Experimental Arrangement and Procedure

Our experimental apparatus consisted of two $100 \; \text{k}\Omega$ resistors, an amplifier and filter (Stanford Research SR560) and a computer. The resistors were connected to the amplifier via coaxial cables and were immersed in four different media: liquid helium (4.22 K), liquid nitrogen (77.36 K), carbon dioxide (dry ice) (195 K) and air (296 K). The temperatures of liquid He, N and CO2 were taken to be the boiling point temperatures, while the room temperature was measured with a thermometer to be 23 oC.

The major purpose of the amplifier is to magnify the thermal noise. It was connected, through an appropriate card, to the computer running NI LabVIEW. The software measured 218 samples of the voltage data from the amplifier at a frequency of 105 Hz and calculated the spectral density via a Fast Fourier Transform1. For better results, the final spectral density analyzed was an average over 20 different spectral densities obtained in this way, the average performed by the software.

Schematics of the apparatus
Fig. 2.1 Simple schematic of the apparatus. The capacitor here represents
the capacitance of the cable connecting the resistor to the amplifier.

The amplifier gain was set to 5,000. For the purpose of obtaining a rough estimate of Boltzmann's constant in this experiment, one could safely operate with a frequency cut-off at about 5 kHz. Assuming that the coaxial cable has a capacitance per length of order ~ 50 pF/m, one can estimate that the frequency at which the cable's capacitance becomes relevant, $f = 1/2\pi RC$, is of order 104. Hence, to study the capacitance effect we set up the low pass filter with a cut-off frequency of 100 kHz and set up the software to calculate the spectral density up to 50 kHz.

Two resistors of same resistance were used for a faster data acquisition. In order to check that the two different resistors and their respective coaxial cables could be used as datasets for a single $100 \; k \Omega$ resistor. That is, since the different temperature spectra could be combined in the same linear graph for SV(T), the following were measured and compared for both:

  1. the spectral density of the noise of the coaxial cables connected to an effective ground (a much higher resistance than the amplifier impedance)
  2. the spectral density of the resistors at room temperature

1 and 2 were no more different than what one would expect from random errors that would be present in two different measurements of the same resistor, so we may safely combine the results of the two resistors as one.

Since we are interested in the voltage spectral density, it is important to check if the amplifier gain remains constant as the frequency of the input signal changes. To this purpose, we fed the amplifier with a signal from a function generator and using an oscilloscope we took note of the peak-to-peak reading of the output of the amplifier. The result of this test is presented in Fig. 3.2 in our results section. As can be seen from the graph, in the region of 1kHz to 3kHz the gain is roughly constant and is about 5% higher than the nominal value (the value set up in the amplifier)2.

We also checked that the noise from the computer card that the amplifier is connected is negligible in comparison to the amplitude of the voltage produced by the amplifier.

The experiment was completed in two days, though CO2 was available only on the second day. For each day, two or three spectral densities (the average of ten runs mentioned before) were measured for each temperature.

Notes regarding the heat baths

This experiment is very sensitive to the quality of the heat bath, so it is difficult to overemphasize here the care that must be taken in preparing it.

To actually use CO2 , we immersed dry ice in acetone in the standard technique to promote a better heat transfer of the dry ice vapor to the resistor. The dry ice/acetone slush was kept inside of an open (i.e., in contact with the atmosphere) vessel.

Helium and nitrogen were used inside large thermally insulated containers (about 1 m in diameter). Previous attempts at using a vessel to store nitrogen resulted in spectral densities that drifted too much in time, possibly due to bubbles in the medium. It should go without saying that in all cases it is necessary for a large amount of material (in comparison to the size of the resistor) to create an effective heat bath, and to wait until the system (resistor and medium) attains thermal equilibrium. By observing the spectral density behavior as a function of time we decided that equilibrium was achieved when the spectral density was approximately time-independent.

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