shows the third term in the equation (7). The
modification factors due to the bandpass property and the
signal noise of the multichannel spectroradiometer is
shown in this figure.
Figure 2. The relation between dT and dC λ0 , and a
measured C λ0 and a fitting curve by Planck’s low. The
dC λ0 has been evaluated from the bandpass property and the
signal noise of the multichannel spectroradiometer.
To evaluate the modification factors, we have measured
some spectral lines by the multichannel spectroradiometer
and estimated the spectral radiance responsivity S(λ 0 , λ).
In this estimation, we assumed the S(λ 0 , λ) was close to a
symmetric function with respect to λ 0 and we neglected the
term that has Λ m whose m is odd number in the equation
(5).
Figure 2 shows that the uncertainty of the ratio C λ0
increases at a wavelength below 600 nm, and hence we
have determined HTBB temperature from the measured
C λ0 , whose wavelength is above 600 nm.
Figure 2 also shows a measured ratio C λ0 and a fitting
curve by Planck’s low. The C λ0 data, whose wavelength is
from 600 nm to 780 nm, is used for this fitting. The fitting
curve agreed well with the measured C λ0 . The HTBB
temperature has determined from this fitting is 2965.7 K
and the uncertainty of this fitting is under 0.1 K.
Conclusions
We have determined the temperature of a blackbody
based on a spectral comparison with a
copper-freezing-point blackbody. A multichannel
spectroradiometer was used for this comparison. From the
evaluation of the signal noise and the bandpass property of
the multichannel spectroradiometer, we determined an
appropriate wavelength range for the spectral comparison.
The results of the spectral comparison agreed well with
Planck’s low.
References
van der Ham, E. W. M., Bos, H. C. D., Schrama, C. A., Primary
realization of a spectral irradiance scale employing
monochromator-based cryogenic radiometry between 200nm
and 20µm, metrologia, 40, S177-S180, 2003.
282