Ústav fyzikální elektroniky
Přírodovědecká fakulta, Masarykova univerzita, Brno
Fyzikální praktikum 4
Planck’s radiation law and measurement of
temperature
podzim 2014
1 Planck’s law
The emission of radiation from heated bodies (incandescence) is manifested with a continuous
optical spectrum. Besides real heated bodies, a model of a so-called black body, absorbing all
the light at all wavelengths, plays a fundamental role, since its spectral dependence of intensity
is directly described by Planck’s distribution. This distribution expressed particularly either as
energy density ρ (energy of radiation present in unit volume), radiance L (radiation power emitted
from unit projected perpendicular surface into unit solid angle) or irradiance M (radiation power
emitted from unit surface). Since for isotropic radiation and ideal diffuse surface
L =
1
4π
ρc, M = πL,
the expressions of Planck’s law have the forms
ρb(λ) =
8πhc
λ5
1
e
hc
kbT λ
− 1
(Jm−3
m−1
),
Lb(λ) =
2hc2
λ5
1
e
hc
kbT λ
− 1
(Wm−2
sr−1
m−1
),
Mb(λ) =
2πhc2
λ5
1
e
hc
kbT λ
− 1
(Wm−2
m−1
).
where T is thermodynamic temperature, λ is wavelength, h is Planck’s constant (6.625 × 10−34
J s), c vacuum speed of light (2.998 × 108 m s−1) and kb Boltzmann constant (1.38 × 10−23 J K−1).
Here, the energy is expressed per unit wavelength interval (last m−1 or nm−1 in units) in a form
convenient for the measurements in the visible spectral region. From the formulas usually derived
for energy per frequency interval (Hz−1) they are obtained as
L(λ) = L(ν)
dν
dλ
.
The differentiation of Planck’s law with respect to the wavelength and the integration of
Planck’s law over all wavelengths give two important laws:
1. Wien displacement law of intensity maximum
λmaxT = konst = 2.88 × 10−3
m K,
2. Stefan–Boltzmann law of total radiation intensity
Mb = σT4
, σ = 5.67 × 10−8
Wm−2
K−4
.
Fyzikální praktikum 2
Radiance(Wm-2sr-1nm-1)
0
200
400
600
800
1000
Wavelength(nm)
0 500 1000 1500 2000 2500 3000
1000 K
2000 K
3000 K
(a)
Radiance(a.u.)
0.0
0.2
0.4
0.6
0.8
1.0
Wavelength(nm)
0 500 1000 1500 2000 2500 3000
1000 K
2000 K
3000 K
(b)
Figure 1: Planck’s law plotted (a) as radiance L(λ) in absolute values and (b) normalized to unit
intensity maximum for three temperatures accessible with common incandescent sources. Region
of wavelengths accessible with spectrometer is demarked in the plot.
Fyzikální praktikum 3
Emissivity
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Wavelength (nm)
0 1000 2000 3000 4000 5000 6000
1600 K
2200 K
2600 K
3000 K
3400 K
Figure 2: Emissivity of tungsten as a function of wavelength and temperature. The data were
taken from [2].
Note, that Stefan–Boltzmann law is obtained by integration over the wavelengths and also over the
relevant solid angle and therefore σ of this value is the only constant in expression for irradiance.
The spectral radiance Lb(λ) for three temperatures accessible with common incandescent
sources is shown in figure 1 (a), the normalized distributions with unit intensity maxima are
shown in figure 1 (b). Obviously, only at temperatures of ≈ 3000 K and higher can the distribution
maximum be registered with a spectrometer range 200 – 1100 nm. Using Wien displacement law,
the intensity maximum lies at 2898 nm for 1000 K, at 1440 nm for 2000 K, at 966 nm for 3000 K
and at 781 nm for 3687 K (melting point of pure tungsten [1]).
Without an absolute intensity measurement, the registration of intensity maximum simplifies
principally the correct determination of temperature from the measured spectra. Anyway, the
relative spectrometer sensitivity must be known.
The deviation of emission of radiation of real bodies from Planck’s law is described by an
emissivity ε
L(λ; T) = ε(λ; T)Lb(λ; T),
where L(λ, T) is the radiance of a real-body surface with thermodynamic temperature T at wavelength
λ and Lb(λ, T) is the radiance of black body with the same temperature and at the same
wavelength. Emissivity of black body is always a unit, otherwise ε < 1.
With a constant emissivity with respect to the wavelength, the radiator is called a grey body.
Otherwise it is called a selective radiator. E.g., in the case of tungsten the emissivity in visible
light weakly depends on the wavelength, decreasing towards the longer wavelengths (see figure 2).
Therefore, the tungsten filament lamp is a slightly more efficient light source than the black
body at the same temperature, although emitting only fraction of the total blackbody radiation
intensity.
Without absolute intensity measurement, the constant emissivity < 1 of tungsten cannot be
determined. The only important influence on the measured spectra may therefore arise from
the spectral dependence of the emissivity. In order to resolve, whether the tungsten may be
handled as grey body (no emissivity correction is needed) or selective radiator (the measurements
must be compared with LW(λ) = εW(λ)Lb(λ)), Planck’s distribution with and without emissivity
correction was calculated for 3000 K and plotted normalized in figure 3 (a). The spectral emissivity
Fyzikální praktikum 4
Radiance(a.u.)
0.0
0.2
0.4
0.6
0.8
1.0
Wavelength (nm)
0 500 1000 1500 2000 2500 3000
blackbody 3000 K
tungsten 3000 K
(a)
Relativeerror(%)
−10.00
−5.00
0.00
5.00
10.00
15.00
20.00
Wavelength (nm)
200 400 600 800 1000
(b)
Figure 3: Comparison of black body (Lb(λ)) and tungsten (ε(λ)Lb(λ)) for temperature 3000 K.
The curves were normalized according to their maxima (a). A relative difference between the
normalized distributions (b).
of tungsten for temperature 3000 K was taken in these calculations (see again figure 2). A relative
difference between the curves
Ln
W(λ) − Ln
b(λ)
Ln
W(λ)
is shown in figure 3 (b) with superscript n denoting normalizated distribution with unit intensity
maximum. The relative differences are generally lower than 20 %.
2 Experimental set-up
Tungsten-filament halogen lamps belong to the the most available and applicable incandescent
light sources operated at high temperatures. The halogen lamp is preferred to standard tungsten
lamp, since the halogen cycle ensures the transport of tungsten atoms from the neighbourhood of
the glass walls back to the filament and thus enables the operation of the lamp at a higher filament
temperature. In our experiment it is probably better to avoid types such as ECO or with a UV-
Fyzikální praktikum 5
Resistivity(10-8Ωm)
0
20
40
60
80
100
120
140
Temperature (K)
0 500 1000 1500 2000 2500 3000 3500 4000
uncorrected for thermal expansion
corrected for thermal expansion
Figure 4: Temperature dependence of tungsten resistivity. The uncorrected curve, showing in
principle the dependence of resistance on the temperature, differs from the true resistivity dependence
due to thermal expansion.
stop filter. The infrared coating on glass bulb of the halogen lamp reflects IR radiation back to the
filament, reducing the necessary power input. On the other end of visible spectrum, the UV-stop
filter doped quartz glass absorbs mostly UV-C and UV-B radiation, but also substantially reduces
UV-A radiation [3]. Although absorption from both coatings in the visible spectrum is probably
negligible, it is better to use a bulb without such modifications. There are also several types of
lamps with the same voltage and wattage, differing in output luminous fluxes and lifetimes. In
this case, the lamp with the highest luminous flux and the lowest lifetime is expected to have the
highest filament temperature at nominal voltage. Osram HLX 64642 24 V/150 W halogen lamp
for slide and overhead projectors with no reflector nor luminaire can be taken as a convenient
source for investigation.
The large advantage of the tungsten lamp is that the temperature of the filament may be
deduced independently from the temperature dependence of tungsten resistance. The temperature
dependence of tungsten resistivity is approximated in range 100 – 3600 K with formula [4]
(T∗) = −0.968 + 19.274 T∗ + 7.826 T∗
2
− 1.8517 T∗
3
+ 0.2079 T∗
4
[T∗] = 1000 K, [ ] = 10−8
Ω m.
The estimated uncertainty of the data used for polynomial expression was about ±3% and the
dependence was corrected for thermal expansion. Since in our case the measured quantity is the
resistance, in which thermal expansion is included, the uncorrected resistivity dependence should
be used instead. However, in the case of tungsten the difference is about 2 % (see figure 4).
Dividing the resistivity dependence with the room temperature value
293 = 5.28 × 10−8
Ω m (293 K),
an inverse function to (T)/ 293 may be fitted with a simple parabolic formula
T = 154.6 + 186.4( / 293) − 1.314( / 293)2
K,
from which the temperature is evaluated when the ratio of measured resistances is substituted
instead of / 293. Note, that due to the large temperature range the formula is only approximate
and does not provide precise values especially at low temperatures.
Fyzikální praktikum 6
The optical spectra in 200 – 1100 nm wavelength range can be measured with small USB
grating spectrometer Avaspec 3648 with 300 lines/mm grating and linear CCD array with 3648
pixels. The fixed slit width is 10 µm. The light from the studied sources can be collected with
an optical fibre. The spectrometer is also equipped with filters providing the removal of higher
spectral orders.
The instrument must be intensity-calibrated with a deuterium and tungsten halogen lamp
with tabulated true spectrum.
Problems
1. Measure the optical emission spectra of heated tungsten filament. Use supplied calibration
data to correct the spectra for a non-uniform spectral sensitivity of the spectrometer.
2. Estimate the blackbody temperature by fitting the corrected spectra with Planck’s law
f(λ) =
A
λ5
1
e
hc
kbT λ
− 1
,
T is thermodynamic temperature, A is scalling coefficient. Use e.g. QtiPlot [5] equipped
with Marquardt-Levenberg algorithm of the least squares method.
3. Compare the obtained blackbody temperature with the temperature derived from the measurement
of filament resistance.
4. Study the effect of spectral dependence of emissivity on the resulting blackbody temperature.
References
[1] Lide D R 2005 CRC Handbook of Chemistry and Physics, Boca Raton: CRC Press.
[2] Flesch P 2006 Light and Light Sources, High-Intensity Discharge Lamps. Berlin: Springer.
[3] Halogen lamps – professional and compact technical knowledge. [On-line] http://www.osram.
com.
[4] White G K and Minges M L 1997 Int. J. Thermophys. 18 1269.
[5] QtiPlot. [On-line] http://www.qtiplot.ro.