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 l at a given temperature. A typical thermal radiation spectrum R(l) 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 lmax 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.
Radiation from this body occurs over a continuous distribution of wavelengths, and as such, it is useful to consider the distribution function R(l) such that R(l)×dl is the intensity of radiation due to wavelengths lying between l and l+dl. The expression for the distribution R(l), for a given temperature T, as empirically proposed by Planck, is given by:
, (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 lmax of the spectrum to the corresponding blackbody temperature. It has been observed that the peak wavelength lmax tends to decrease as the temperature is raised. This suggests the following inverse relationship:
. (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:
. (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:
, (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
. (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:
(6)
The numerical constant s is expressed as:
(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.
3 Apparatus
A block diagram illustrating the experimental apparatus is shown below in Figure 2.
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.
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 mm. 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.
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 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 represent the human error associated with measurement with the monochromator, and vertical error bars result from simple instrument error.
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.
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.
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.
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,
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.
Fig. 8: Wein’s Law: Fitted Data
Generally, the data showed a trend of an inverse relationship, with an exponent of peak wavelength lman 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.
The mean value and standard deviation of the peak wavelength-temperature product was computed, and yielded a value of .
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.