Results:

Figure 9 - Frequency Drift of TDO measured at room temp.

Figure 10 - Frequency Drift of TDO measured at 77 Kelvin

Figure 11 - Frequency Drift of TDO measured at 4.5 Kelvin

One of the most pressing sources of systematic error in this experiment is due to frequency drift. Since measuring the susceptibility requires two different measurements; one with the sample in the inductor and one measuring the background frequency. Any drift in frequency during this period will lead to significant error. We measured the frequency drift at several different temperatures, specifically at room temperature (figure 9), at 77 K(figure10) and at 4.5 Kelvin (figure 11). One can easily see that the room temperature drift is quite large, varying by 900 Hz over a 20 minute data run. However when the circuit is cooled to 77K the frequency only drifts by 300 Hz during a 15 minute data run. For this data set the circuit was still cooling down in the first few data points, therefore they do not contribute to our frequency drift measurements. The drift at 4.5 Kelvin is even more impressive, over a 30 minute data run the frequency drifts only 20 Hz. However from the graph one can see anomalous 100 Hz dips in our data; we have been unable to explain these dips in frequency. Since the dip always returns to the original frequency we do not believe that it is frequency drift, but rather some unexplained noise. Since we will be measuring changes much greater than the frequency drift at low temperatures, this effect will not inhibit our measurements.

Before taking measurements at low temperature we took a background frequency vs. temperature measurement. This is shown in figure 12. In the region between 2 and 30 K the frequency can be approximated by four different functions, to find these functions we used LSQ Fit, a least squares fitting program. Between 2 and 5.4 K the background frequency can be approximated by f(T) = 31081459(±16.17) - 352(±4.19)T with a reduced of 1.35; between 5.4 and 7 Kelvin f(T) = 31085455(±129) - 1105(±20) T with a reduced of 0.99; between 7 and 20 K f(T) = 31081458(±7.4) - 523(±0.533) T with a reduced of 1.4; between 20 and 30 f(t) = 31081000(±100) -514(±4)T with a reduced of 3.25.

A. Measurements of Bismuth

We initially attempted to measure the magnetic susceptibility of Bismuth at room temperature. For this experiment we used 99.5% to 100% purity Bismuth samples manufactured by Merck & Co. The samples were massed before being placed within the inductor coil. Their density at 25 C is 9.79g/cm^{3 (8)}.

Four different samples were measured, and our results are tabulated in Table 1. The values have such high error because the frequency drift in this region was quite high (Figure 9) Furthermore, as shown in Table 1, the calculated values of are much higher than the expected value of ⁽⁹⁾. The data ranges from 32 to 447 standard deviations away. We believe this is caused by an induced current in the Bismuth samples. These values are similar to values obtained with Aluminum which is weakly paramagnetic at room temperature; therefore we assume that the diamagnetism is being caused by the induced current on these samples.

B. Measurements of CrO₈S₂ 12H₂O

In order to measure the magnetic susceptibility of CrO₈S₂ 12H₂O we placed 0.0163 grams of the substance within a 0.11±0.005 cm inner diameter, 0.221±0.001 cm length of Teflon tubing; this equals 0.0020 ±0.0002 cm³, which corresponds to a filling factor of 0.046±0.009. The ends were capped with five minute epoxy, and the entire tubing was placed within the core of the inductor. The core was held snugly in place using 0.0138± 0.0001 grams of Teflon tape. The measurements were all made within the Quantum Devices Physical Properties Measurement System. We took two frequency vs. temperature measurements, one from 2-11.3 K, and the other from 13-30K. A graph of these measurements is shown in figure 13. After subtracting the background frequency data we can easily see that our data is not what we expected (Figure 14). As one can see there is actually an increase in frequency after putting the sample into the inductor. This is opposite the effect that would indicate paramagnetism. Furthermore as the temperature increases the frequency also increases. After calculating our magnetic susceptibility vs. temperature one can easily see that the data does not show the expected xm = C/T. We have yet to determine the cause of these readings.

C. Measurements of Niobium

In order to measure the transition the transition temperature of Niobium we placed 0.0083±0.0001g of 0.14±0.005 mm diameter Niobium wire into the core of the inductor. We then measured the frequency and amplitude vs. temperature from 4.9K to 14K. The frequency vs. temperature data is shown in figure 18, and the amplitude vs. temperature data is shown in figure 19. From this data one can easily see that the transition temperature for this sample of Niobium starts at 8.2±0.1K and ends at 8.6±.1K. After subtracting the background frequency from our data one can see that the frequency is constant above the critical temperature, and below the critical temperature the frequency is slightly increasing. Therefore since the susceptibility must be constant once the critical temperature has been reached, the only changing property of the material is the penetration depth. Therefore plotting vs. temperature we are able to observe the change in penetration depth vs. temperature (Figure 20).