Resonance frequency

The capacitance of the gap of the LGR is given by

${\displaystyle C=\varepsilon _{0}{\frac {w\,\ell }{t}}\,,}$

where ${\displaystyle \varepsilon _{0}}$ is the permittivity of free space, ${\displaystyle w}$ is the thickness of the bore wall, ${\displaystyle t}$ is the gap width, and ${\displaystyle \ell }$ is the length of the resonator. The resonator bore acts as a single-turn solenoid with inductance given by

${\displaystyle L=\mu _{0}{\frac {\pi \,r_{0}^{2}}{\ell }}\,,}$

where ${\displaystyle \mu _{0}}$ is the permeability of free space and ${\displaystyle r_{0}}$ is the inner radius of the LGR bore. For a high-${\displaystyle Q}$ resonator, the resonant frequency is, to an approximation, given by

${\displaystyle f_{0}\approx {\frac {1}{2\pi }}{\frac {1}{\sqrt {LC}}}={\frac {c}{2\pi r_{0}}}{\sqrt {\frac {t}{\pi w}}}\,,}$

where ${\displaystyle c=1/{\sqrt {\varepsilon _{0}\mu _{0}}}}$ is the vacuum speed of light. Therefore, the resonant frequency of the LGR is determined from its geometry and is, to first approximation, independent of its length.

Quality factor

For a highly underdamped series ${\displaystyle LRC}$ circuit, the quality factor, which determines the sharpness of the resonance, is given by

${\displaystyle Q\approx {\frac {1}{R}}{\sqrt {\frac {L}{C}}}\,.}$

The effective resistance of a LGR can be estimated by considering the length of conductor through which the current travels and the cross-sectional area available to it. The relevant conductor length is the circumference ${\displaystyle 2\pi r_{0}}$ of the conductor's inner surface. The depth that the current penetrates into the inner surface of the LGR bore is determined by the electromagnetic skin depth ${\displaystyle \delta }$. Therefore, the cross-sectional area through which charge flows is ${\displaystyle \delta \,\ell }$. Combining these results gives an effective resistance

${\displaystyle R_{\rho }\approx \rho {\frac {2\pi r_{0}}{\delta \,\ell }}\,,}$

where ${\displaystyle \rho }$ is the resistivity of the conductor. The effective capacitance, inductance, and resistance then lead to a simple expression for the expected quality factor of the LGR

${\displaystyle Q\approx {\frac {r_{0}}{\delta }}\,,}$

where, for a good conductor, the electromagnetic skin depth at the resonance frequency is given by

${\displaystyle \delta \approx {\sqrt {\frac {2\rho }{\mu _{0}\omega _{0}}}}\,,}$

and ${\displaystyle \omega _{0}=2\pi f_{0}}$. For an aluminum resonator with ${\displaystyle r_{0}=1\,\mathrm {cm} }$ and ${\displaystyle f_{0}=1\,\mathrm {GHz} }$ the above analysis predicts ${\displaystyle Q\approx 3900}$.^[1][6]

Radiative losses

In practice, the measured quality factor a cylindrical LGR, without additional electromagnetic shielding, will be much less than the predicted value of ${\displaystyle r_{0}/\delta }$. The suppression of the quality factor is due to radiative power loss from magnetic field lines that extend out of LGR bore and into free space. An order-of-magnitude estimate of the effective radiation resistance can be made by treating the LGR as a conducting loop. In the limit that the wavelength of the radiation is much larger than the loop radius ${\displaystyle r_{0}}$, the radiation resistance is

${\displaystyle R_{\mathrm {r} }\approx {\frac {\pi }{6}}{\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}\left({\frac {\omega _{0}r_{0}}{c}}\right)^{4}\,,}$

and can be much larger than the resistance ${\displaystyle R_{\rho }}$ due to the resistivity of the LGR conductor.^[8][9] The radiative losses can be suppressed by placing the LGR inside a circular waveguide. Provided that the cutoff frequency of the lowest TE₁₁ waveguide mode is well above the resonant frequency of the LGR, the magnetic field lines will be prevented from propagating into free space. The presence of the electromagnetic shield will alter the resonant frequency and quality factor of the LGR, but typically by only a few percent.^[1][6]

Toroidal LGR

Drawing of a toroidal LGR with a section cut out so as to expose the bore and gap of the resonator.

Photograph of the two halves of a copper toroidal loop-gap resonator. Also visible are an inductive coupling loop suspended within the resonator bore and a so-called extended split-ring resonator placed in the LGR bore.^[10]

In some applications requiring high quality factors, the electromagnetic shielding provided by a concentric circular waveguide surrounding a cylindrical LGR can be bulky and awkward to work. A toroidal LGR can be used for high-${\displaystyle Q}$ measurements without requiring additional electromagnetic shielding. In the toroidal geometry the two ends of a cylindrical LGR are joined to form a completely closed structure. In this case, the magnetic field is completely confined within the bore of the resonator and there is no radiative power loss. The toroidal LGR consists of two halves that are bolted together along the outer diameter of the structure.

Like the cylindrical LGR, the toroidal LGR can be modelled as a series ${\displaystyle LRC}$ circuit. In general, the effective capacitance, inductance, and resistance of the toroidal LGR will differ from the expressions given above for the cylindrical LGR. However, in limit that the radius of the torus is large compared to the bore radius ${\displaystyle r_{0}}$, the capacitance, inductance, and resistance of the toroidal LGR are approximated by the expressions above if one takes ${\displaystyle \ell }$ to be equal to the circumference of the torus.

The toroidal LGR is particularly convenient when characterizing the electromagnetic properties of liquid samples or particles suspended in a liquid. In these cases, the bore of the toroidal LGR can be partially filled with the liquid sample without requiring a special sample holder. This setup allows one to characterize the magnetic properties of, for example, a ferrofluid. Alternatively, if the liquid sample is nonmagnetic, the entire toroidal LGR can be submerged in the liquid (or gas). In this case, the dielectric properties of the sample only modify the effective capacitance of the resonator and the changes in ${\displaystyle f_{0}}$ and ${\displaystyle Q}$ can be used to determine the complex permittivity of the sample.^[7][9]

Theory

The inductance of the LGR can be expressed as ${\displaystyle L=\mu _{0}V_{\mathrm {r} }/\ell ^{2}}$, where ${\displaystyle V_{\mathrm {r} }}$ is the volume of the LGR bore. Since the resonant frequency ${\displaystyle f_{0}}$ of the LGR is proportional to ${\displaystyle L^{-1/2}}$, a small change in the effective volume of the resonator bore will result in a change in the resonant frequency given by

${\displaystyle \delta f_{0}=-{\frac {f_{0}}{2}}{\frac {\delta V_{\mathrm {r} }}{V_{\mathrm {r} }}}\,.}$

Due to the Meissner effect, when a superconducting sample is place in the bore of a LGR, the magnetic flux is expelled from the interior of the sample to within a penetration depth ${\displaystyle \lambda }$ of its surface. Therefore, the effective volume of the resonator bore is reduced by an amount equal to the volume from which the magnetic flux has been excluded. This excluded volume is given by

${\displaystyle V_{\mathrm {ex} }\approx ab\left(c-2\lambda _{a}\right)\,,}$

where ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$ are the sample dimensions along the three crystallographic directions and ${\displaystyle abc}$ is the sample volume ${\displaystyle V_{\mathrm {s} }}$. In the above expression, it has been assumed that the microwave magnetic field is applied parallel to the ${\displaystyle b}$-axis of the sample. Since the presence of the superconductor reduces the LGR volume, ${\displaystyle \delta V_{\mathrm {r} }=-V_{\mathrm {ex} }}$ and

${\displaystyle \delta f_{0}={\frac {f_{0}}{2V_{\mathrm {r} }}}\left(V_{\mathrm {s} }-2ab\lambda _{a}\right)\,.}$

Solving this expression for the ${\displaystyle a}$-axis penetration depth yields

${\displaystyle \lambda _{a}={\frac {c}{2}}-{\frac {V_{\mathrm {r} }}{ab}}{\frac {\delta f_{0}}{f_{0}}}\,.}$

Generally, it is not possible to use LGR frequency-shift measurements to determine the absolute value of the penetration depth because it would require knowing the sample thickness ${\displaystyle c}$ very precisely. For example, in fully doped YBa₂Cu₃O₇, ${\displaystyle \lambda _{a}\approx 100~\mathrm {nm} }$ at low temperature.^[21] Therefore, to use the LGR measurement to determine ${\displaystyle \lambda _{a}}$ to within 10%, one would have to know the value of ${\displaystyle c}$ with an accuracy of ${\displaystyle 10~\mathrm {nm} }$ which is typically not possible.

Instead, the strategy is to track the changes in frequency as the sample temperature varies (while keeping the LGR at a fixed temperature). The absolute penetration depth can be expressed as

${\displaystyle \lambda _{a}(T)=\lambda _{a}(T_{0})+\Delta \lambda _{a}(T)\,,}$

where ${\displaystyle T}$ is temperature, ${\displaystyle T_{0}}$ is the experimental base temperature, and ${\displaystyle \Delta \lambda _{a}(T)}$ is the change in penetration depth as the sample temperature is increased above the base temperature. One can, therefore, express the change in penetration depth as

${\displaystyle \Delta \lambda _{a}(T)=\lambda _{a}(T)-\lambda _{a}(T_{0})=-{\frac {V_{\mathrm {r} }}{ab}}{\frac {1}{f_{0}}}\left[\delta f_{0}(T)-\delta f_{0}(T_{0})\right]\,.}$

Finally, defining ${\displaystyle \Delta f_{0}(T)=\delta f_{0}(T)-\delta f_{0}(T_{0})}$, one has

${\displaystyle \Delta \lambda _{a}(T)=-{\frac {V_{\mathrm {r} }}{ab}}{\frac {\Delta f_{0}(T)}{f_{0}}}\,.}$

This final expression shows how the LGR shifts in resonant frequency can be used to determine the temperature dependence of the magnetic penetration depth in a superconducting sample.

Experimental details

In a d-wave superconductor, the penetration depth typically changes by a few ångströms per degree kelvin, which corresponds to ${\displaystyle \Delta f_{0}/f_{0}\sim 10^{-10}}$ for a ${\displaystyle 1~\mathrm {mm} ^{2}}$ platelet sample in a LGR with a bore volume of ${\displaystyle 1~\mathrm {cm} ^{3}}$. Measuring such small changes in relative frequency requires an extremely high-${\displaystyle Q}$ resonator. The ultrahigh quality factors are obtained by coating the LGR surfaces with a superconducting material, such as a lead-tin alloy. The resonator is then cooled below the superconducting transition temperature of the coating using a bath of superfluid liquid helium. Quality factors of ${\displaystyle 10^{6}}$ have been achieved using copper LGRs coated with lead-tin and cooled to ${\displaystyle 1~\mathrm {K} }$.^[20]

Complex permittivity

Now suppose that the gap of resonator has been completely filled with a dielectric material that has complex relative permittivity ${\displaystyle \varepsilon _{\mathrm {r} }=\varepsilon ^{\prime }-j\varepsilon ^{\prime \prime }}$. In this case, the effective capacitance becomes ${\displaystyle \varepsilon _{\mathrm {r} }C}$ and the impedance of the LGR is given by

${\displaystyle Z_{\varepsilon }=R+j\omega L+{\frac {1}{j\omega \left(\varepsilon ^{\prime }-j\varepsilon ^{\prime \prime }\right)C}}\,.}$

Separating the real and imaginary terms leads to

${\displaystyle Z_{\varepsilon }=\left[R+\left({\frac {\varepsilon ^{\prime \prime }}{\left(\varepsilon ^{\prime }\right)^{2}+\left(\varepsilon ^{\prime \prime }\right)^{2}}}\right){\frac {1}{\omega C}}\right]+j\left[\omega L-\left({\frac {\varepsilon ^{\prime }}{\left(\varepsilon ^{\prime }\right)^{2}+\left(\varepsilon ^{\prime \prime }\right)^{2}}}\right){\frac {1}{\omega C}}\right]\,.}$

This expression shows that a nonzero ${\displaystyle \varepsilon ^{\prime \prime }}$ enhances the effective resistance of the LGR and, therefore, lowers its quality factor. A nonzero ${\displaystyle \varepsilon ^{\prime }}$, on the other hand, alters the imaginary part of the impedance and modifies the resonant frequency. Written in terms of the empty-resonator resonant frequency and quality factor, the above impedance can be expressed as

${\displaystyle {\frac {Z_{\varepsilon }}{QR}}=\left[{\frac {1}{Q}}+\left({\frac {\varepsilon ^{\prime \prime }}{\left(\varepsilon ^{\prime }\right)^{2}+\left(\varepsilon ^{\prime \prime }\right)^{2}}}\right){\frac {\omega _{0}}{\omega }}\right]+j\left[{\frac {\omega }{\omega _{0}}}-\left({\frac {\varepsilon ^{\prime }}{\left(\varepsilon ^{\prime }\right)^{2}+\left(\varepsilon ^{\prime \prime }\right)^{2}}}\right){\frac {\omega _{0}}{\omega }}\right]\,.}$

Provided that ${\displaystyle \omega _{0}}$ and ${\displaystyle Q}$ are known before hand, a measurement of the frequency dependence of ${\displaystyle Z_{\varepsilon }}$ can be used to determine ${\displaystyle \varepsilon ^{\prime }}$ and ${\displaystyle \varepsilon ^{\prime \prime }}$ of the material filling the gap of the LGR. This analysis gives the values of ${\displaystyle \varepsilon ^{\prime }}$ and ${\displaystyle \varepsilon ^{\prime \prime }}$ at the resonant frequency of the filled LGR.^[5][6][7]

Complex permeability

Next, suppose that the bore of a LGR is filled with a magnetic material have complex relative permeability ${\displaystyle \mu _{\mathrm {r} }=\mu ^{\prime }-j\mu ^{\prime \prime }}$. In this case, the effective inductance becomes ${\displaystyle \mu _{\mathrm {r} }L}$ and the impedance of the LGR is given by

${\displaystyle Z_{\mu }=R+j\omega \left(\mu ^{\prime }-j\mu ^{\prime \prime }\right)L+{\frac {1}{j\omega C}}\,.}$

Separating ${\displaystyle Z_{\mu }}$ into its real and imaginary components and writing the impedance in terms of ${\displaystyle \omega _{0}}$ and ${\displaystyle Q}$ of the empty LGR yields

${\displaystyle {\frac {Z_{\mu }}{QR}}=\left[{\frac {1}{Q}}+\mu ^{\prime \prime }{\frac {\omega }{\omega _{0}}}\right]+j\left[\mu ^{\prime }{\frac {\omega }{\omega _{0}}}-{\frac {\omega _{0}}{\omega }}\right]\,.}$

Once again, ${\displaystyle \mu ^{\prime \prime }}$ contributes additional dissipation which lowers the quality factor of the filled resonator and ${\displaystyle \mu ^{\prime }}$ shifts the resonant frequency. A measurement of the frequency dependence of ${\displaystyle Z_{\mu }}$ can be used to extract the values of ${\displaystyle \mu ^{\prime }}$ and ${\displaystyle \mu ^{\prime \prime }}$ at the resonant frequency of the filled LGR.^{[5][10][12][22]}