Resistivity is usually symbolised with the greek letter 'rho' (ρ) and is measured in ohm metres.

What does the (electrical) resistance (R) of a sample substance depend on?

1. The Type of Material
2. The Cross-Sectional Area (a) of the sample
3. The Length (l) of the sample

Resistance is proportional to length
Resistance is inversely proportional to cross-sectonal area
Combine these two and you get:
Resistance is proportional to length divided by cross-sectonal area.

Resistance = (ρl)/a
ρ = resistivity constant for the material

ie. The resistance of the sample equals the length divided by the cross-sectional area, all multiplied by the resistivity constant for the material.

One way to measure the bulk electrical resistivity of a solid sample is by the so-called Q-method. This method generally gives values correct within a factor of two and good temperature dependencies.1,2 This means that the number you get may not be very accurate, but a plot of resistivity vs temperature will correctly indicate whether the sample is metallic (conducting) or semiconducting (insulating). It is also a good method to use when the sample is not available in the form of a large single crystal (to which wires may be easily attached to measure the resistivity directly).

The basic apparatus is a wire coil connected to a high frequency Q-meter3 (the Q stands for "quality factor"). The coil is placed within a double-walled glass tube which is partially immersed in a liquid nitrogen-filled dewar. From the bottom, a small heating element is also inserted into the glass tube. Pressurized nitrogen gas is directed through the tube to blow cold nitrogen vapor from the surface of the liquid onto the coil. A thermocouple inside the glass tube near the coil measures the temperature. Peripheral equipment includes a temperature contoller for the heater, a temperature readout from the thermocouple, and a Faraday cage to enclose the entire assembly.

A small quantity (40-60 mg) of powdered sample is sieved to obtain a uniform particle size. For example, an average particle diameter of 200 microns is obtained by putting the powder through 250- and 150-micron meshes, in that order, and collecting only the portion that remains on top of the smaller mesh. The sample is diluted with dry chromatographic-grade alumina (Al2O3) to a total volume of about 1 cm3 to minimize contact between the particles. Then the diluted sample is sealed into an evacuated 10 mm o.d. Pyrex glass tube about 5 cm long. A hook or open loop of glass is formed on one end of the tube.

A string and pulley is used to lower and raise the sample tube into and out of the coil. The pulley should be attached above the coil. The string should be tied to the glass loop on the sample tube (perhaps with a slipknot) and extended through the front of the Faraday cage within reach of the operator. When the sample is dropped within the coil, it is possible to measure the quality factor (Qi) of a LC circuit induced by rf skin absorption in the uniformly sized grains. The quality factor is also measured when the sample is out of the coil (Q0). Both measurements are taken about every 10 ° C from room temperature to about the boiling point of nitrogen (77 K; -196 °C)

The change in the quality factor is used to calculate the sample resistivity (rho) with units of ohm-meters according to the equation:

rho = (BVa2)/Δ(1/Q)

where Δ(1/Q)=(1/Qi)-(1/Q0), a is the average particle radius in meters, V is the sample volume (the quotient of the mass of the sample and the density of the compound), and B is a constant calibrated to the particular coil.

A plot of resistivity vs temperature will have negative slope for semiconductors and positive slope for metals. The room temperature resistivity can be extrapolated from the plot, but it may not be very accurate. The method assumes that the sample is diamagnetic or only weakly paramagnetic.

Notes:

1. El-Hanany, U. Rev. Sci. Instrum. 1973, 44, 1067.

2. Shinar, J.; Dehner, B.; Beaudry, B. J.; Peterson, D. T. Physical Review B 1988, 37, 2066.

3. The model we used was a Hewlett-Packard 4342A Q-meter (now produced by Agilent Technologies). This model has specifications: 5 to 1000 Q, 22 kHz to 70 MHz, 0.09 micro-H to 1.2 H, 25 to 470 pico-F. I have seen some recently offered for sale at \$1300-1700. Ours was operated at 34 MHz and had a coil constant (B) equal to 4.84 x 105.

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