35Cl NQR measurements in sodium chlorate, para-dichlorobenzene

Pulsed nuclear quadrupole resonance studies were carried out in sodium chlorate, para-dichlorobenzene (35Cl), and hexamethylene-tetramine (14N). All studies were conducted using the home-made superheterodyne spectrometer. The apparatus was largely unchanged for the three studies except for the tank section and the pulse protection circuitry between the power amplifier and the receiver section.

NQR measurement in para-dichlorobenzene

A number of combinations were used for the bridge and tank circuit sections for trials with various samples. Below you see the bridge circuit used in the case of para-dichlorobenzene, which includes the use of a hybrid tee as well as matching network B.

The bridge circuit configuration used for measurement of NQR in para-dichlorobenzene.



Matching network B was employed for the para-dichkorobenzene measurmenets only. Matching newtwork A was used for NAClO3 as well as for NQR measurements in HMT.

These earliest measurements display a lower S/N than those made in NaClO3 despite the NQR amplitude being larger in para-dichlorobenze because of two disadvantages of the experimental design used at that time. (1) The hybrid tee junction for isolation is inferior to a quarter wave section because the signal power is divided both for the pulse in and NMR out. (2) a short length of RG-58 coaxial was connected with BNC from the sample coil to the matching circuit. This is inferior to having the sample coil connected directly to the matching capacitors because of the ambient radiation that is shielding in the latter case as well as the characteristic inductance and loss in the cable. Nevertheless NQR was measured in para-dichlorobenze at 34.2534 MHz. The pulse width was 20 us and the time between pulses was 7 ms. The RF amplifier used was Minicircuits ZFL-1000N, and hybrid tee Marrimac HJ-55.


FID of paradichlorobenzene polycrystal at 34.2534 MHz. The pulse width was 20 us and time between pulses was 2.7 ms with signal averaging N=100.

NQR measurement in NaClO3

In the summer of 2013 visiting scientist Mikolaw Baranowski assisted with several improvements to the spectrometer that produced a high S/N for measurement of NaClO3 35Cl NQR. Matching network A was chosen instead. Additionally, crossed diodes were added at the output of the power amp. An audio frequency amplifier was used after the mixdown, and better RF amplifiers were used for signal detection. The coil was placed inside a shielded gunbox along with the matching caps.

The most important alteration is the use of a quarter wave transformer for isolation instead of a hybrid tee. Though this is unfeasible for low frequency NQR in 14N, as the wavelength is inconveniently long, a quarter wave transformer can be fashioned quickly with a length of RG-58 coaxial cable. The length of the cable is just one quarter of the wavelength at the design frequency times .67, the velocity factor for RG-58. Crossed diodes D1 and D2 provide optimal power delivery to the sample with no power allowed to leak back into the power amp. D3 and D4 short when the pulse is on, ensuring maximal power transfer to the tank. When the pulse is off, Dall diodes look like an open circuit, so the NMR signal is delivered to the fast recovery amplifiers.

The bridge circuit configuration used in measurements of sodium chlorate 35Cl NQR. Notice the absence of hybrid tee junction, which is supplanted by the quarter wave section and diode switches.

FID of NaClO3 powder at 29.9260 MHz. The pulse width was 40 us and time between pulses was 10 ms with signal averaging N=1000.

The SNR shown by the power spectrum of the above FID.


The NaClO3 tank. S/N was improved by having the sample connected directly to the matching circuit and placed in the aluminum enclosure without use of coaxial connector.

The tank circuit used for NaClO3 also differed from that of dichlorobenzene in the matching circuit configuration. For the NAClO3 test, matching configuration A was used. In this configuration a series capacitor matches a parallel resonance with the coil, shown in figure 2.

Calculation of capacitances for matching network A

A precise analysis of the above L-network for impedance matching is given now. We will examine the impedance of this reactive L network to characterize the parameter space of the variables Cm, Ct, ω, L, r and produce some useful set of tables for lab, in which the relationship between Cm and Ct is known for a given ω L and r, where r is a pure resistive parameter in series with the coil inducatance.

For impedance matching conditions to be satisfied, which for us means 50 ohms real we must enforce

Im_Z = 0

Re_Z = R := 50 ohms

with

Z_coil = j ω L + r

So the total impedance is

Z_tot = -j / (Cm ω) + Z_coil || Z_Ct = Re_Z + j Im_Z = R + j 0 = 50

Note:

* Z_m is the impedance of the matching cap only

* Z_t is the impedance of the tuning cap only

* the notation A || B means “A parallel B” and A || B = ( 1/A + 1/B)^(-1)

Since Z_coil has a real and an imaginary part, the expression for total impedance is .

Z_tot = -j / (Cm ω) + Z_coil || Z_Ct = Re_Z + j Im_Z = R + j 0 = 50

So I did it by hand, and with mathematica, and iteratively found what I consider decently short code with reasonably concise expressions. I prefer using mathematica because it allows the dynamic adjustment of the parameters L, and r to instantly output the capacitances as a function of frequency. Calculation is on the internet here and printed below.

(*You may copy and paste all of this text including the comments.Just \

run this entire block in mathematica in a single input.It should \

output the capacitances and sliders for you to adjust the other \

parameters.##### 2014 #####*)Clear[w, L, r, f, Z, \

Zform, ImZ, ReZ, T, M, Tminus, Tplus, Mminus, Mplus, Ma, ALL, tunem, \

tunep, matchm, matchp]

(*To avoid subscripts,we denote the tuning capacitance as T and the \

matching capacitance as M.The coil inductance is L.For the NMR probe \

topology shown in the image \

the general \

form for the impedance looking into the tank while r is in series \

with L is*)

(* The type

Z = 1/(I*w*M) + 1/(1/(I*w*L + r) + I*w*T);

(*Looking at the output of*)

Zform = ComplexExpand[Z];

(*Will yield the real and imaginary parts.The rest comes from \

enforcing Re{Z}=Z0 and Im{Z}=0*)

ReZ = r/((r^2 +

L^2 w^2) (r^2/(r^2 +

L^2 w^2)^2 + (T w - (L w)/(r^2 + L^2 w^2))^2));

ImZ = (-(1/(M w)) - (T w)/(r^2/(r^2 +

L^2 w^2)^2 + (T w - (L w)/(r^2 +

L^2 w^2))^2) + (L w)/((r^2 +

L^2 w^2) (r^2/(r^2 +

L^2 w^2)^2 + (T w - (L w)/(r^2 + L^2 w^2))^2)));

Solve[ReZ == Z0, T];

Solve[ImZ == 0, M];

(*The resulting quadratic for the capacitance T requires us to \

discard the value with a plus sign before the square root.These \

solutions lead to negative values of capacitance for M (the matching \

cap).I have kept the variables defined,but excluded them in the \

output.*)

Tminus = Dynamic[(L w^2 Z0 -

Sqrt[r^3 w^2 Z0 + L^2 r w^4 Z0 -

r^2 w^2 Z0^2])*1.00/(r^2 w^2 Z0 + L^2 w^4 Z0)];

Tplus = Dynamic[(L w^2 Z0 +

Sqrt[r^3 w^2 Z0 + L^2 r w^4 Z0 - r^2 w^2 Z0^2])/(r^2 w^2 Z0 +

L^2 w^4 Z0)];

Ma[T_] := (-1 + 2 L T w^2 - r^2 T^2 w^2 -

L^2 T^2 w^4)/(w^2 (-L + r^2 T + L^2 T w^2));

Mminus = Dynamic[(

(-1 + (2 L w^2 (L w^2 Z0 -

Sqrt[r^3 w^2 Z0 + L^2 r w^4 Z0 -

r^2 w^2 Z0^2]))/(r^2 w^2 Z0 +

L^2 w^4 Z0) - (r^2 w^2 (L w^2 Z0 -

Sqrt[r^3 w^2 Z0 + L^2 r w^4 Z0 -

r^2 w^2 Z0^2])^2)/(r^2 w^2 Z0 +

L^2 w^4 Z0)^2 - (L^2 w^4 (L w^2 Z0 -

Sqrt[r^3 w^2 Z0 + L^2 r w^4 Z0 -

r^2 w^2 Z0^2])^2)/(r^2 w^2 Z0 +

L^2 w^4 Z0)^2)/(w^2 (-L + (r^2 (L w^2 Z0 -

Sqrt[r^3 w^2 Z0 + L^2 r w^4 Z0 -

r^2 w^2 Z0^2]))/(r^2 w^2 Z0 +

L^2 w^4 Z0) + (L^2 w^2 (L w^2 Z0 -

Sqrt[r^3 w^2 Z0 + L^2 r w^4 Z0 -

r^2 w^2 Z0^2]))/(r^2 w^2 Z0 + L^2 w^4 Z0)))

)*1.00];

Mplus = Dynamic[(-1 + (2 L w^2 (L w^2 Z0 +

Sqrt[r^3 w^2 Z0 + L^2 r w^4 Z0 -

r^2 w^2 Z0^2]))/(r^2 w^2 Z0 +

L^2 w^4 Z0) - (r^2 w^2 (L w^2 Z0 +

Sqrt[r^3 w^2 Z0 + L^2 r w^4 Z0 -

r^2 w^2 Z0^2])^2)/(r^2 w^2 Z0 +

L^2 w^4 Z0)^2 - (L^2 w^4 (L w^2 Z0 +

Sqrt[r^3 w^2 Z0 + L^2 r w^4 Z0 -

r^2 w^2 Z0^2])^2)/(r^2 w^2 Z0 +

L^2 w^4 Z0)^2)/(w^2 (-L + (r^2 (L w^2 Z0 +

Sqrt[r^3 w^2 Z0 + L^2 r w^4 Z0 -

r^2 w^2 Z0^2]))/(r^2 w^2 Z0 +

L^2 w^4 Z0) + (L^2 w^2 (L w^2 Z0 +

Sqrt[r^3 w^2 Z0 + L^2 r w^4 Z0 -

r^2 w^2 Z0^2]))/(r^2 w^2 Z0 + L^2 w^4 Z0)))];

f = Dynamic[w/(6.28319)];

{"f = " Dynamic[f] }

{Slider[Dynamic[w], {10^6, .2001*10^9, .001*10^6}, ImageSize -> 800],

"w = " Dynamic[w],

Slider[Dynamic[L], {.2*10^-6, 50*10^-6, 1.0*1.0010*10^-7},

ImageSize -> 800], "L = " Dynamic[L],

Slider[Dynamic[r], {.001, 300, .001}, ImageSize -> 800],

"r = " Dynamic[r],

Slider[Dynamic[Z0], {1, 300, 1}, ImageSize -> 800],

"Z0 = " Dynamic[Z0]}

r = 7.000;

Z0 = 50;

L = 26.000*10^-6;

w = 2.00*10^7;

Q = Dynamic[(L*1.00)*w/r];

{"tunecap=", Tminus, "matchcap=", Mminus, "Q=", Q}