Theoretical:

Basic EPR Theory:

Using the Uncertainty Principal for quantum systems, the only thing that can be measured with certainty is the square of the angular momentum (S*S) and one component of the angular momentum vector, usually the z component (Sz). The magnitude of angular momentum (S) for an electron has been determined by experimentation and the z component of the angular momentum (Sz) can be determined by the spin quantum number (s). For free electrons in a magnetic field there are only two possible orientations, thus

The spin quantum number only has two possible values, ½ and –½. From eq-1 it is apparent that the z component of angular momentum associated with these orientations are Sz=hbar/2, spin up, or Sz=-hbar/2, spin down. The two different spin states of the electron have different energies associated with them, which can be calculated.

An electron within a magnetic field has a magnetic energy associated with it that is determined by eq-3. The magnetic moment of the electron is then related to the angular momentum by eq-4. The energy of the electron in a given spin state is then determined by combining eq-3 and eq-4 to obtain eq-5. From eq-5 it is determined that the energy of the electron can only have two values of E = ± ½geuoHZ.

To obtain an EPR spectrum, transitions between the spin states must be induced. This is achieved by the introduction of an electromagnetic field, usually an RF field. The energy required for the transition to occur is equivalent to the difference in energy between the spin states and can be determined using eq-6. The energy supplied by the

RF field is calculated by E=hv, where h is Planck’s constant and v is the frequency of the field. Equating eq-6 and the energy of the RF field reveals the fundamental equation for EPR, eq-7.

The EPR spectrum is then obtained by sweeping the magnetic field, when the energies are equal an absorption line is observed, this allows for the calculation of the g-value. The g-value is a proportionality constant between the energy of the RF field and the magnetic field. For free electrons g = 2.002319, but for unpaired electrons within atoms, molecules, and crystals the g-value deviates from the free electron. This deviation is due to the internal magnetic interactions between the unpaired electrons and their surroundings, thus allowing for the study of that environment. The line widths and intensities of the absorption lines also produce useful information. Magnetic interactions between the electrons and their surroundings within a sample will prevent the electrons from transitioning all at once. The width and intensity of an absorption line is then dependent on the environment and concentration of the EPR active species. The signal shape and area below the absorption curve is proportional to the concentration of the EPR active species. The shape of the curve indicates the rate of excitation and de-excitation of the unpaired electrons.

Cavity Resonance:

The cavity resonator is an essential part of the EPR microwave bridge circuit. Standing waves within the cavity store the microwave energy over time intervals longer than the wave period, which allows for the amplification of the microwave power within the cavity.

A cavity resonator consists of a cavity completely enclosed by conducting walls. The geometry of the cavity will dictate the resonance frequency and quality factor (Q).

The cavity walls are machined from of a material with a high conductivity and are assumed to be perfect conductors. Losses from the penetration of the fields into the walls are assumed to be negligible.

To find the resonant frequency of a cavity resonator, solutions to Maxwell’s equations must be found which satisfy the boundary conditions imposed by the resonator. Boundary conditions for cylindrical cavities nominally require that no tangential electric fields or no normal magnetic fields will exist at the surface of the cavity (6). Although the equations and curves are valuable in showing the general characteristics of simple resonators, they should only be used as guides in the design of the resonator. In most cases, they should be supplemented by a thorough laboratory analysis. 

In a simple cylindrical cavity resonator, only two normal modes are possible; transverse electric (TE) and transverse magnetic (TM). The transverse electric is a mode where the electric field vector is normal to the direction of propagation, while the transverse magnetic mode has its magnetic field vector normal to the direction of propagation. The resonance wavelength can be calculated using eq-8, which is derived from a combination of trigonometric and Bessel functions. The mode that the cavity resonates in is determined by the number of standing wavelengths allowed by the cavity geometry. These boundary conditions are referenced to the z-axis of the cylinder and are labeled as l, m, and n. The normal field modes for a cylindrical cavity are defined using eqs 9 - 20. Figure 2 details graphically how the TE and TM modes are distributed within a circular waveguide. These distributions will be similar to the distributions within a circular cavity. For the free space wavelength of the resonant frequencies, the

l: the number of full period variations of Er with respect to theta
m: the number of half-period variations of Etheta with respect to r
n: the number of half-period variations of Er with respect to z

Figure 1: Geometric Parameter for a right cylindrical cavity.

Figure 2: Field distributions for lower E-modes and H-modes in circular waveguide (7).

Equations for normal field modes for a cylindrical cavity (4):

TE - modes:

For n > 0 and m > 0

TM – modes:

For m > 0

Where:

values of Xl,m are defined as the mth root of J'l (x) = 0 for TE modes and the mth root of Jl (x) = 0 for TM modes (8). Table 1 lists the roots for some of the lower modes. 

Table 1: Roots of Jl (x) and J'l (x) for right circular cylinders (4)

Most cavity resonators are made to be tunable in one dimension. In order to calculate the resonate frequency of the cavity, one must know what modes are attainable. A mode chart similar to the one in figure 3 can be used to find the dominant modes that the cavity will support. Since not all cavities act alike, the mode chart should only be used as a guide, where the measured resonance frequency should lie close to the target mode line. Using the mode chart in conjunction with eq-8 reveals the resonance frequency for an ideal cavity, which serves as a starting point when searching for the cavity resonance. In most cases fine tuning the cavity is required, which should be done in the laboratory with the cavity installed in the circuit.

Figure 3: Mode chart for simple cylindrical cavities (5). Boundary box indicates
the possible modes attainable by the cavity used in the experiment.

The Unbalanced method:

A simple method for obtaining EPR spectrums in a homodyne system is to match the impedance of the cavity to the circuit impedance. In this configuration, when the cavity is properly tuned, there is no power reflected to the detector. A measure of the quality factor (Q), eq-25, indicates how well the microwave bridge circuit is

tuned into resonance. The Q-factor is the ratio of the resonance frequency (fr) to the separation between the full width half power points (Delta f). A well tuned circuit has a Q-factor greater than 1000. When the sample begins to resonate, the impedance of the cavity changes proportionally. This change in cavity impedance unbalances the circuit, thus power is reflected to the detector allowing for the EPR signal to be observed as a voltage rise. As the sample comes into and out of resonance, the reflected power rises and falls proportionally to the degree of mismatch between the source impedance and the load impedance.

The Synthetic Ruby:

Synthetic ruby is no different from natural ruby on the atomic level. It is a hexagonal crystal that belongs to space group R3c, and has the same dimensions as Al2O3 (Corondum/pure ruby), though some of the aluminum positions are filled instead with Cr+3 atoms. The Al-O or Cr-O atomic distances are 1.85 or 1.97 Å.

When looking at the valence electrons for Cr+3 a single electron occupies the 3d shell. When the same is done for oxygen, it was determined that two electrons exist in its 2p shell. Work published by D. Freude (9) states that the 4s orbital is empty. This would lead to a remainder in the Cr+3 3d shell of three valence electrons. The stability of the crystal lattice is sustained because of the repulsion of the three chromium valence electrons with the two oxygen valence electrons from each of the surrounding oxygen atoms. 

Symbol Reference:

D: Cavity Diameter
L: Cavity Length 
E: Magnetic energy
ge: g-value for electrons
g: g-value
h: Planck’s Constant
hbar: h/(2*pi)
H: Magnitude of the magnetic field
H: Magnetic field vector
HZ: The z component of the magnetic field vector
S: Intrinsic angular momentum vector
SZ: The z component of the intrinsic angular momentum vector
theta: Angle between the magnetic moment and magnetic field
lamda: Wavelength
v: Microwave Frequency
ub: The Bohr magneton
ue: Magnitude of the magnetic moment
ue: Magnetic moment vector