# Cavity Perturbation Technique

Introduction

The Cavity Perturbation Technique (CPT) is one of the contact-free technique most widely used because of its high sensitivity and relative simplicity. A microwave resonant cavity is a box fabricated from high-conductivity metal (usually from oxygen-free copper) with dimensions comparable to the wavelength. The cavity has simple geometrical form (rectangle or cylinder) to make it easier to calculate the distribution of electromagnetic fields inside. At resonance, the cavity is capable of sustaining microwave oscillations, which form an interference pattern (standing wave configuration) from superposed microwaves multiply reflected from the cavity walls. Each particular cavity size and shape can sustain oscillations in a number of standing wave configurations called modes. In this technique one measures adiabatic change of the characteristics of a resonator upon the introduction of a foreign body (the sample under investigation). The foreign body must be small compared to the spatial variation of the field (quasi-static limit) and its disturbing influence must not be strong enough to force a jump from the unperturbed cavity mode. The perturbation is viewed as a serial expansion in powers of the filling factor (the volume of the sample ${V_s}$ divided by the volume of the cavity ${V_c}$) and only first order effects are accounted for in this analysis.

The principles to determine the sample properties when inserted inside a cavity are simple. In this technique one simply takes the difference of the measured quality factor Q (or a halfwidth of the resonant curve ${\Gamma =f_0/Q}$) and center frequency ${f_0=\omega_0}$ with the sample in and out. However, the relative changes of quality factor (${\Delta {1 \over 2Q}}$) and center frequency (${{\Delta \omega \over \omega_0}}$) upon introducing the sample are connected to the complex conductivity in a nontrivial way, depending on the sample “electrodynamical” geometry. Namely, the sample can either be a good conductor and microwaves penetrate only in the length scale of skin depth into it which is called skin depth regime, or it can be a bad conductor (or even a dielectric) and microwave radiation penetrates completely through it (since the skin depth exceeds the principal dimensions of a sample) which is called depolarization regime.

Cavity Characteristics

A resonant cavity can sustain many modes above a lower cut-off frequency. Near each resonant frequency, the power absorption spectrum has a Lorentzian shape

$\displaystyle A(\omega)={1 \over 4(\omega -\omega_0)^2+(2\pi \Gamma)^2}, \ \ \ \ \ (1)$

where ${f_0=\omega_0/2\pi}$ is the center frequency and ${\Gamma}$ is the bandwidth or full frequency width at half maximum. ${f_o \text{ and }\Gamma}$ are the two characteristics of the resonator and their ratio gives the quality factor ${Q}$, defined as

$\displaystyle Q\equiv {f_0 \over \Gamma}={\omega_0 \over L}, \ \ \ \ \ (2)$

where ${\textless\emph{W}\textgreater}$ is the time-averaged energy stored in the cavity and ${L}$ the energy loss per cycle. The simplest formulation of this problem can be made with the use of a complex frequency notation

$\displaystyle \hat{\omega} \equiv \omega - i{\omega_0 \over 2Q} \ \ \ \ \ (3)$

The Polarizability

The principle of the cavity perturbation technique is to measure separately the cavity characteristics both before (o) and after (s) a small sample has been inserted. The change in the complex frequency is

$\displaystyle \Delta \hat{\omega}=\hat{\omega}_s-\hat{\omega}_o \ \ \ \ \ (4)$

If the change ${\Delta \hat{\omega}}$ is adiabatic, then the product and the time-averaged energy stored is invariant (Boltzmann-Ehrenfest theorem)

$\displaystyle {\textless W\textgreater \over \hat{\omega}}=\text{constant} \ \ \ \ \ (5)$

This implies that

$\displaystyle {\Delta \textless W\textgreater\over \textless W\textgreater}= {\Delta \hat{\omega} \over \hat{\omega}}\sim {f_s-f_o \over f_o}-{i \over 2}\left({1 \over Q_s}-{1 \over Q_o}\right), \ \ \ \ \ (6)$

$\displaystyle \textless W\textgreater={1 \over 16\pi}\int_{V_c}\left(|\mathcal{E} (\mathbf{r})|^2+|\mathcal{H} (\mathbf{r})|^2 \right) d\nu, \ \ \ \ \ (7)$

$\displaystyle \Delta \textless W\textgreater= -{1 \over 4}\int_{V_s} (\mathbf{P}\cdot \mathcal{E}^*+\mathbf{M}\cdot \mathcal{H}^*)d\nu. \ \ \ \ \ (8)$

In our definition, ${\Delta}$ is the variation caused by the introduction of a foreign body into the resonating structure, ${\Delta f=f_s-f_o}$ is the frequency shift and ${\Delta \Gamma = 1/Q_s-1/Q_o}$ the change in the width of the resonance.

For ellipsoidal samples, the spatial dependence of the depolarization field can be omitted and the quantity should be evaluated at the center of the sample. If we put the sample in the antinode of the magnetic field (${\mathcal{E}=0}$), then

$\displaystyle \mathbf{M}=\hat{\chi}_m\mathcal{H} \ \ \ \ \ (9)$

$\displaystyle {\Delta \hat{\omega} \over \omega}=-{\alpha_m \over 4} V_s {|\mathcal{H}|^2 \over \textless W\textgreater}=-4\pi \gamma \hat{\alpha}_m \ \ \ \ \ (10)$

where ${\gamma = \gamma_0 V_s/V_c}$ and ${\gamma_0}$ is a constant that depends only on the resonance mode of the cavity.

$\displaystyle \gamma_0={|\mathcal{H}|^2\over16\pi \textless W\textgreater}V_c = {|\mathcal{H}|^2\over 2 \textless |H|^2 \textgreater} \ \ \ \ \ (11)$

$\displaystyle ={1 \over V_c}\int_{V_c}|\mathcal{H}(\mathbf{r})|^2 d\nu \ \ \ \ \ (12)$

If the sample is in the antinode of the electric field, Eqs. (9)-(12) will be equivalent but with ${\mathcal{H}}$ repalced by ${\mathcal{E}}$ and M by P.

In conclusion, the absorption of electromagnetic waves by small particles is proportional to the polarizability of the sample

$\displaystyle {\Delta \hat{\omega} \over \omega}=-4\pi \gamma \hat{\alpha}. \ \ \ \ \ (13)$