Theorem for impedance matching in NMR tanks using L-networks facilitating broadband frequency sweeps for unknown quadrupolar resonances in solids.

 

Locating pure NQR spectra precisely would in many cases clarify NMR studies.  Furthermore NQR is indicative of internal field geometry in solids and is thus useful in the identification of quantum phase transitions.

The pursuit of pure NQR is difficult however because the resonant frequency is sample specific and is often unknown. Unlike in the case of NMR, the frequency cannot be controlled in the laboratory, but is rather a property of a material that is a fingerprint of the local environment of the nucleus in question.  

In general, operating a pulsed spectrometer at various frequencies requires the corresponding adjustment of the two capacitors shown below. Reducing the parameter space to a single value would make sweeping much more efficient. Any shortcuts  and tricks to allow easy sweeping could greatly accelerate understanding of NQR in yet unstudied samples.  

This general probe topology is common in the practice of nuclear resonance.

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The inductive load is tuned and matched to the characteristic impedance of a transmission line Z0 (usually 50 ohms) by the two variable capacitors C1 and C2. 


Postulate: If the series losses in the coil are set to

R = Z0 / 4, 

then 

C_1 =~ C_2

regardless of the value of Z0, and for any reasonable and f where f is frequency of operation and > 500 KHz. For < 500 KHz the approximation begins to break down for feasible values of L.

Suppose we can utilize transmission line transformers to reduce the effective Zo from 50 ohms to something lower, allowing a higher Q.




If for example the effective characteristic impedance of the tank Z0 = 20 ohms, then one could set r = 5 ohms externally. This results in nice agreement for the caps with L = 30 uH down to around 1 MHz. This would be excellent for sweeping and snooping for unknown quadrupolar resonances in this band, as 14N NQR often appears below 5 MHz.

 

Source tree on Github

https://github.com/Altoidnerd/NMR-Tank-Circuits

Raw copy pasta for mathematica 

https://raw.githubusercontent.com/Altoidnerd/nmr-tank-circuits/master/mathematica/A-network.m