Analysis and Discussion

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# Filtering the data

As one can see from our results, the frequency region above ~ 4 kHz is not simple thermal noise, or 'white noise'. As discussed previously (What is Johnson noise?), this is due to the capacitance of the line that connects the resistor to the amplifier. However, we may still use the approximate expression for Johnson noise independent of frequency in the low frequency part of the spectrum. Therefore, we will consider only the data from 1 kHz to 3 kHz. Four samples of the typical data in this frequency range are shown in Fig. 4.1. Most of the data from the He tank had a marked peak in S at about 2.1 kHz and this peak was also excluded from the analysis, since it was probably due to some other non-thermal source. 296 K 195 K 77 K 4 K Fig. 4.1 From top to bottom: filtered spectra for room temperature, 195 K, 77 K and 4 K.

# Obtaining SV(T)

After selecting the frequency range, the average value and standard deviation of S were calculated for each data set. All the averages and standard deviations are shown in Table 2 at the Appendix. This provides a list of various values of S for each temperature. The uncertainty in each value of S was taken to be the standard deviation from the mean. All the values for a given temperature were weight-averaged to give the final S for that temperature and its uncertainty. In Figs. 4.2 and 4.3 are shown the final results for SV(T) for the first and second day of the experiment, respectively, as well as the linear fit to the data.

Clearly, SV(T) behaves linearly, in agreement with Nyquist's prediction. The parameters of the fits are shown in Table 1 and the r parameter (see Tab. 1), that measures the fit quality, proves quantitatively that the linear behavior is a good approximation. Fig. 4.2 Linear fit for the spectral density of the voltage S in V2/Hz as a function of temperature for the data of the first day. Fig. 4.3 Linear fit for the spectral density of the voltage S in V2/Hz as a function of temperature for the data of the second day.
 a b r1 $\chi^2_\text{red}$ Fig. 4.2 1.86(19) x 10-10 1.75(33) x 10-9 0.9998 0.033 Fig. 4.3 1.72(21) x 10-10 3.57(39) x 10-9 0.984 1.67 Tab. 1 Linear regression parameters for Figs. 4.2 and 4.3 in the form y(x) = ax + b

# Extracting Boltzmann's constant

From the previously derived relation for SV(T), we find from Fig. 4.2

(1)
\begin{align} k_\text{B} = \frac{a}{4RG^2} = 1.86(19) \times 10^{-23} \; \text{J} \; \text{K}^{-1} \end{align}

and from Fig. 4.3,

(2)
\begin{align} k_\text{B} = \frac{a}{4RG^2} = 1.72(21) \times 10^{-23} \; \text{J} \; \text{K}^{-1} \end{align}

where G is the amplifier gain (5,000), a is read from Table 1 and R is the resistor's resistance ($100 k\Omega$). The weighted average of the two results is

(3)
\begin{align} k_\text{B} = 1.80(14) \times 10^{-23} \; \text{J} \; \text{K}^{-1} \end{align}

This is not in perfect agreement with the reference value of Boltzmann constant, 1.381 x 10-23 J/K, being off by a factor of 30%, though it is roughly close to it. One of the possible sources of error is the actual value of the gain; it seems to be higher than the nominal value by a factor of about 5%. If we correct the gain by this factor, then kB must be corrected by a factor of 0.91 and Eq. (3) would give kB = 1.63(14) x 10-23 J/K, that shows a good agreement with the reference value in less than two standard-deviations. We are therefore led to conclude that Nyquist's relation is approximately correct. However we note that this result is not very precise since it furnishes only three significant figures for kB.

It seems that there is some systematic error in the experiment, since our values for Boltzmann constant are all above the reference value. In a similar apparatus, K L McKirahan  also obtained a high value: $k_\text{B} = 1.88 \times 10^{-23}\; \text{J}/\text{K}$ “with large errors” not quoted explicitly. However Kittel et al.  were able to obtain $k_\text{B} = 1.369 \times 10^{-23}\; \text{J}/\text{K}$ in a very good agreement with the reference value, in spite of the fact that their value also has “large uncertainties” not calculated explicitly.

## Appendix: average values of SV

T (K) Day $\langle S_V \rangle$ $\sigma_S$
296 1 5.63 1.25
5.59 1.29
5.67 1.27
5.66 1.30
2 5.66 1.27
5.69 1.30
5.67 1.28
5.67 1.28
5.67 1.29
T (K) Day $\langle S_V \rangle$ $\sigma_S$
195 2 3.52 8.02 -1
3.13 7.18 -1
3.08 6.84 -1
77.4 1 1.64 3.83 -1
1.69 4.00 -1
2 2.18 5.41 -1
2.13 7.26 -1
2.01 5.22 -1
T (K) Day $\langle S_V \rangle$ $\sigma_S$
4.22 1 2.40 -1 5.61 -2
2.60 -1 6.62 -2
2.61 -1 6.31 -2
2.59 -1 6.99 -2
2 2.96 -1 7.50 -2
2.88 -9 7.07 -2

Tab. 2: list of the average values of the spectral density S in the region 1kHz to 3kHz for each day and each temperature followed by the standard deviation from the mean. All values on this table are in 10-8 V2/Hz. A superscript n right to a number is a shorthand for 10n.

Bibliography
1. K L McKirahan. unpublished report (2005), available here
2. P Kittel, W R Hackleman and R J Donnelly. Am. J. Phys. 46, 1 (1978)
page revision: 169, last edited: 09 Apr 2007 02:55