Measurement of radiated blackbody spectra and comparison
with predictions of the Stefan and Wein laws
Kirk Lamont and Truitt Wiensz
March 9, 2001
1
Objectives
There are four main objectives in this experiment:
1. Measurement of the spectrum radiated from a thermal cavity as a function of
temperature.
2. Comparison of peak wavelength temperature dependence with Wein’s displacement
law.
3. Measurement of output power from a thermal cavity as a function of temperature.
4. Comparison of output power temperature dependence with Stefan’s Law.
2
Theory
Thermal radiation is the name given to electromagnetic radiation emitted by all objects as a
consequence of their finite temperature. Radiation from a thermal body forms a continuous
spectrum, with the spectral shape and maximum being dependent on the body’s temperature
and surface characteristics of the body.
The radiancy, R, of a thermal body is defined as the power emitted per unit area, and is a
function of wavelength  at a given temperature. A typical thermal radiation spectrum R()
is shown below in Figure 1.
Fig. 1: Typical thermal radiation spectrum
A body radiates with a unique spectrum at each distinct temperature. Several spectral
characteristics are of interest for this experiment, notably the peak wavelength max and the
radiated power (the integral under the curve). Wein’s displacement law and Stefan’s law,
respectively, make quantitative predictions of the temperature dependence of these
characteristics.
In order to minimize contributions of factors other than temperature on the radiated
spectra, an ideal radiator is introduced.
2.1
Blackbody Radiation
The ideal blackbody is described as follows: its surface is completely black, and as such it
absorbs all of the radiation that strikes it. As well, all radiation emitted from a blackbody
leaves through an infinitesimally small hole in the surface. Thus this hole is actually the
radiator, and not the body itself. In this configuration, radiation from outside that enters the
hole gets lost inside the box, and thus has a negligible probability of reemergence. It is
under these assumptions that the blackbody is said to absorb all incident radiation.
2
Radiation from this body occurs over a continuous distribution of wavelengths, and as such,
it is useful to consider the distribution function R() such that R()d is the intensity of
radiation due to wavelengths lying between  and +d. The expression for the distribution
R(), for a given temperature T, as empirically proposed by Planck, is given by:
R ( ) 
2hc 2

5
1
e
hc kT
1
,
(1)
Here h is Planck’s constant, c is the speed of light, k is the Boltzmann constant. This
relationship effectively describes the functional dependence shown in Figure 1 at the
specified temperature. It is assumed in this experiment that the radiating body behaves as a
perfect blackbody.
Having isolated the effects of the temperature on a body’s radiated spectra, it is now possible
to consider the temperature dependence of several spectra characteristics. Of interest here
are the peak wavelength of the spectra and the observed intensity as a function of
temperature.
2.2
Wein’s Displacement Law
Wein’s law relates the peak wavelength max of the spectrum to the corresponding blackbody
temperature. It has been observed that the peak wavelength max tends to decrease as the
temperature is raised. This suggests the following inverse relationship:
 max 
1
.
T
(2)
Based on the measurements of many spectra over a wide temperature range, the product of
peak wavelength and temperature has been found to be approximately 2.89810-3 mK.
Thus Wein’s displacement law may be stated as:
 max T  2.898  10 3 m  K .
(3)
A less heuristic approach is to calculate the wavelength in the Planck radiation formula
(equation 1) at which the distribution function is maximum, as:
 1 hc

dR
e hc kT
5
1
2
 2hc  5 2

0
2
hc kT
d
,
6 e hc kT  1
1
   kT e
  max T  2.898 10 3 m  K

3

(4)
giving the same result as previously obtained.
This relationship may be examined
experimentally by determining the wavelength at which the emitted intensity reaches a
maximum for a set temperature, over a large range of temperatures.
2.3
Stefan’s Law
It has been observed experimentally that the total intensity radiated over all wavelengths
tends to increase as the temperature of the radiating body is increased. This total intensity
corresponds to the integral of the radiancy of the body over all wavelengths, or the integral
of the curve shown in Figure 1. Stefan’s law provides a relationship between the total
radiated intensity I, and the blackbody temperature T.
From measurement, it has been observed that intensity depends on temperature as
I  T 4 .
(5)
Once again, a more mathematical approach may be taken by calculating the integral of the
Planck radiation formula (1) in order to determine the total radiated intensity as a function of
temperature. This is evaluated as follows:

I   R( )d  2hc

2
0
1
1
5
hc kT

0
e
1
d



2h
3
d
c 2 0 e h kT  1
2h  kT 
 
c2  h 
(6)
4
x3
0 e x  1 dx
 T 4
The numerical constant  is expressed as:

2 5 k 4 h 3
 5.671  10 8 W  m  2  K  4
2
15c
(7)
The expression (5) may be examined experimentally by taking output intensity measurements
as a function of temperature, and examining logarithmic plots to determine the experimental
exponent of temperature.
4
3
Apparatus
A block diagram illustrating the experimental apparatus is shown below in Figure 2.
Power corresponding
to a temperature
Temp. Control
System
Blackbody
Radiation
spectrum
emitted
Voltage proportional
to intesity of wavelength
Thermopile
Detector
Monochromator
Feedback to give
proper temperature
Intensity at the
specified wavelength
Preamp
DMM
Amplifed voltage
Fig. 2: Experimental Apparatus
The blackbody radiator used in this experiment is a Graseby IR-508 Blackbody.
Temperature of the blackbody is controlled with an IR-201 Digital Temperature Controller.
Selection of wavelengths is performed with an Oriel 77250 Grating Monochromator, and
intensity measurements from the thermal cavity are then taken with an Oriel Thermopile
Detector.
3.1
IR-508 Blackbody and IR-201 Digital Temperature Controller
The IR-201 DTC is a power source and PID (proportional-integral-differential) control
system for the blackbody. A temperature value is entered into the DTC, which corresponds
to an electrical current output to the blackbody. This current causes resistive heating in the
blackbody, which is measured by two thermocouples inside the IR-508. This temperature
signal is sent back to the DTC and displayed to show the actual temperature to within 0.1%
error. The heaters are constructed such that the cavity area is heated uniformly, giving the
IR-508 its blackbody characteristics. A small aperture allows emission of the blackbody
radiation, which is then output to the monochromator. The interior configuration of the
blackbody can be seen in Figure 3.
5
Figure 3: Interior Configuration of Blackbody
3.2
Oriel 77250 Grating Monochromator
The monochromator is composed of two flat mirrors, a concave mirror, and a diffraction
grating. The path of the radiation through the monochromator is such that the diffraction
grating splits the continuous spectrum into orders of light, each individually satisfying the
grating equation, as illustrated in Figure 4. The grating used is specified to have 300
lines/mm and a range of 1-4 m. The grating is blazed at 2 microns, corresponding to peak
grating efficiency occurring at 2 microns, within the range of wavelengths used in this
experiment. The grating breaks up the radiation into its constituent wavelengths at angular
separations satisfying the diffraction condition, allowing for measurement of the wavelength
output, based on separation between diffraction peaks.
A display window on the monochromator provides a calibrated measure of the wavelength
measured, by which is performed by measuring the angular separation between successive
orders of spectral maxima. Due to the type of grating used in the monochromator, the
actual wavelength being measured is 4 times the reading shown on the display window.
6
Figure 4: Monochromator Optics
3.3
Oriel Thermopile Detector
The thermopile detector consists of a thermocouple array, which is used to measure
intensities by detecting a rise in temperature due to the incident thermal radiation. An
output voltage is proportional to the intensity, which is output to a preamp. This is used to
increase the signal to a reasonable level, which is then monitored with a digital multimeter.
4
Procedure
4.1
Monochromator Calibration
Calibration of the monochromator was performed with a red neon laser of wavelength 632.8
nm. The laser was input to the monochromator, and a photomultiplier tube was used to
precisely measure the spacing between the diffraction maxima locations by intensity
measurements. The monochromator gave a corresponding wavelength measurement of 632
 2 nm for the red laser. The first-order maximum of the red laser was taken as the baseline
for any following measurements. This angular position on the diffraction grating was then
taken to correspond to 632 nm, or a reading of 158 on the monochromator window.
4.2
Measurement of Peak Wavelengths
Peak wavelength was measured as a function of temperature in the first section of this
experiment.
The digital temperature controller was used to regulate the blackbody
7
temperature, which output a power spectrum to the monochromator. The thermopile
detector was then used to measure the output intensity. The thermopile detector voltage
was then fed into a preamp, allowing intensity measurements with a digital voltmeter.
The peak wavelength of the spectrum was measured at a given temperature by finding the
wavelength at which the measured intensity reached a maximum value. Peak wavelength
was measured at temperatures from 625 C to 1050 C in 25 increments.
4.3
Radiation Intensity - Temperature Dependence
In the second section of this experiment, the total intensity from the blackbody, radiated
over all wavelengths, was measured as a function of temperature. The digital temperature
controller was used to regulate the blackbody, which was in turn directly coupled to the
thermopile detector.
The thermopile signal was output to a preamp, which allowed
measurement of the signal with a digital multimeter.
Since only the functional dependence of radiated intensity on temperature was to be
examined in this experiment, the amplified thermopile signal was taken as a measure of
normalized intensity. An experimental value for Stefan’s constant could be obtained if the
calibration curve of the thermocouple detector was known. Total intensity radiated from the
blackbody was measured at temperatures from 80 to 1040 C, in increments of 20.
5
Observations
5.1
Monochromator Defects
A large intensity peak was measured on the monochromator. Several characteristics of this
peak have led us to believe that it is a defect on the diffraction grating. First, the peak
occurred at the same angular location, regardless of the input signal type, indicating that this
does not depend on the input spectrum. Also, the width of the peak seemed independent of
both the input spectrum and the temperature, leading us to the conclusion that this spike is a
defect, likely a scratch on the grating caused by handling.
A profile of the angular
dependence of the intensity response is shown in Figure 5 below. Horizontal error bars
8
represent the human error associated with measurement with the monochromator, and
vertical error bars result from simple instrument error.
Spike Spectral Profile
8.0
Intensity (dimensionless)
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0
550
570
590
610
630
650
670
690
710
730
750
Wavelength (nm)
Fig. 5: Spectral/Angular Distribution of Grating Defect
The observed spectral distribution in Fig. 5 shows a somewhat Gaussian profile, indicating
that the defect likely resulted from a sharp impact on the grating surface.
5.2
Peak Wavelength – Temperature Dependence
Some difficulty was encountered in measuring the peak wavelength, as there seemed to be
no sharp maximum on the spectrum, as was expected. A more full description of the
problems encountered is given in the Discussion. Several “trends” (of decreasing peak
wavelength with increasing temperature) were noted as the temperature was varied. The
best data obtained in this section may be seen in Figure 6 below. Errors in wavelength are
similar to those described above in Section 5.1, and the temperature error bars reflect the
instrument accuracy, and are too small to be displayed.
9
Peak Wavelegnth -vs- Temperature
4000
3500
Peak Wavelength (nm)
3000
2500
2000
1500
1000
500
0
1000
1050
1100
1150
1200
1250
1300
1350
Temperature (K)
Fig. 6: Peak wavelength as a function of temperature
Generally, the data followed the trend that as the blackbody temperature was increased, the
peak wavelength tended to decrease.
5.3
Radiation Intensity Temperature Dependence
Total intensity radiated over all wavelengths was measured as a function of temperature in
this section. The data obtained may be seen in Figure 7 below. Temperature errors are due
solely to instrument error, and the intensity error bars on this plot are too small to be seen.
10
Stefan's Law
180.0
160.0
Intensity (dimensionless)
140.0
120.0
100.0
80.0
60.0
40.0
20.0
0.0
0
100
200
300
400
500
600
700
800
900
1000
Temperature (K)
Fig. 7: Output intensity temperature dependence
The data shown in Figure 7 clearly show that the intensity measured increased with
temperature at a polynomial rate.
11
6
Analysis
6.1
Wein’s Law: Temperature Dependence of Peak Wavelength
Wein’s law makes two predictions:
1. A blackbody’s peak wavelength and temperature are inversely proportional.
2. The temperature-peak wavelength product is constant,  max T  2.898  10 3 m  K
The first prediction would result in a power of -1 on a plot of the peak wavelength versus
the temperature. A least-squares power fit of the best data obtained in this section is shown
below in Figure 8.
Peak Wavelegnth -vs- Temperature
4000
3500
Peak Wavelength (nm)
3000
y = 433529x-0.6984
R2 = 0.9615
2500
2000
1500
1000
500
0
1000
1050
1100
1150
1200
1250
1300
1350
Temperature (K)
Fig. 8: Wein’s Law: Fitted Data
Generally, the data showed a trend of an inverse relationship, with an exponent of peak
wavelength man being proportional to T-0.693. А full tabulation of the data obtained in this
section may be seen in Appendix A.1. As mentioned in Section 5.2, several trends of peak
wavelength versus temperature were measured, with the best results being displayed on the
fit shown above in Fig. 8.
12
The mean value and standard deviation of the peak wavelength-temperature product was
computed, and yielded a value of  max T  3.28  0.5810 3 m  K .
6.2
Stefan’s Law: Temperature Dependence of Radiation Intensity
Stefan’s law predicts that the total intensity radiated over all wavelengths is proportional to
temperature to the fourth power.
By examining a log-log plot of intensity versus
temperature, the exponent in this relationship may be obtained. This log-log plot may be
seen in Figure 9.
ln Intensity -vs- ln Temperature
7.0000
6.0000
y = 3.5059x - 18.601
R2 = 0.9989
ln(Intensity)
5.0000
4.0000
3.0000
2.0000
1.0000
0.0000
6.1000
6.3000
6.5000
6.7000
6.9000
7.1000
7.3000
ln(Temperature)
Fig. 9: Stefan’s Law: Determination of temperature exponent
The slope of the graph in Figure 9 was calculated to be 3.51  0.04, corresponding to a
relationship of the form I  T 3.506 .
13
7
Discussion
7.1
Wein’s Law: Temperature Dependence of Peak Wavelength
Peak spectral wavelength was measured as a function of temperature in this experiment.
Assuming that peak wavelength is inversely proportional to temperature, a power fit to the
experimental data is expected to give a dependence of wavelength on temperature to the
power -1. The value of the power found by a least-squares fit was -0.693. Generally, the
data followed the trend of an inverse relationship. This inverse relationship gave a mean
value of their product of  max T  3.28  0.5810 3 m  K , which agrees within
experimental error to the accepted constant of 2.89810-3 mK. The experimental error was
relatively high due to difficulties encountered in determining the peak wavelength of the
spectrum. There were several reasons for this. First, the thermopile detector provided poor
differentiation between peak signals and background intensities. The ratio of peak intensity
signal to the background readings, or the signal-to-noise ratio, was typically on the order of
1.5. This caused great difficulty in determining where the middle, or peak, of the intensity
band was. A more accurate thermopile detector would help to eliminate these problems.
Also, a large spike in the data blocked the first order wavelength. It is believed that this
spike was due to a defect, possibly a scratch, in the grating, as is discussed in Section 5.1.
This grating should be inspected before further experiments are performed. As well, some
residual heating of the blackbody housing and the thermopile detector cause heating effects,
which are not reflected in the DTC temperature display.
7.2
Stefan’s Law: Temperature Dependence of Radiation Intensity
Stefan’s law relates the intensity of a blackbody to the temperature through a proportionality
of temperature to the fourth power. Therefore, a log-log plot of intensity and temperature
for the blackbody was made to find this exponent. The slope of this plot was expected by
Stefan’s law to be four, where it experimentally came out to be 3.51  0.04. Generally, the
data follows the expected trend of a fourth order power. The Stefan-Boltzmann constant
could not be found, since the normalized intensity measurements were voltage readings,
which were directly proportional to the intensity. Direct experimental measurement of the
Stefan-Boltzmann constant requires that intensity measurements be taken.
14
In this
experiment, the proper calibration of the voltage to intensity was not available. Therefore
the proportionality of the intensity and the temperature could be found. A possible source
of error is the fact that Stefan’s law requires use of an ideal blackbody. We did not have
such a device, resulting in the data collected having some deviation from the expected
values.
15
Appendices
A.1
Peak Wavelength as a function of Temperature – Data (Wien’s Law)
Wein's Law
Temp
Temp
Mono
Wavelength Constant
(celcius) (K)
Reading (nm)
+/-1K
(+/-1)
(+/-4nm)
625
898
1154
2516 0.00226
650
923
1235
2840 0.00262
675
948
1193
2672 0.00253
700
973
1188
2652 0.00258
725
998
1173
2592 0.00259
750
1023
1149
2496 0.00255
775
1048
1355
3320 0.00348
800
1073
1347
3288 0.00353
825
1098
1340
3260 0.00358
850
1123
1336
3244 0.00364
875
1148
1332
3228 0.00371
900
1173
1304
3116 0.00366
925
1198
1292
3068 0.00368
950
1223
1281
3024 0.00370
975
1248
1272
2988 0.00373
1000
1273
1265
2960 0.00377
1025
1298
1252
2908 0.00377
1050
1323
1227
2808 0.00371
Mean:
0.00328
Standard Deviation:
0.00056
16
A.2
Total Radiated Intensity as a function of Temperature - Data (Stefan’s Law)
Stephan's Law
Temp
Temp
Natural Log of Error in Voltage
Natural Log of Error in
(celcius) (K)
Temp
ln(temp) (mV)
Voltage
ln(V)
+/-5K
(+/-0.3mV)
80
353
5.8665 0.0142
15.9
2.7663 0.0189
100
373
5.9216 0.0134
16.6
2.8094 0.0181
120
393
5.9738 0.0127
17.2
2.8449 0.0174
140
413
6.0234 0.0121
18.2
2.9014 0.0165
160
433
6.0707 0.0115
19.4
2.9653 0.0155
180
453
6.1159 0.0110
21.2
3.0540 0.0142
200
473
6.1591 0.0106
23.5
3.1570 0.0128
220
493
6.2005 0.0101
26.0
3.2581 0.0115
240
513
6.2403 0.0097
28.5
3.3499 0.0105
260
533
6.2785 0.0094
32.0
3.4657 0.0094
280
553
6.3154 0.0090
35.3
3.5639 0.0085
300
573
6.3509 0.0087
39.3
3.6712 0.0076
320
593
6.3852 0.0084
43.7
3.7773 0.0069
340
613
6.4184 0.0082
48.3
3.8774 0.0062
360
633
6.4505 0.0079
54.0
3.9890 0.0056
380
653
6.4816 0.0077
59.7
4.0893 0.0050
400
673
6.5117 0.0074
66.1
4.1912 0.0045
420
693
6.5410 0.0072
73.3
4.2946 0.0041
440
713
6.5695 0.0070
81.3
4.3981 0.0037
460
733
6.5971 0.0068
89.2
4.4909 0.0034
480
753
6.6241 0.0066
98.4
4.5890 0.0030
500
773
6.6503 0.0065
107.8
4.6803 0.0028
520
793
6.6758 0.0063
118.3
4.7732 0.0025
540
813
6.7007 0.0062
129.5
4.8637 0.0023
560
833
6.7250 0.0060
141.3
4.9509 0.0021
580
853
6.7488 0.0059
154.2
5.0383 0.0019
600
873
6.7719 0.0057
167.3
5.1198 0.0018
620
893
6.7946 0.0056
181.2
5.1996 0.0017
640
913
6.8167 0.0055
196.2
5.2791 0.0015
660
933
6.8384 0.0054
212.3
5.3580 0.0014
680
953
6.8596 0.0052
229.0
5.4337 0.0013
700
973
6.8804 0.0051
247.9
5.5130 0.0012
720
993
6.9007 0.0050
266.6
5.5857 0.0011
740
1013
6.9207 0.0049
286.4
5.6574 0.0010
760
1033
6.9402 0.0048
308.6
5.7320 0.0010
780
1053
6.9594 0.0047
331.0
5.8021 0.0009
800
1073
6.9782 0.0047
355.6
5.8738 0.0008
820
1093
6.9967 0.0046
378.1
5.9352 0.0008
840
1113
7.0148 0.0045
403.7
6.0007 0.0007
860
1133
7.0326 0.0044
421.3
6.0433 0.0007
880
1153
7.0501 0.0043
441.9
6.0911 0.0007
900
1173
7.0673 0.0043
480.5
6.1748 0.0006
920
1193
7.0842 0.0042
519.6
6.2531 0.0006
17
A.3
Equipment Listing
Instrument
Quantity
Graseby IR-508 Blackbody
Graseby Model 201 Digital Temperature
Controller and manual.
Oriel 77250 grating monochromator
Oriel Thermopile Detector
Oriel Preamp
Hewlett-Packard 3438A digital multimeter
18
1
1
1
1
1
1