The procedure that follows provides a step by step method for determining the critical temperature of the superconductor sample contained within the superconducting susceptibility probe included in the Superconducting Magnetic Susceptibility Kit. The experimenter will simultaneously measure the magnetic susceptibility of the sample as well. This probe consists of a 0.5 inch long coil of wire (approximately 400 turns) around a 0.5 inch diameter superconductor rod. A thermocouple is also attached for temperature measurement.
This experiment is based on the expulsion of a magnetic field by the superconducting sample (the Meissner Effect). A current introduced into the coil will generate a magnetic field. When cooled below the Critical Temperature, the sample expels the induced field which can be seen by a distinct change in the inductance of the coil. Analysis of the data collected in this procedure reveals a sharp transition point at the Critical Temperature, T_{c}, of the sample, and can also be used to compute the magnetic susceptibility of the superconductor.
The resistance of the coil (R_{L}) is temperature dependent, therefore we now need to determine the value of R_{L} as a function of temperature. The following procedure accomplishes this task:
Precautions:
The critical temperature (T_{c}) at which the superconducting transition takes place can be seen clearly by plotting the inductance of the coil vs. temperature. An example plot is given in figure 12 on the following page. The inductance of the coil (wL ) can be computed using the following equation:
wL = ((V_{L}/I)^{2}-R_{L}^{2})^{1/2}
Where V_{L} is the voltage of inductance (the ac voltage measured across the black coil leads), and R_{L} is the resistance of the coil. Since ceramic superconductors do not have sharply defined critical temperatures (the transition is visible over a range of 5K in figure 12), the midpoint of the range is usually taken as T_{c}.
An ideal superconductor screens the B-field completely at B-fields lower than the critical field. This makes a superconductor perfectly diamagnetic and thus the magnetic susceptibility (X) is equal to -1.
If the coil contains a sample with X not equal to -1, the magnetic flux through the coil will change, resulting in a change in the inductance of the coil. The inductance of a coil measured in a medium of susceptibility X is given by:
L = L_{o}(1+X)
Where L_{o} is the inductance of the coil in a vacuum. This equation holds for any sample provided that the sample occupies all the space in which the coil produces a field. Since our sample occupies most of the volume (sample diameter = 1.26 cm, coil diameter = 1.36cm) we can approximate the inductance (L) by adjusting the above equation to account for this difference:
L = L_{o}(1+fX)
Where f is the fraction of the coil volume occupied by the sample.
It follows that X is given by the equation:
X = (L/L_{o} - 1)/f
Calculate X with the data collected in the previous exercises and plot X vs. T. You will notice that the plot yields the same type of graph as wL, as well as the same critical temperature.
You may notice that although the superconducting transition is clearly visible, the susceptibility in the superconducting state is not exactly -1. This is due in part to geometrical corrections which were neglected in our equation for X, which can result in an error in susceptibility. In addition, polycrystalline samples of YBa_{2}Cu_{3}O_{7} contain grains of material with large lower critical fields which undergo a transition at 90 Kelvin, but material between the grains has smaller lower critical fields and not all of this material will become superconducting in the apparatus. This can be verified by noticing that the fraction of the sample which screens the B-field is smaller in an applied field of 30 Gauss than in an applied field of 3 Gauss, and the transition temperature is depressed in the stronger field.