Abstract:

    In this experiment a Tunnel Diode Oscillator was created for use in magnetic susceptibility measurements. Measurements of the susceptibility of Bismuth at room temperature and the change in susceptibility of CrO₈S₂ 12H₂O were attempted. The susceptibility of Bismuth could not be measured due to induced eddy currents on the sample, creating a diamagnetic effect. Furthermore the measurements of CrO₈S₂ 12H₂O were inconclusive because the effect observed was opposite the effect expected. The Tunnel Diode oscillator apparatus was also used to measure the nuclear magnetic resonance (NMR) of Hydrogen and Fluorine, however no resonance was observed due to large noise in our apparatus in the presence of a magnetic field. This apparatus was successfully used to measure the transition temperature of Niobium. This was measured to begins at 8.2±0.1K and ends at 8.6±.1K, furthermore a change in skin depth was also observed.

Theory:

Because we propose to study the magnetic susceptibility of various materials, it is necessary to include a brief description of magnetic susceptibility. While many atoms have a permanent non-zero magnetic moment, the net magnetic moment of an assembly of atoms is generally zero due to the random orientation of atoms. However when materials are brought within an external magnetic field H the magnetic moment can be changed. The effect of H can expressed as M = cH, where M = ·m/V. Here M is the magnetization (net dipole moment per unit volume), and V is the volume of the system, and c is the magnetic susceptibility of the material. Therefore the external magnetic field exerts a torque on the atoms, forcing the atomic moments to align with H. However temperature will serve to counteract the alignment because of Brownian motion, therefore the degree of alignment depends on temperature and magnetic field. The alignment of the dipoles with H gives a positive contribution to c which is called paramagnetic susceptibility. However the external field also induces a change in the orbital motion of the electrons, this produces a magnetic field opposite the external magnetic field. This gives a negative contribution to c, and is called diamagnetic susceptibility. This effect is independent of temperature, and will be present in all atoms. Therefore materials may be classified as paramagnets if the total magnetic susceptibility c is greater than zero, and diamagnets if c is less than zero.

Paramagnets can be described by Curie's law which states ⁽¹⁾

(1)

where M is the magnetization, C is the Curie constant, and B is the magnetic field, and T is the temperature. Furthermore

(2)

Therefore we will measure the Curie constant by measuring the change in vs. temperature.

A. Tunnel Diode

Since the tunnel diode is essential to our experiment, a more in-depth investigation of the properties and physics surrounding tunnel diodes is required. Tunnel diodes are different from normal diodes due to their extreme doping. Normal diodes are lightly doped with one impurity atom per ten million semiconductor atoms, however the tunnel diode is doped to the extent of 1000 times as much, with one-thousand impurities per ten-million semiconductor atoms. This heavy doping causes an extremely narrow depletion zone. Figure 1a shows the Fermi levels of the P-type and N-type materials with no bias voltage applied. In this condition equal numbers of electrons from the N-type materials tunnel into the P-type as into the N-type. However, if a negative bias is applied (Figure 1b) electrons tunnel exclusively into the N type, thereby causing a sharp negative I/V curve. When a small positive bias is applied, tunneling occurs from the N-type to the P-type (Figure 1c). This tunneling current continues until the Energy level of the N-type is completely above the free states in the valence band of the P-type. After reaching this maximum tunneling current, increase in forward bias pushes the N-type into the forbidden zone, corresponding to a decrease in our current (Figure 1e). This results in a negative slope resistance, the property that we will use in our circuit. The decrease in tunneling current will continue until some minimum value is reached. After this occurs electrons will be able to jump to the conduction band of the P-type using the normal thermal excitation (Figure 1f). A graph Current vs. Voltage for the BD-4 tunnel diode which we used for this experiment is shown in figure 2.

B. Tunnel Diode Oscillator

Clover and Wolf ⁽²⁾ have shown that for small changes in inductances in a tunnel diode oscillator circuit, ÆL/L<<1, the chance in resonance frequency is given by

(3)

Where Æw/w is the relative change in frequency, F is the filling factor of the sample in the coil, and is the real part of the ac magnetic susceptibility. A positive change in Æw/w indicated diamagnetism, and a negative change means paramagnetism.

By measuring the change in frequency, we will be able to measure the change in magnetic susceptibility. We will do this by first measuring the resonant frequency of the tank circuit with nothing in the core of the inductor, and then measure the resonant frequency again with our sample placed into the core of the inductor. By measuring these two frequencies we will be able to measure the real part of the magnetic susceptibility.

C. Nuclear Magnetic Resonance

When a material is placed within a magnetic field, an energy difference in the nuclear spin orientations of the nucleus is induced. If the material is then irradiated with energy equal to the difference between the two energy levels, flipping between the two levels will be induced.⁽³⁾ This energy is dependent on the magnetic field B₀ around the nucleus

(4)

where h is Planck's constant. Using the Bohr condition () the frequency of the nuclear transition is

(5)

where is the gyromagnetic ratio and is a constant for any particular type of nucleus. In this experiment our frequency is constant, therefore to observe NMR we changed the magnetic field in which the sample was placed. Because of coupling between the frequency of oscillation and the nuclear magnetic resonance we expected to see a shift in frequency around the resonance frequency, and also the power required to induce these oscillations should cause a decrease in the amplitude of our tunnel diode oscillations. Therefore we attempted to measure the gyromagnetic ratio of fluorine and hydrogen by changing the magnetic field strength. The gyromagnetic ratio for hydrogen is and for Fluorine it is (4)(Web elements.com) For our experiment operating at 30 MHz this corresponds to a magnetic field of 0.7045 for Hydrogen, and 0.7485 for Fluorine.

D. Transition Temperature of Niobium

In order to measure the transition temperature of Niobium we will use a fundamental property of superconductivity. This property is called the Meissner Effect, where magnetic fields do not penetrate a superconducting sample. Therefore in an ideal superconductor this would mean that the material has zero magnetic permeability, which is a perfect diamagnet. This comes about because inside the magnetic field a current is induced on the surface layer of the superconductor in such a way that its magnetic field cancels the applied field. The depth that the magnetic field penetrates into the superconductor is called the penetration depth.⁽⁵⁾ The Transition temperature for Niobium is quoted as 9.25 ±0.02 K ⁽⁶⁾