Author ]to whom correspondence should be addressed

# A Cryogenically-Cooled High-Sensitivity Nuclear Quadrupole Resonance
Spectrometer

Jarred Glickstein<EMAIL_ADDRESS>Soumyajit Mandal  [<EMAIL_ADDRESS>Electrical, Computer, and Systems Engineering Department, Case Western Reserve
University, Cleveland, OH 44106, USA.

###### Abstract

The paper describes a radio frequency (RF) spectrometer for 14N nuclear
quadrupole resonance (NQR) spectroscopy that uses a detector coil cooled to 77
K to maximize measurement sensitivity. The design uses a minimally-intrusive
network of active duplexers and mechanical contact switches to realize a
digitally reconfigurable series/parallel coil tuning network that allows
transmit- and receive-mode performance to be independently optimized. The
design is battery-powered and includes a mixed-signal embedded system to
monitor and control secondary processes, thus enabling autonomous operation.
Tests on an acetaminophen sample show that cooling both the detector and
sample increases the signal-to-noise ratio (SNR) per scan by a factor of
approximately 88 (in power units), in good agreement with theoretical
predictions.

##  I Introduction

Nuclear quadrupole resonance (NQR) spectroscopy is a quantitative and non-
destructive radio frequency (RF) analytical technique with applications in
solid-state chemistry [1, 2], detection of explosives [3], characterization of
pharmaceuticals [4], and authentication of manufactured objects [5, 6]. While
a variety of quadrupolar nuclei generate NQR spectra, ${}^{14}N$ is often of
primary interest due to its widespread occurrence in organic compounds,
pharmaceuticals, and explosives. However, the applications of ${}^{14}N$ NQR
spectroscopy are often limited by its intrinsically low sensitivity compared
to both optical methods (e.g., near-infrared and Raman spectroscopy) and
nuclear magnetic resonance (NMR) spectroscopy. The key issue is that the
separations between nuclear energy levels involved in NQR transitions are
small (in the low-MHz range for ${}^{14}N$), which results in small population
differences in thermal equilibrium and thus poor signal-to-noise ratios (SNR)
for the transitions. Moreover, these separations are intrinsic to the sample
and its crystal structure and cannot be increased by applying an external
field, unlike for NMR.

The low sensitivity of ${}^{14}N$ NQR spectroscopy makes it particularly time-
consuming to discover the spectra of hitherto-unknown samples, since this
generally requires broadband high-resolution rfrequency sweeps. Reducing the
temperature of the sample and/or detector is an effective way to decrease
overall experimental time by increasing the SNR per scan. Accordingly,
laboratory applications of NQR spectroscopy have long relied on cryostats to
cool the sample [7]. In [8], the authors describe a conventional tuned (i.e.,
narrowband) inductive detector within a cryostat; the tuning frequency can be
programmed via by a mechanically-variable capacitor controlled by a stepper
motor. Other work has focused on broadband superconducting detectors
(typically operating at 4 K), such as SQUIDs [9] and superconducting-magneto-
resistive hybrid sensors [10]. However, mechanical tuning methods are slow
($\sim 10-100$ ms per frequency step), thus making it difficult to use
frequency-interleaved scans [11] to speed up frequency sweeps. On the other
hand, broadband detectors generally require extensive magnetic shielding to
protect against external radio frequency interference (RFI).

In this paper, we address these issues by developing a high-sensitivity
${}^{14}N$ NQR spectrometer that uses a custom cryostat to cool both the
sample and an inductive detector (coil) to liquid-$N_{2}$ temperatures (77 K).
In addition, a flexible series/parallel electronic tuning network enables
rapid probe re-tuning and allows the transmit- and receive-mode performance of
the coil to be independently optimized. The rest of the paper is organized as
follows. Section II discusses the sensitivity of NQR spectroscopy and its
dependence on temperature. The proposed tunable probe is discussed in Section
III. Section IV and V discuss the custom RF receiver and transmitter used by
the proposed spectrometer. System-level implementation and test results are
described in Section VI, while Section VII summarizes our conclusions.

##  II Theoretical Analysis

###  II.1 Sensitivity of NQR Spectroscopy

The sensitivity of NQR spectroscopy can be theoretically estimated as follows.
We begin by estimating the magnetization density within the sample immediately
after an RF excitation pulse with a nutation angle of $\theta$. Assuming that
i) the sample was originally in thermal equilibrium, and ii) the RF pulse has
enough bandwidth to excite the entire width of an NQR line centered around
$\omega_{0}$, the result is [12, 11]

$M_{0}(\theta)=s_{exc}(\theta)\rho_{s}\frac{\gamma\hbar^{2}\omega_{0}}{(2I+1)k_{B}T_{s}}\left(\frac{I(I+1)-mm^{\prime}}{I^{2}}\right),$
(1)

where $s_{exc}(\theta)$ is known as the nutation function, $\rho_{s}$ is the
spin density (i.e., the number of target nuclei per unit volume), $\gamma$ is
the gyromagnetic ratio of the nucleus, $\hbar$ is the reduced Planck’s
constant, $k_{B}$ is Boltzmann’s constant, $T_{s}$ is the sample temperature,
$I\geq 1$ is the total angular momentum (i.e., nuclear spin), and $m$ and
$m^{\prime}$ are the spin quantum numbers of the two nuclear states involved
in the NQR transition. Due to quantum selection rules, these values must
satisfy $|m-m^{\prime}|=1$. For spin-1 nuclei such as 14N (i.e., nuclei with
$I=1$), the frequencies of the three allowed transitions are conventionally
denoted by $\nu_{+}$, $\nu_{-}$, and $\nu_{0}$, respectively.

For polycrystalline materials, the nutation function $s_{exc}(\theta)$ must be
calculated via a so-called powder average over all possible relative
orientations of the crystallographic principal axes to the RF magnetic field
generated by the pulse, $\vec{B}_{1}$. For spin-1 nuclei, the result can be
shown to be [13]

$s_{exc}(\theta)\approx\sqrt{\frac{\pi}{2\theta}}J_{3/2}(\theta),$ (2)

where $J_{3/2}(\theta)$ denotes the Bessel function of the first kind. The
peak value of this function is $\approx 0.426$, which occurs at the optimum
nutation angle of $\theta_{opt}\approx 2.09$ rad (i.e., $119.5^{\circ}$).

The induced magnetization, $M_{0}$, represents the initial amplitude of a
quantum coherence that oscillates sinusoidally at the resonant frequency,
$\omega_{0}$. Here we assume that it is measured using a conventional
inductive detector (i.e., coil). By Faraday induction, the resulting time-
varying magnetic flux generates a RF voltage in the detector coil. Using the
principle of reciprocity for electromagnetic fields [14], the amplitude of
this voltage is given by

$V_{coil}=\omega_{0}\int_{V_{s}}\left(\frac{B_{1}(\vec{r})}{I_{1}}\right)M_{0}(\theta(\vec{r}))dV_{s},$
(3)

where the integral is carried out over $V_{s}$, the sample volume;
$\theta(\vec{r})=\sqrt{3}B_{1}(\vec{r})t_{p}$ is the position-dependent
nutation angle (with $t_{p}$ being the RF pulse length); and
$B_{1}(\vec{r})/I_{1}$ is the position-dependent _coil sensitivity function_ ,
i.e., the amplitude of the RF field generated by a unit current flowing in the
coil.

Consider a pulsed NQR experiment with spin echo detection, often known as a
spin-locked spin echo (SLSE) or pulsed spin locking (PSL) pulse sequence [15].
Ignoring transverse relaxation, the signal-to-noise ratio (SNR) per echo (in
power units) is then given by

$SNR_{e}=\frac{V_{coil}^{2}}{2\sigma^{2}_{n}},$ (4)

where $\sigma_{n}$ is the root-mean-squared (rms) noise in the coil and the
factor of 2 arises from converting amplitude to rms. In the absence of
external sources (such as RFI), $\sigma_{n}$ is dominated by the thermal noise
of the coil and given by

$\sigma^{2}_{n}=4k_{B}T_{c}R_{c}F\Delta f,$ (5)

where $T_{c}$ is the coil temperature, $R_{c}$ is the series coil resistance,
$F>1$ is the noise factor of the receiver, and $\Delta f$ is the final
detection bandwidth. Note that the coil resistance can be written as
$R_{c}=\omega_{0}L_{c}/Q_{c}$, where $L_{c}$ and $Q_{c}$ are the coil
inductance and quality factor at $\omega_{0}$, respectively. Also $\Delta
f\approx\left(1/T_{2}^{\ast}+1/T_{acq}\right)$, where $T_{2}^{\ast}$ is the
decay constant due to inhomogeneous broadening of the NQR line (which
determines the width of each spin echo) and $T_{acq}$ is the duration of the
echo acquisition window.

In practice, the SNR obtained after averaging across several scans is reduced
by i) signal decay due to transverse relaxation during each SLSE sequence
(defined by the time constant $T_{2,eff}$), and ii) wait times $t_{W}$ between
SLSE sequences to allow for longitudinal ($T_{1}$) relaxation. This effect has
been analyzed in detail in our earlier work [5]; the SNR available per echo
after averaging is found to be

$SNR_{e,av}=SNR_{e}\frac{\left(1-e^{-\alpha}\right)^{2}\left(1-e^{-\beta}\right)^{2}}{\beta\left(\beta+\alpha\delta\right)},$
(6)

where $\alpha=t_{W}/T_{1}$, $\beta=t_{SLSE}/T_{2,eff}$, and
$\delta=T_{1}/T_{2,eff}$ are dimensionless parameters, and
$t_{SLSE}=N_{E}t_{E}$ is the total duration of a single SLSE sequence
(consisting of $N_{E}$ echo periods, each of length $T_{E}$).

Given the measured relaxation time constants of the sample, the expression in
eqn. (6) can be numerically maximized to find the measurement parameters that
maximize $SNR_{e,av}$, i.e., to find the optimum values of $t_{W}$ (which
controls $\alpha$) and $t_{SLSE}$ (which controls $\beta$) 111This process is
complicated by the fact that $T_{2,eff}$ is a combination of several
relaxation time constants, including $T_{1}$, $T_{2}$, $T_{1\rho}$ ($T_{1}$ in
the rotating frame, which itself depends on SLSE pulse sequence parameters
such as the $t_{E}$ and the peak RF power level), and $T_{2^{\ast}}$ (which
depends on sample preparation). We will ignore the dependence of $T_{2,eff}$
on pulse sequence parameters for convenience; this is an acceptable
approximation over commonly-used ranges of parameter values.. For example, the
$\nu_{+}$ transition of acetaminophen (in the monoclinic form) has
$T_{1}\approx 9$ s and $T_{2,eff}\approx 2.5$ s for $t_{E}=1200$ $\mu$s at
room temperature [17, 5]. The optimum parameter values are then found to be
$\alpha_{opt}\approx 1.53$ and $\beta_{opt}=1.01$, resulting in
$SNR_{e,av}\approx 0.038\times SNR_{e}$.

###  II.2 Effects of Temperature on Sensitivity

Eqn. (1) shows that the magnetization density is inversely proportional to
sample temperature, i.e., $M_{0}\propto 1/T_{s}$. Note that this relationship
is ultimately derived from the relative populations of nuclear spin states in
thermal equilibrium: $M_{0}\propto(n-n^{\prime})/n$ where $n$ and $n^{\prime}$
are the populations of the lower and upper energy states involved in the NQR
transition. Boltzmann statistics predicts that
$n^{\prime}/n=\exp(-\hbar\omega_{0}/k_{B}T_{s})$, such that $M_{0}$ is
proportional to
$1-\exp(-\hbar\omega_{0}/k_{B}T_{s})\approx\hbar\omega_{0}/(k_{B}T_{s})$. The
latter approximation is valid when the energy difference between the states is
much smaller than the thermal energy, i.e., $\hbar\omega_{0}\ll k_{B}T_{s}$,
which is always true for NQR transitions (with $\omega_{0}$ in the low-MHz
range) for sample temperatures greater than a few mK.

Since the nutation function and coil sensitivity function are both independent
of temperature, the induced coil voltage $V_{coil}$ is proportional to
$M_{0}$. Thus, it also decreases with temperature as $V_{coil}\propto
1/T_{s}$. As result, cooling the sample increases the NQR signal power (and
thus $SNR_{e}$) $\propto 1/T_{s}^{2}$.

Cooling the detector can further increase $SNR_{e}$ by reducing the noise
power $\sigma^{2}_{n}$. Since the power spectral density (PSD) of coil thermal
noise $\propto k_{B}T_{c}$, eqn. (5) predicts that $\sigma^{2}_{n}\propto
T_{c}$ such that $SNR_{e}\propto 1/(T_{s}^{2}T_{c})$. However, in reality the
dependence is more complex since both the coil resistance $R_{c}$ and the
receiver noise figure $F$ are temperature-dependent.

First we analyze the coil resistance $R_{c}$, which is proportional to the
resistivity of the coil winding material (typically copper). The resistivity
$\rho$ of metals such as copper is dependent on the presence of impurities,
which result in scattering of the carriers [18]. The contribution of such
impurities to resistivity is approximately temperature-independent, such that
the total resistivity versus temperature can be written as

$\rho(c,T)=\rho_{0}(c)+\rho_{i}(T),$ (7)

where $c$ is the impurity concentration and $\rho_{0}$ is the residual
resistivity at 0K due to the impurities. For example, good-quality annealed
copper wire has $\rho_{0}\approx 2\times 10^{-11}$ $\Omega$-m. In addition,
$\rho_{i}$ is the intrinsic resistivity, which goes to zero at 0K
approximately $\propto T^{5}$ and becomes $\propto T$ at high temperatures,
i.e., as $T\rightarrow\infty$ [18]. The resulting total resistivity
$\rho(c,T)$ for annealed copper wire is shown in Fig. 1; it is approximately
proportional to temperature (i.e., $\rho(T)\propto T$) above $\sim 55$ K.

Figure 1: Typical resistivity of annealed copper wire as a function of
temperature, with the intrinsic resistivity $\rho_{0}=2\times 10^{-11}$
$\Omega$-m. Data from [18]. The open circles denote room temperature (300 K)
and liquid-$N_{2}$ temperature (77 K).

The observed behavior of $\rho$ was included in a numerical model (referred to
as Approx. A) that predicts the inductance and series resistance $R_{c}$ of
typical solenoid sample coils as a function of their temperature $T_{c}$. The
results can also be used to predict the coil Q, but the latter estimate
becomes inaccurate at high frequencies due to the parasitic self-capacitance
$C_{p}$ of the coil. The self-capacitance consists of both inter-turn
capacitance, which can be reliably modeled [19], and poorly-modeled capacitive
coupling to the shielding enclosure and other nearby conductors. Including an
estimate of this term in the numerical model (referred to as Approx. B)
provides a more accurate estimate of coil Q versus frequency, as shown in Fig.
2 for a typical coil geometry at 77 K and 300 K. The model shows that cooling
the coil using liquid-$N_{2}$, such that $T_{c}=77$ K, reduces $R_{c}$ (and
thus increases $Q$) by approximately $3\times$ compared to room temperature.

Figure 2: Evaluation of two numerical models for a typical sample coil $Q$ at
77 K and 300 K. The two models treat the coil as i) an ideal inductor with
series resistance $R_{c}$ (Approx. A), and ii) an ideal inductor with series
$R_{c}$ and parallel capacitance $C_{p}$ (Approx. B).

Next, we consider the receiver noise factor $F$. Defining the input-referred
voltage and current noise PSDs of the receiver as $e_{n}^{2}$ and $i_{n}^{2}$,
respectively, $F$ may be written as

$F=\left(1+\frac{e_{n}^{2}+i_{n}^{2}\left|Z_{p}\right|^{2}}{4k_{B}T_{c}R_{c}G_{p}^{2}}\right),$
(8)

where we have assumed that $e_{n}$ and $i_{n}$ (also known as the series and
parallel noise, respectively) are uncorrelated for simplicity. In addition,
$Z_{p}$ and $G_{p}$ are the output impedance and voltage gain of the NQR probe
(which consists of the detector coil and an impedance matching or
transformation network) around the resonant frequency $\omega_{0}$. Note that
the coil thermal noise (i.e., the denominator in eqn. (8)) decreases strongly
as the detector is cooled, making it more challenging to maintain a low
receiver noise figure ($NF\equiv 10\log_{10}(F))$. In this case, one option is
to reduce $e_{n}$ and $i_{n}$ by also cooling the receiver front-end
electronics [20, 21].

Here we consider the performance of the following common probe designs: i) un-
tuned or broadband, with passive voltage gain provided by a transformer (turns
ratio $1:N$); ii) tuned to $\omega_{0}$ with a parallel capacitor; and ii)
matched to a reference impedance $Z_{0}$ (typically 50 $\Omega$) with a two-
capacitor network. These cases, which have been analyzed in detail in [22],
result in the values of $Z_{p}$ and $G_{p}$ summarized in Table 1; here
$L_{c}$ is the coil inductance.

Table 1: Properties of common NQR probe designs Probe topology | Voltage gain | Output impedance
---|---|---
Un-tuned | $N$ | $j\omega_{0}L_{c}N^{2}$
Tuned | $Q$ | $\omega_{0}L_{c}Q$
Matched | $\frac{1}{2}\sqrt{Z_{0}/R_{c}}$ | $Z_{0}$

Table 1 shows that tuned probes tend to have the largest voltage gain: typical
values of $N$ (gain for un-tuned probes) and $\sqrt{Z_{0}/R_{c}}$ (gain for
matched probes) range from 3-6 and 4-10, respectively, while $Q$ (gain for
tuned probes) can reach values $>200$ at room temperature (see Fig. 2). Eqn.
(8) then predicts that tuned probes have the lowest receiver NF. Equivalently,
the high voltage gain of tuned probes enables their output impedance
$\omega_{0}L_{c}Q$ to approach the (usually) high noise resistance
($R_{n}\equiv e_{n}/i_{n}$) of FET-input receivers; as is well-known,
satisfying this “noise matching” condition minimizes NF.

We define the SNR enhancement factor as the ratio of $SNR_{e}$ at a specified
temperature to its value at room temperature (300 K). Fig. 3 summarizes the
expected enhancement factor versus temperature for two cases: i) only the
detector is cooled, and ii) both the detector and sample are cooled. For
simplicity, we have assumed that receiver NF remains low over the entire
temperature range; this is generally true for tuned probes using high-Q coils.
The plot shows that even modest amounts of cooling can significantly increase
$SNR_{e}$, and thus the number of scans required to obtain a given SNR.

Figure 3: Estimated SNR enhancement factor as a function of experimental
temperature.

Finally, we note that the relationship between $SNR_{e}$ and $SNR_{e,av}$, as
defined by eqn. (6), also varies with sample temperature. This is because the
relaxation times $T_{1}$ and $T_{2,eff}$ (which enter into the parameters
$\alpha$, $\beta$, and $\delta$) are strongly temperature-dependent. In
general, both decrease with temperature $\propto e^{E_{a}/(RT_{s})}$ where
$E_{a}$ is an activation energy parameter and $R$ is the molar gas constant
[23]. Thus, obtaining the full benefits of sample cooling requires the SLSE
pulse sequence parameters $t_{w}$ and $t_{SLSE}$ to be varied with temperature
such that $\alpha$ and $\beta$ remain close to their optimal values.

##  III Tunable Probe Design

###  III.1 Series/Parallel-Tuned Probe

One of the significant challenges of probe design for 14N NQR spectroscopy is
the fact that the resonant frequencies are unique to each molecule and can
vary over a broad range (1-5 MHz). As a result, tuned or matched probes must
be capable of rapid and reliable re-tuning, ideally under software control.
Broadband un-tuned probes provide an appealing alternative, but result in
significant performance compromises in both transmit mode (reduced peak RF
power handling capability) and receive mode (increased NF) [24]. Thus, here we
focus on digitally-tuned narrowband probes. In earlier work, Chen and Ariando
have demonstrated such probes for NQR and low-field NMR that use networks of
relays to program the capacitor values of a traditional two-capacitor
impedance-matching network [11, 25]. However, two independent banks of relays
are required to tune the probe, which greatly increases the complexity of the
tuning algorithm. Impedance-matched probes at such low frequencies also suffer
from limited bandwidth in transmit mode (which results in long pulse rise/fall
times) and low voltage gain in receive mode (which makes the receiver NF
sensitive to probe tuning errors).

The usual solution, which is often found in low-field NMR systems such as
well-logging tools [26], is to use a single tuning capacitor $C_{1}$ in
parallel with the sample coil $L_{c}$. This tuned (but not impedance-matched)
design allows low-impedance switching power amplifiers to be over-coupled to
the probe in transmit mode (which increases the transmit bandwidth) and also
provides high voltage gain in receive mode (which reduces the effects of probe
mistuning by providing low NF over a broad bandwidth, as discussed in the
previous section). As a result, the AFE does not itself have to be cooled to
maintain low NF over a wide range of probe temperatures, which greatly
simplifies the receiver design. Moreover, a single bank of relays can be used
to program $C_{1}$, resulting in straightforward tuning. However, the absence
of impedance matching eliminates voltage gain during transmit, so high-voltage
(HV) transmitters are required to obtain adequate $B_{1}$ amplitudes.

HV operation is problematic for electrical safety, and is particularly
undesirable for portable devices. In addition, the figure of merit (FoM) for
switching devices within the transmitter (e.g., MOSFETs or IGBTs) is a
strongly-decreasing function of their voltage rating. Thus, it is beneficial
to replace the high-impedance parallel-tuned load with a low-impedance series-
tuned load that allows the use of lower transmit voltages for the same coil
current (and thus $B_{1}$). However, series-tuning provides no voltage gain in
receive mode, making it unsuitable for use with high-input-impedance pre-
amplifiers. These issues can be addressed by using separately-tuned transmit
and receive coils that are orthogonal to each other to minimize mutual
coupling. However, the associated geometric constraints generally result in
reduced coil sensitivity and fill factor, and may also complicate access to
the sample during experiments.

Figure 4: Schematic of the dynamically reconfigurable tuned transmit-receive
probe.

Fig. 4 presents an alternative approach in which high-performance gallium
nitride field-effect transistors (GaN FETs) are used to rapidly switch a
single-coil probe between two configurations during pulse sequences, namely
series-tuned (for transmit) and parallel-tuned (for receive).

The reconfigurable tuning network uses three active bidirectional switches
$S_{1}$, $S_{2}$, and $S_{3}$: all switches are “off” in transmit mode, and
“on” in receive mode. GaN FET based switches are used due to their high
switching FoM. For example, $S_{2}$ is realized using two back-to-back EPC2037
GaN FETs (EPC, El Segundo, CA), resulting in a typical on-resistance of
$R_{on}=800$ m$\Omega$ and an off-capacitance of $C_{off}=3.25$ pF. The low
value of $C_{off}$ minimizes its effect on resonant frequency during transmit
mode. In receive mode, the relatively high impedance of the parallel resonant
circuit minimizes the noise contribution of $S_{2}$ and $S_{3}$, but not
$S_{1}$. Thus, $S_{1}$ uses devices with very low $R_{on}$. The same low-
capacitance device as $S_{2}$ is suitable for $S_{3}$. Gate drive voltages for
each switch are supplied by floating H-bridges with optically-isolated control
inputs.

In the realized receiver, a fourth switch (not shown) with series resistor is
routed in parallel with $S_{1}$; the resulting current shunt decreases the
ringdown time at the end of transmit pulses. The effect is realized by
selecting a resistor which critically damps oscillations of the resonant
network at the end of transmit pulses. The optimal value of this resistor is
strongly dependent on coil $Q$, and thus both $\omega_{0}$ and $T_{c}$. To
simplify the system, we used a fixed resistor optimized for nominal values of
these parameters ($2\pi\times 2.5$ MHz and 300 K, respectively).

###  III.2 Optimization of the Sample Coil Geometry

The available probe tuning range is heavily dependent on the parameters of the
physical spectrometer. Selection of receiver coil dimensions, its position
relative to the sidewalls of the enclosure, and distance to the receiver are
the dominant influences. In addition, secondary effects include parasitic and
stray effects in the realized PCB layout, component selection, and conductive
enclosures. The potentially detrimental effects of nearby shielding (or lack
thereof) coupled to the highly sensitive network are analyzed later.

Figure 5: Impedance sweep shown for transmit (-$\cdot$-) and receive (—) modes
with a typical coil and stepped values for $C_{p}$.

Fig. 5 presents simulated impedance $\left|Z_{\rho}\right|$ of the
reconfigurable resonant network in both transmit and receive modes for a
typical sample coil. Each of fifteen positions displayed in the figure are
obtained by varying the tuning capacitance $C_{t}\in[100~{}{pF},1500~{}{pF}]$
in increments of $100$ pF, while $L_{c}$ is fixed at 8.36 $\mu$H with
$R_{s}=$370 m$\Omega$ and $C_{p}=10$ pF.

Figure 6: Measurement of network impedance in parallel and series modes while
adjusting tuning capacitance.

Fig. 5 figure predicts a good match (within 40 kHz) between the resonant
series- and parallel-tuned configurations over the entire tuning range.
However, this model is only accurate when the coil is in free space away from
nearby conductors. In a shielded enclosure (necessary to minimize RFI),
capacitive coupling to the walls of the box amplifies the mismatch between
resonant frequencies and thus makes it difficult to optimally tune both modes
simultaneously. The effect is visualized in Fig. 6. Using the same coil as for
simulation and tuning to 2.564 MHz 222One of the 14N resonant frequencies in
acetaminophen, tuning for optimum receive performance (a “parallel-dominant”
approach) results in a somewhat high corresponding series mode impedance 27
$\Omega$. Tuning for optimum transmit performance (a “series-dominant”
approach) nearly eliminates the beneficial passive gain of the tuned network
for the receiver, though the series impedance drops to 2.2 $\Omega$. Moving
the coil to a safe distance away from nearby shielding and enclosing surfaces
reduces the effective tuning gap and enables a compromise position which
yields $>70$% of peak receive mode $\left|Z_{\rho}\right|$ and agrees with
simulations.

Detector coil designs must balance between maximizing the sensitivity function
(which is proportional to $V$ and $N/l$) and minimizing $C_{p}$ (which
determines the gap between series- and parallel-mode resonant frequencies). A
large sample volume is desirable for greater nuclear spin population but
requires relatively large (thus long) coils for which $C_{p}$ becomes large.
Sensitivity to $C_{p}$ is significantly alleviated by reducing the inductance
of the detector coil via either the turns density or the total number of
turns.

Various coil designs for the cryogenic detector were wound by hand and
experimentally characterized using a vector network analyzer (VNA) to evaluate
the effects described above. The VNA converts two-port measurements into the
equivalent series R-L model $Z=R_{s}+sL_{s}$, which would be accurate if the
device under test was truly just an inductor with series resistance. However,
this does not account for parasitic capacitance. Approximating the latter as a
parallel capacitance $C_{p}$, consider the values reported by the network
analyzer to be $Z=R_{net}+sX_{net}$. The real values of coil resistance
$R_{s}$ and inductance $L_{s}$ are then related by

$R_{net}+sX_{net}=\frac{(sL_{s}+R_{s})s^{-1}C_{p}^{-1}}{sL_{s}+R_{s}+s^{-1}C_{p}^{-1}}.$
(9)

The system was reduced to two variables by approximating inductance using a
low-frequency measurement. An estimator for $C_{p}$ was then found as that of
the series R-L-C circuit with self-resonant frequency $f_{sr}$, such that
$C_{p}=1/\left(4\pi^{2}f_{sr}^{2}L_{s}\right)$. The model was used to compute
actual coil parameters by iterative error minimization. Table 2 summarizes the
estimated R-L-C parameters for some of the test coils used in the setup.

Table 2: Measurement parameters of four solenoidal test coils for the cryo-system. Here TPC denotes the turn density, i.e., turns/cm. Coil Name | Parameters
---|---
Nturns | TPC | R@1 MHz [$\mathbf{\Omega}$] | Ls [$\mu$H] | Cp [pF]
A | 46 | 6 | 0.823 | 22.5 | 9.21
C | 32 | 5.5 | 0.494 | 13.1 | 8.72
D | 32 | 5 | 0.534 | 11.6 | 8.67
E | 27 | 4 | 0.369 | 8.3 | 9.22

Table 2 suggests that coupling to nearby metallic objects is the dominant
effect on the measured $C_{p}$, with stray capacitance between the turns
contributing less than expected. Thus, increasing the turns packing ratio
$N/l$ is a convenient option for increasing coil sensitivity since it has
limited effect on $C_{p}$. However, for a fixed length (fixed $V$) it also
increases $R_{coil}$ (thus source noise) and $L_{s}$. Cryogenic cooling
reduces the impact of $R_{coil}$ on measurement sensitivity if a high turns
ratio is desired; that said, it is beneficial to have relatively small coils
with low inductance. Minimum inductance coils reduce the impact of small
variances $\Delta C$ on the resonant frequency of the tuned detector, thus
improving robustness against parasitic effects and allowing a greater degree
of control over tuning.

The test results summarized in Table 2 were used to select the optimum coil
geometry for our setup (design “A” in this case). Models for automated
optimization of the detector coil could be developed using a cost function
based on measurement sensitivity, but were not pursued during this study.

###  III.3 Optimization of the Tuning Network

Options for realizing an electronically-controllable tuning capacitor include
i) voltage-controlled capacitors (varactors), and ii) arrays of fixed
capacitors $C_{t,i}$, $i\in[1,N]$, each enabled/disabled by a switch.
Varactors enable continuous tuning and can be cryogenically cooled along with
the coil, but suffer from limited breakdown voltages (which results in a
severe trade-off between power handling capability and tuning range). Thus, we
opt for a switch-based tuning network (Fig. 7). Our choice of switching device
is guided by their FoM, which is defined as $R_{on}\times C_{off}$ where
$R_{on}$ is the resistance in the ‘on’ state and $C_{off}$ is the capacitance
in the ’off’ state. Since $R_{on}$ degrades SNR by contributing noise while
$C_{off}$ limits the tuning resolution, it is important to pick devices with
minimal FoM. As in our earlier work [11], we used miniature electromechanical
switches (reed relays) since they offer both low FoM ($\sim 100\times$ lower
than semiconductor switches) and relatively fast switching times (100 $\mu$s –
1 ms).

Figure 7: The proposed switch-based tuning network. There are a total of $N$
tuning capacitors with series switches, of which $n$ are in the ‘on’ state
(blue) and $m=N-n$ are in the ‘off’ state (green).

The number of tuning capacitors, $N$, was chosen to ensure complete frequency
coverage over a range $1-5$ MHz, which is sufficient for most 14N sites in
pharmaceutical compounds. By complete coverage, we imply that the spacing
$\Delta f_{0}$ between available tuning frequencies is always smaller than the
bandwidth $f_{0}/Q$ of the tuned network, i.e., $\Delta f_{0}<f_{0}/Q$, such
that arbitrary RF frequencies can be chosen for NQR experiments. To determine
the minimum value of $N$, we developed a model of the tuning network with an
arbitrary number of capacitors $n<N$ in the on-state. The latter are treated
as ideal capacitors $C_{t,i}$ in series with small resistances
$R_{t}=R_{on}+ESR_{C}$ where $ESR$ is the equivalent series resistance of each
tuning capacitor. In addition, the $m=N-n$ off-state capacitors are
approximated as $C_{off}$, the open-state capacitance between the relay
contacts. The quality factor of the network, which determines the probe
bandwidth, is then estimated as $Q\equiv X(\omega)/R(\omega)$ where
$X(\omega)$ and $R(\omega)$ are the imaginary and real components of the probe
impedance $Z(\omega)$, respectively.

Since $Q$ increases with frequency, smaller step sizes are required as $f_{0}$
increases. Thus, using a single binary-weighted array of tuning capacitors, as
in previous work [11], is inefficient since it results in a fixed value of
$\Delta f_{0}$ that must be set based on the worst case (i.e., to ensure
coverage at the highest tuning frequencies). Instead, we use two separate
banks of binary-weighted capacitors with different values of the least
significant bit (LSB). Specifically, we use two sets of $N_{1}=10$ capacitors,
one with values in the range $C_{t,i}\in
5~{}\mathrm{pF}\times\\{1,2,\,...\,,2^{10}-1\\}$ and the other in the range
$C_{t,i}\in 1~{}\mathrm{pF}\times\\{1,2,\,...\,,2^{10}-1\\}$. The resulting
frequency spacing for the chosen sample coil is shown in Fig. 8; the worst
case value of $\sim$14 kHz is adequate at room temperature but too large for
cryogenic operation due to the higher value of $Q$.

Figure 8: Simulated spacing $\Delta f_{0}$ between tuning frequencies for a
20-channel dual-bank tuning network with an additional fixed capacitance (100
pF) in parallel with the switched capacitors. The following relay parameters
were assumed: $R_{on}=0.1$ $\Omega$, $C_{off}=0.1$ pF.

To ensure adequate coverage at cryogenic temperatures, one option is to simply
increase the resolution of the tuning network to reduce $\Delta f_{0}$, i.e.,
by reducing the LSB capacitance and adding more tuning channels. However, the
off-state relay capacitance $C_{off}=0.1$ pF places a lower bound on the
achievable step size; the tuning function becomes non-monotonic when the LSB
capacitance becomes comparable to $C_{off}$. Instead, we relax the requirement
on $\Delta f_{0}$ by using a noiseless feedback network to increase the probe
bandwidth, i.e., reduce its effective $Q$ without adding noise [24]. The
method uses a capacitor of value $C_{fb}$ in series with a $90^{\circ}$ phase
shifter, for instance an integrator with transfer function
$H(j\omega)=A/(j\omega\tau)$. The phase shift effectively generates a
resistance by shifting the phase of the capacitor’s impedance into the real
plane, but does not add noise since $C_{fb}$ is (ideally) noiseless. The
capacitor thus behaves like a grounded resistor of value

$R_{damp}=\frac{\tau}{AC_{fb}}$ (10)

in parallel with the tuning network. The effective $Q$ of the network is then
given by

$Q_{eff}=\frac{Q}{\left(\omega_{0}L_{c}Q/R_{damp}\right)+1},$ (11)

where $Q\gg 1$ is the original (un-damped) quality factor of the tuning
network. Fig. 9 summarizes the quality factor ($Q_{eff}$) and bandwidth
($f_{0}/Q_{eff}$) of the damped network for various values of $C_{fb}$ and
typical integrator parameters ($A$,$\tau$). The figure shows that a relatively
small capacitance ($C_{fb}\approx 0.1$ fF) suffices to provide sufficient
bandwidth over the entire tuning range. Larger values of $C_{fb}$ are not
recommended since the voltage gain $Q_{eff}$ of the damped probe also
decreases, eventually resulting in higher receiver NF as predicted by eqn.
(8).

Figure 9: Simulated quality factor ($Q_{eff}$) and bandwidth of the damped
network at 77 K for various values of the feedback resistor $C_{fb}$.

##  IV A Custom Broadband NQR Receiver

The receiver is organized into two primary components: an analog front-end
(AFE) and a digital controller. The AFE consists of the signal path between
the detector coil and measurement data acquisition; the latter utilizes a
commercial benchtop NMR spectrometer (Magritek Kea2). The digital controller
facilitates experiment sequencing and inter-experiment preparation, including
tuning of the resonant detector, monitoring ambient climate, control over
polarization enhancement, and enabling/disabling the analog receiver. Digital
input/output (I/O) pins are also used to check status of battery charge and
pause the experiment if any battery packs are low. These components are shown
in the block diagram of Fig. 10.

Figure 10: Block diagram of the proposed NQR receiver, which consists of an
analog front-end and a digital controller.

###  IV.1 Analog Front-end Design and Circuit Performance

The AFE consists of two main components: the duplexer and the preamplifier.
The duplexer uses FET switches to enable broadband operation. It was initially
designed using cascode topology GaN FETs (Transphorm, USA) to tolerate a
greater (650 V) input amplitude, at the cost of increased input capacitance in
the off state. However, the additional capacitance resulted in significant de-
tuning, even for small coils. Thus, the circuit was redesigned using smaller
100 V rated GaN FETs (EPC, USA) to minimize off-resonance effects in the tuned
detector when it is mated to the receiver.

Apart from the tuned detector, the AFE has a broadband frequency response. The
input stage of the preamplifier uses a silicon JFET configured as a source
follower to ensure both low voltage and current noise and high input impedance
[24]. A small 1:4 step-up transformer is used to realize nearly noiseless
broadband voltage gain before the input JFET, thus further improving the noise
figure (NF). The simulated worst-case NF (assuming noiseless feedback damping
of the probe) is $<0.01$ dB over the entire temperature and frequency ranges
of interest; such excellent performance is due to the high passive voltage
gain $Q_{eff}\gg 1$ provided by the tuned probe.

As noted in prior art [28], such broadband preamplifier designs are
susceptible to environmental noise and transformer self-resonance.
Fortunately, the feedback damping network (which is mainly used to increase
probe bandwidth, as described in the previous section) also mitigates these
effects. The $90^{\circ}$ phase shifter required by the damping circuit was
realized using an op-amp integrator. The gain of various stages in the
receiver was adjusted to compensate for the reduced gain resulting from the
damping circuit. The gain and width of the passband are only mildly impacted
by these changes.

###  IV.2 Mixed Domain Isolation

Several means of mixed signal domain isolation were implemented in the
spectrometer. The intent is to achieve good mixed signal noise immunity by
reducing the coupling between domains when passing control signals. Note that
these techniques are not effective in isolation. The digital controller and
analog front-end were powered by separate DC supplies, with a star ground
topology used to avoid coupling of digital switching noise into the amplifier.
The receiver uses latching relays to tune the detector while maintaining
electrical isolation from the digital controller. The DC supply (battery pack)
used by the relay coils is enabled through a momentary digital switch which is
disconnected while not in use.

Four floating gate drivers were used to implement the detector described in
Section III.1. Each driver is supplied by an isolated battery pack and the
state of each gate is controlled through optical isolators. Note the use of
such isolators is not a complete solution, since they do not prevent digital
switching noise from crossing into the analog domain of the preamplifier
without careful filtering of the input signals which drive the LED in each
isolator. Furthermore, the floating supplies create a new challenge in
ensuring noise immunity. The lack of a defined ground in each duplexer supply
and control circuit with respect to the spectrometer ground makes each
duplexer highly susceptible to noise coupling from external sources. Thus,
these supplies are well shielded inside the spectrometer, and wires between
each duplexer battery pack and the receiver PCB are kept minimal in length.

Communication between the digital controller and an external computer uses a
serial port. A USB signal isolator using monolithic air core transformers is
used to electrically isolate the computer from the spectrometer. This device
is primarily intended to isolate the computer from damage resulting from
unanticipated circuit failures. Noise at the communication frequency is not
isolated, though this occurs at 480 MHz and is high pass filtered by the air
core transformers to prevent coupling low frequency noise from the external
computer system.

###  IV.3 Fully-Assembled Receiver

Figure 11: Completely assembled receiver in the experimental setup at room
temperature. The tuning network is labeled.

An assembled receiver board is shown in Fig. 11. General purpose digital and
analog IO near the bottom of the board allow versatile adaptation of the
digital controller to evolving design intent. Flexible transmitter and coil
connectivity (at top left in Fig. 11) provide similar functionality.

##  V A Broadband RF Pulse Transmitter

RF pulses are delivered by a broadband class-D amplifier (a H-bridge) similar
to that used in our previous work [29]. However, the present design is series-
tuned in transmit mode, which results in significantly better power transfer
from the amplifier [30, 31]. Specifically, the low impedance of the series-
tuned coil ($\approx R_{c}$ on resonance) is now relatively well-matched to
the low output resistance of the amplifier ($\approx 2R_{on}$, where $R_{on}$
is the on-state resistance of the switches in the H-bridge). As a result, the
system no longer requires a HV power supply to deliver sufficient current to
the coil.

For improved power efficiency, the H-bridge was implemented using discrete
GaNFETs (Transphorm, USA). The board layout and components was designed to
support pulse currents up to 5 A. However, our experiments used lower current
levels. Specifically, a $\approx 9.5$ V supply was used to develop typical
pulse amplitudes of $1$ Apk. Note that the required supply voltage is
unusually high for a tuned load due to off-resonance effects (see Fig. 6); the
coil was always tuned to be on-resonance in the parallel configuration to take
full advantage of the high Q in receive mode.

Figure 12: Completely assembled transmitter in the experimental setup at room
temperature.

##  VI System Implementation and Test Results

###  VI.1 Power Supplies

The spectrometer was powered using battery packs to eliminate power supply
noise and ripple. Two variations were utilized, a 4-cell split rail supply
with output voltage $\pm$7.4 V and a 2-cell single rail supply with output
voltage 3.7 V. The battery packs used 18650 size lithium-ion cells because
they offer reasonably large capacity and are readily available at reasonable
prices due to their popularity. The nominal cell voltage is 3.7 V and the
typical fully charged voltage is 4.2 V. Each cell has dimensions $18\times 65$
mm ($H\times D$). The battery packs implement two state charge monitors (“OK”
and “RECHARGE”) which interface with the receiver digital controller using an
optically isolated output. Isolation is required not only to maintain noise
immunity, but to maintain separation of floating gate drivers from the
receiver ground. In the event of a low battery signal, the receiver indicates
a fault condition using a colored LED, signals to a control program over its
serial interface, and halts data acquisition. In addition, a sealed lead acid
battery provides the nominally 12 V unregulated supply to switch the H-bridge
in the transmitter.

###  VI.2 Design of the Cryogen Tank

The cryogen tank (i.e., cryotank) has interior dimensions of $30.5\times
71\times 30.5$ cm ($H\times W\times D$) and was made of welded aluminum panels
to ensure mechanical robustness. Specifically, it was constructed using 6.35
mm 6061 aluminum sheets. However, the high thermal conductivity of aluminum
requires the addition of an outer insulating layer. Wood is inexpensive, easy
to assemble, and a poor conductor of heat due to its high porosity. The outer
layer of the cryotank was therefore produced from birch plywood, which has a
low thermal conductivity of 0.11 W/mK [32]. The structure was then insulated
using closed cell insulation foam packed within a 8.9 cm gap between the
aluminum sheets and outer wooden walls.

Despite the insulation, testing revealed that the outer walls were being
cooled below the dew point of the local environment, causing water vapor to
condense in the wood. Since water is an excellent conductor of heat (0.6
W/mK), further enhanced when frozen (1.6 W/mK), such condensation limits the
operating lifetime of the tank. Specifically, the tank was measured to have a
useful operating lifetime (before refilling) of 3-4 hours. While this is
sufficient for most experiments, revised tank designs may nevertheless replace
the wood with a plastic (e.g., polyvinyl chloride) to trade off somewhat
higher thermal conductivity for considerably lower porosity.

Figure 13: Full implementation of the cryogenic spectrometer, including
control terminal, samples, instrumentation, auxilliary power supplies, and
climate controls: (top) full view, (bottom) zoomed-in view.

The complete spectrometer system is shown in Fig. 13 (top). It is operated
from a control terminal running on a dedicated desktop computer. Due to the
known long duration of experiments, the control terminal was supported by a
backup battery. Backup batteries also protect the actuator and VTX power
supplies and the ambient air blower’s AC induction motor. Spectrometer
instrumentation was also powered by battery packs to minimize noise, as
described in Section VI.1. Thus loss of AC outlet power for up to one hour can
be sustained without interruption to the experiment. Continuous data
synchronization with a remote file server was implemented to ensure reliable
access to data.

Spectrometer instrumentation was maintained at room temperature in a shielded
steel enclosure mounted to the lid of the cryotank with a small ($\sim$2.5 mm)
air gap for thermal insulation. During the experiment, the box experiences
minimal cooling, though small amounts of ice have been observed to accumulate
on the leads of the coil; this was removed after every $\sim$4-6 hours at
cryogenic temperature using a compressed air blower at 40 psi (not shown). A
particulate filter and dessicant drier were included in the air supply.

Figure 14: U-bend conduit constructed by solvent welding of PVC pipes and
fittings. The detector coil is wrapped around the lower section of one side.

A PVC conduit formed into a U-bend was installed under the lid of the
cryotank, Fig. 14 shows the general layout. The coil was wrapped around the
lower section of the right side; uniformly-spaced threads were cut in this
section of pipe using a lathe before assembly of the U-bend. Both ends of the
conduit are accessible from outside the cryotank to enable inspection and
maintain some degree of control over temperature in the region where the
sample is installed.

Additional spectrometer features are indicated in Fig. 13 (bottom). When fully
assembled, the cryotank lid is estimated to have a mass of over 10 kg. A
pulley system and manual winch is used to lift this lid in order to access the
U-bend to wrap the coil and for visual inspection of the cryogen level. The
secondary temperature monitor uses two K-type thermocouples to monitor the
temperature of the sample and ambient space in the conduit. An ambient air
blower produces a continuous air current through the U-shaped PVC pipe
submerged in cryogen to reduce the temperature drop in the sample. A
computerized linear actuator was used to apply polarization enhancement
between scans. The actuator was also used to remove the sample from the
cryogen between scans, thus reducing heat loss from the sample between runs to
maintain consistent temperature.

###  VI.3 Coil Characterization in the Cryogen Tank

The coil under test is a solenoid (design “A” in Table 2). It uses AWG 18
copper wire wrapped around a coil form with a 0.5 mm groove, yielding a ID of
31.75 mm, and consists of 48 turns with $\sim$1.7 mm pitch (6 turns per cm).
Coil resistance and inductance was estimated from VNA data using the approach
described in Section III.2. Resistance scales linearly $\propto f$ and
inductance is constant (found to be $\sim 23.5$ $\mu$H) as expected. The self
resonant frequency of the coil was $f_{sr}\approx 7$ MHz. The VNA data was
also used to estimate Q as a function of coil temperature $T_{c}$. The
results, which are summarized in Fig. 15, show that Q increases by
approximately $3\times$ between 300 K and 77 K, in agreement with our
theoretical analysis.

Figure 15: Measured quality factor for the test coil submerged in
liquid-$N_{2}$. The highest curve in the plot is at 77 K. Subsequent lower
curves are from data recorded at intervals as the nitrogen boiled off in the
tank. The lowest curve is at room temperature. Impedance data ($R_{s}$ and
$L_{s}$) was computed by de-embedding the effects of $C_{p}$.

###  VI.4 Detector Tuning and Quality Factor

Automated measurement of the probe network impedance at all possible
combinations of tuning capacitor was used to produce a tuning table which
reports the resonant frequency and bandwidth at all points. Impedance
measurement data was also recorded for use in data visualizations.

Figure 16: Network impedance measurement at various tuning frequencies
collected at room temperature (dashed lines) and at 77 K with the detector in
parallel resonant configuration.

Fig. 16 is a plot of select measurement results across a broad range of
operating frequencies. There is an unusual drop in the network quality factor
around 2.9 MHz which is observed to be consistent at room temperature and 77
K. This might be the result of secondary resonant networks formed between the
network analyzer and the probe as long wires are required to reach the probe
in the setup. Another (more likely) possibility is that some of the capacitors
in the tuning network have significantly higher ESR than others.

Figure 17: Bandwidth computed for tuned network at resonant frequencies
corresponding to the impedance data shown in Fig. 16.

A resonant bandwidth measurement corresponding to selected points in the
previous sweep is shown in Fig. 17. The bandwidth decreases at all points at
77 K, indicating an increase in Q as expected.

###  VI.5 Room Temperature Measurements

Experiments were performed using SLSE pulse sequences on a 15 g acetaminophen
sample in the form of 30 CVS-brand 500 mg tablets. As the first step, a pulse
duration sweep at room temperature (RT) was used to determine the RF pulse
length corresponding to the optimum flip angle $\theta_{opt}$ that maximizes
the echo signal amplitude. It is known that $\theta_{opt}=119.5^{\circ}$ for
$I=1$ in crystalline powders 333This value assumes that the asymmetry
parameter $\eta\neq 0$, which is true for the vast majority of samples. The
measured nutation curve, which is shown in Fig. 18, also provides a valuable
tool to confirm that acquired signals which appear to be echoes are in fact a
result of quadrupole resonance. External environmental noise near the
resonance frequency (denoted by $\omega_{0}$) or piezoelectric ringing may
appear similar to an NQR echo, but the response of these sources does not vary
with RF pulse duration.

Figure 18: RT nutation curve produced from pulse duration sweep at 2.564 MHz
on 15 g acetaminophen (CVS brand tablets). Experimental parameters: $N=64$
scans per point, inter-experiment time of $t_{R}=20$ s, echo period
$t_{E}=900$ $\mu$s, and $N_{E}=1000$ echoes per scan.

Figure 19: Time domain echo (left) and corresponding spectrum (right) for a RT
experiment on 15 g acetaminophen at the optimal flip angle. Experimental
parameters: $N=64$ scans with pulse duration $t_{p}=250$ $\mu$s at 2.564 MHz.
Blue and red traces correspond to the real and imaginary components of the
data, respectively.

The measured time domain echo sum and the corresponding spectrum for
measurements of the $\nu_{+}$ transition at the optimal flip angle are shown
in Fig. 19. Note that the complex acquired data was phase rotated to ensure
that all the signal energy lies in the real channel; such phase sensitive
detection improves $SNR_{e}$ by a factor of 2.

###  VI.6 Low Temperature Measurements

NQR resonant frequencies are known to be temperature-dependent. Acetaminophen
is an unusual sample as it is one of the few compounds for which the
documented frequency shift for 14N NQR has a positive temperature coefficient.
Barras et al. have measured a positive temperature coefficient for the
$\nu_{+}$ line of approximately $+69$ Hz/K at temperatures between 280 and 310
K [34]. However there is no reference data on this compound at lower
temperatures. Using the known trend around room temperature 444Supplemental
data in [34] suggest the relation $\omega(T)=2562+0.069$ where $\omega$ is in
kHz. we expect $\nu_{+}$ would decrease to 2.547 MHz at 77 K assuming
linearity. However, prior measurements of other compounds shows that the
magnitude of the temperature coefficient decreases below $\sim$120 K [36, 37],
so this result is expected to be an under-estimate.

SLSE experiments were repeated in dry ice (194.7 K) and liquid-N2 (77 K); both
were chosen for their low cost and wide availability. Preventing heat flux in
the sample was found to be difficult due to the large temperature gradient
between ambient air and the cryogen, particularly as ice collects gradually
while an experiment runs. Initially the sample was inserted into the detector
for 2 s and removed between scans during the 20 s repetition time. However,
the sample temperature was noted to drop an average of 20∘ from ambient and
vary by as much as 10∘ during the experiment, which is enough for $\omega_{0}$
to vary by 1-2 kHz during acquisition. Thus, we allow the sample to cool to
thermal equilibrium inside the U-bend and perform a frequency sweep to search
for the new resonant frequency. As discussed in Section II.2, the reduced
sample temperature further increases SNR by increasing the NQR signal
amplitude. The T1 relaxation time is also expected to increase at lower
temperatures (approximately $\propto E_{a}/(RT_{s})$), so the inter-experiment
time was increased from 20 s to 60 s.

Measurements in dry ice (194.7 K) used a frequency sweep with a step size of 1
kHz. The results reveal a resonant frequency of 2.552 MHz, which is slightly
lower than expected from a linear extrapolation of near-RT data. Additionally,
the SNR per scan increased by $2.47\times$ compared to RT, which is about
$2\times$ lower than the expected increase of $4.64\times$. This discrepancy
is probably due to the additional resistance in the resonant network, which
consists of the ESR of the tuning capacitors, $R_{DS,ON}$ of the active
duplexer switches, contact resistance of the relays, and resistance of the PCB
traces and connecting wires. These components remain at RT, so their noise
does not improve with reduced probe temperature.

Measurements in liquid-N2 reveal a resonant frequency of 2.542 MHz, which in
general agreement with our prediction based on the near-RT and dry ice data.
The measured time- and frequency-domain data after $N=8$ scans is shown in
Fig. 20. The expected total SNR enhancement factor at 77 K is 160$\times$
(when the sample is cooled in addition to the coil). The experimentally
realized enhancement factor is 88.4$\times$, which is again about $2\times$
lower than predicted and likely limited by the additional RT resistance in the
resonant network.

Figure 20: Time domain echo (left) and corresponding spectrum (right) for an
experiment on 15 g acetaminophen with probe and sample at 77 K and the optimal
flip angle. Experimental parameters: $N=8$ scans, RF frequency $=2.542$ MHz.
Blue and red traces correspond to the real and imaginary components of the
data, respectively.

Fig. 21 summarizes the observed resonance frequency of the $\nu_{+}$
transition of acetaminophen versus sample temperature. While the number of
data points below RT is limited, the data suggests that the temperature
coefficient decreases below $\sim$120 K, as seen in prior work on other
compounds [36, 37].

Figure 21: Summary of measured resonance frequency of the $\nu_{+}$ transition
of acetaminophen versus sample temperature. Green squares denote prior data
from Barras et al. [34], while the other points were measured using our
proposed spectrometer. The dashed line shows a numerical fit to the data.

###  VI.7 Further Enhancement using PE-NQR

In previous work, polarization enhancement (PE) by adiabatic population
transfer from nearby protons has been demonstrated to improve $SNR_{e}$ of 14N
NQR by 25-64$\times$ [28]. Unlike the cryogenic approach, signal enhancement
by PE-NQR is not a universally applicable technique; it is not effective if
there are no protons close to the target 14N nucleus. In addition, the
prepolarization time required varies greatly between samples and is dependent
on the 1H NMR spin relaxation time $T_{1}$, which varies widely between
compounds. Furthermore, the enhancement factor is proportional to
$\nu_{H}/\nu_{N}$, meaning that is is less effective for high frequency lines.
Thus we need to know some information about the compound under study in order
to set the pre-polarization time, predict the effective SNR, and determine the
required number of scans. Nevertheless the technique has great value in some
cases, especially for low frequency ($<2$ MHz) lines.

Here we briefly describe the potential combination of PE with cryogenic noise
reduction for even higher SNR per scan. A probe designed for this purpose
should allow the sample to be first pre-polarized in a static field $B_{0}$
and then adiabatically demagnetized to zero-field prior to the NQR
measurement. Since electromagnets generate significant heat (making them
incompatible with cryogenic operation), we instead generate the required time-
varying $B_{0}$ field by using a computerized mechanical actuator to move the
sample outside a permanent magnet, as in our earlier work [28].

For simplicity, consider a mechanically simple design in which both the
actuator and magnet are immersed within the cryogenic liquid. While neodymium
alloy permanent magnets exhibit brittle behavior at all temperatures down to 4
K, their mechanical properties (including elastic modulus $E$ and breaking
stress $\sigma$) improve at low temperatures [38]. Furthermore, the remanent
field strength $B_{r}$ increases with decreasing temperature. However, NdFeB
material undergoes spin reorientation at 135 K, which can decrease the
magnetic flux by up to 14% below this temperature. Fortunately, this effect
does not cripple the effectively increased field strength of NdFeB magnets
when cooled to 77 K. To demonstrate this point, we measured the field strength
of a custom NdFeB Halbach dipole array magnet large enough to accommodate the
sample after immersion in liquid-N2 within the cryotank. The field strength
increased by 10% (from 706 mT to 777 mT) between RT and 77 K. The
corresponding signal enhancement factor (EF) for the $\nu_{+}$ transition of
acetaminophen is predicted to be $16.5\times$ [4]. Thus, combining PE-NQR with
cryogenic operation (at 77 K) is expected to increase SNR per scan by a factor
of $88.4\times(16.5)^{2}\approx 2.4\times 10^{4}$. The result would be
approximately 2 orders of magnitude improvement in sensitivity (i.e., limit of
detection) and/or 4 orders of magnitude reduction in averaging time required
to obtain a given SNR. Similar improvements are expected for 14N sites in
other molecules.

##  VII Conclusion

We have described a custom benchtop spectrometer for 14N NQR measurements. The
system allows both the detector and sample temperature to be reduced to 77 K
(and potentially even lower) to greatly improve the signal-to-noise ratio
(SNR) of NQR scans, thus increasing sensitivity and lowering the limit of
detection for a given experiment duration. Low-loss switches allow the NQR
sample probe to be reconfigured from series-tuned in transmit mode (thus
eliminating HV power supplies) to parallel-tuned in receive mode (thus
minimizing receiver RF). The probe also uses a dual-array tuning network to
enable its resonant frequency to be programmed over a broad frequency range,
while a noiseless feedback damping circuit ensures adequate bandwidth in
receive-mode. The spectrometer implements several self-monitoring features to
enable autonomous and/or remotely-controlled operation. Tests on an
acetaminophen sample confirm the expected increase in SNR per scan at
cryogenic temperatures.

###### Acknowledgements.

The authors wish to thank Dr. Michael Malone for several useful discussions.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# Invariant Properties of Fatou Set, Julia Set and Escaping Set of Holomorphic
Semigroup

Bishnu Hari Subedi Central Department of Mathematics
Institute of Science and Technology
Tribhuvan University
Kirtipur, Kathmandu
Nepal<EMAIL_ADDRESS>/<EMAIL_ADDRESS>and  Ajaya Singh
Central Department of Mathematics, Institute of Science and Technology,
Tribhuvan University, Kirtipur, Kathmandu, Nepal<EMAIL_ADDRESS>/
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove that escaping set of transcendental semigroup is
S-forward invariant. We also prove that if holomorphic semigroup is abelian,
then Fatou set, Julia set and escaping set are S-completely invariant. We see
certain cases and conditions that the holomorphic semigroup dynamics exhibits
same dynamical behavior just like the classical complex dynamics. Frequently,
we also examine certain amount of connection and contrast between classical
complex dynamics and holomorphic semigroup dynamics.

###### Key words and phrases:

Transcendental semigroup, Fatou set, Julia set, escaping set, S-completely
invariant set.

###### 2010 Mathematics Subject Classification:

37F10, 30D05

This research work of the first author is supported by PhD faculty fellowship
from University Grants Commission, Nepal.

##  1\. Introduction

It is quite natural to extend the Fatou-Julia-Eremenko theory of the iteration
of single holomorphic map in complex plane $\mathbb{C}$ or extended complex
plane $\mathbb{C}_{\infty}$ to composite of the family of holomorphic maps.
Let $\mathscr{F}$ be a space of holomorphic maps on $\mathbb{C}$ or
$\mathbb{C}_{\infty}$. For any map $\phi\in\mathscr{F}$, $\mathbb{C}$ or
$\mathbb{C}_{\infty}$ is naturally partitioned into two subsets: the set of
normality and its complement. We say that a family $\mathscr{F}$ is normal if
each sequence from the family has a subsequence which either converges
uniformly on compact subsets of $\mathbb{C}$ or $\mathbb{C}_{\infty}$ or
diverges uniformly to $\infty$. The set of normality or Fatou set $F(\phi)$ of
the map $\phi\in\mathscr{F}$ is the largest open set on which the iterates
$\phi^{n}=\phi\circ\phi\circ\ldots\circ\phi$ (n-fold composition of $\phi$
with itself) is a normal family. The complement $J(\phi)$ is the Julia set. A
maximally connected subset of Fatou set $F(f)$ is a Fatou component.

Semigroup $S$ is a very classical algebraic structure with binary composition
that satisfies associative law. It naturally arose from the general mapping of
a set into itself. So a set of holomorphic maps on $\mathbb{C}$ or
$\mathbb{C}_{\infty}$ naturally forms a semigroup. Here, we take a set $A$ of
holomorphic maps and construct a semigroup $S$ consisting of all elements that
can be expressed as a finite composition of elements in $A$. We say such a
semigroup $S$ by holomorphic semigroup generated by the set $A$. For our
simplicity, we denote the class of all rational maps of $\mathbb{C_{\infty}}$
by $\mathscr{R}$ and class of all transcendental entire maps of $\mathbb{C}$
by $\mathscr{E}$. Our particular interest is to study of dynamics of the
families of above two classes of holomorphic maps. For a collection
$\mathscr{F}=\\{f_{\alpha}\\}_{\alpha\in\Delta}$ of such maps, let

$S=\langle f_{\alpha}\rangle$

be a holomorphic semigroup generated by them. Here $\mathscr{F}$ is either a
collection $\mathscr{R}$ of rational maps or a collection $\mathscr{E}$ of
transcendental entire maps. $\Delta$ represents an index set to which $\alpha$
belongs is finite or infinite. Here, each $f\in S$ is a holomorphic function
and $S$ is closed under functional composition. Thus $f\in S$ is constructed
through the composition of finite number of functions
$f_{\alpha_{k}},\;(k=1,2,3,\ldots,m)$. That is, $f=f_{\alpha_{1}}\circ
f_{\alpha_{2}}\circ f_{\alpha_{3}}\circ\cdots\circ f_{\alpha_{m}}$. In
particular, if $f_{\alpha}\in\mathscr{R}$, we say $S=\langle
f_{\alpha}\rangle$ a rational semigroup and if $f_{\alpha}\in\mathscr{E}$, we
say $S=\langle f_{\alpha}\rangle$ a transcendental semigroup.

A semigroup generated by finitely many holomorphic functions
$f_{i},(i=1,2,\ldots,\\\ n)$ is called finitely generated holomorphic
semigroup. We write $S=\langle f_{1},f_{2},\ldots,f_{n}\rangle$. If $S$ is
generated by only one holomorphic function $f$, then $S$ is cyclic semigroup.
We write $S=\langle f\rangle$. In this case, each $g\in S$ can be written as
$g=f^{n}$, where $f^{n}$ is the nth iterates of $f$ with itself. Note that in
our study of semigroup dynamics, we say $S=\langle f\rangle$ a trivial
semigroup.

The following result will be clear from the definition of holomorphic
semigroup. It shows that every element of holomorphic semigroup can be written
as finite composition of the sequence of $f_{\alpha}$

######  Proposition 1.1.

Let $S=\langle f_{\alpha}\rangle$ be an arbitrary holomorphic semigroup. Then
for every $f\in S$, $f^{m}$(for all $m\in\mathbb{N}$) can be written as
$f^{m}=f_{\alpha_{1}}\circ f_{\alpha_{2}}\circ f_{\alpha_{3}}\circ\cdots\circ
f_{\alpha_{p}}$ for some $p\in\mathbb{N}$.

Let $f$ be a holomorphic map. We say that $f$ iteratively divergent at
$z\in\mathbb{C}$ if
$f^{n}(z)\rightarrow\alpha\;\textrm{as}\;n\rightarrow\infty$, where $\alpha$
is an essential singularity of $f$. A sequence $(f_{k})_{k\in\mathbb{N}}$ of
holomorphic maps is said to be iteratively divergent at $z$ if
$f_{k}^{n}(z)\to\alpha_{k}\;\ \text{as}\;\ n\to\infty$ for all
$k\in\mathbb{N}$, where $\alpha_{k}$ is an essential singularity of $f_{k}$
for each $k$. Semigroup $S$ is iteratively divergent at $z$ if
$f^{n}(z)\rightarrow\alpha_{f}\;\textrm{as}\;n\rightarrow\infty$, where
$\alpha_{f}$ is an essential singularity of each $f\in S$. Otherwise, a
function $f$, sequence $(f_{k})_{k\in\mathbb{N}}$ and semigroup $S$ are said
to be iteratively bounded at $z$.

Based on the definition of classical complex dynamics (that is, on the Fatou-
Julia-Eremenko theory of a complex analytic function), the Fatou set, Julia
set and escaping set in the settings of holomorphic semigroup are defined as
follows.

######  Definition 1.1 (Fatou set, Julia set and escaping set).

Fatou set of the semigroup $S$ is defined by

$F(S)=\\{z\in\mathbb{C}:S\;\ \textrm{is normal in a neighborhood of}\;\ z\\}$

and the Julia set $J(S)$ of $S$ is the compliment of $F(S)$. If $S$ is a
transcendental semigroup, the escaping set of $S$ is defined by

$I(S)=\\{z\in\mathbb{C}:S\;\text{is iteratively divergent at}\;z\\}$

We call each point of the set $I(S)$ by escaping point. Any maximally
connected subset $U$ of the Fatou set $F(S)$ is called Fatou component.

It is obvious that $F(S)$ is the largest open subset of $\mathbb{C}$ on which
the family $\mathscr{F}$ in $S$ (or semigroup $S$ itself) is normal. Hence its
compliment $J(S)$ is a smallest closed set for any semigroup $S$. Whereas the
escaping set $I(S)$ is neither an open nor a closed set (if it is non-empty)
for any semigroup $S$.

If $S=\langle f\rangle$, then $F(S),J(S)$ and $I(S)$ are respectively the
Fatou set, Julia set and escaping set in classical complex dynamics. In this
situation we simply write: $F(f),J(f)$ and $I(f)$.

The fundamental contrast between classical complex dynamics and semigroup
dynamics appears by different algebraic structure of corresponding semigroups.
In fact, non-trivial semigroup (rational or transcendental) need not be, and
most often will not be abelian. However, trivial semigroup is cyclic and
therefore abelian. As we discussed before, classical complex dynamics is a
dynamical study of trivial (cyclic) semigroup whereas semigroup dynamics is a
dynamical study of non-trivial semigroup.

The following characterization of escaping set will be clear from the
definition 1.1 of escaping set and proposition 1.1, which can be an
alternative definition.

######  Theorem 1.1.

If a complex number $z\in\mathbb{C}$ is escaping point of any transcendental
semigroup $S$, then every sequence in $S$ has a subsequence which diverges to
$\infty$ at $z$.

The following immediate relations hold for any $f\in S$ from the definition
1.1. Indeed, it shows certain connection between classical complex dynamics
and semigroup dynamics.

######  Theorem 1.2.

Let $S$ be a semigroup. Then

  1. (1)

$F(S)\subset F(f)$ for all $f\in S$ and hence $F(S)\subset\bigcap_{f\in
S}F(f)$.

  2. (2)

$J(f)\subset J(S)$ for all $f\in S$.

  3. (3)

$I(S)\subset I(f)$ for all $f\in S$ and hence $I(S)\subset\bigcap_{f\in
S}I(f)$ in the case of transcendental semigroup $S$.

Hinkkanen and Martin proved the following results ([4, Lemma 3.1 and Corollary
3.1]).

######  Theorem 1.3.

Let $S$ be a rational semigroup. Then Julia set $J(S)$ is perfect and
$J(S)=\overline{\bigcup_{f\in S}J(f)}$

K. K. Poon proved the following results ([11, Theorems 4.1 and 4.2]).

######  Theorem 1.4.

Let $S$ be a transcendental semigroup. Then Julia set $J(S)$ is perfect and
$J(S)=\overline{\bigcup_{f\in S}J(f)}$

From the theorem 1.2 ((1) and (3)), we can say that the Fatou set and the
escaping set may be empty. For example, the escaping set of semigroup
$S=\langle f,g\rangle$ generated by functions $f(z)=e^{z}$ and $g(z)=e^{-z}$
is empty (the particular function $h=g\circ f^{k}\in S$ (say) is iteratively
bounded at any $z\in I(f)$). There are several transcendental semigroups where
Fatou set and escaping set are non-empty (see for instance [9, Examples 3.2
and 3.3] and [7, Examples 2.6 and 2.7])

##  2\. Invariant features of Fatou set, Julia set and escaping set

The main contrast between classical complex dynamics and semigroup dynamics
will appear in the invariant features of Fatou set, Julia set and escaping
set. Note that invariant feature is considered a very basic and fundamental
structure of these sets.

######  Definition 2.1 (Forward, backward and completely invariant set).

For a given semigroup $S$, a set $U\subset\mathbb{C}$ is said to be
$S$-forward invariant if $f(U)\subset U$ for all $f\in S$. It is said to be
$S$-backward invariant if $f^{-1}(U)=\\{z\in\mathbb{C}:f(z)\in U\\}\subset U$
for all $f\in S$ and it is called $S$-completely invariant if it is both
S-forward and S-backward invariant.

If $S$ is a rational semigroup, then Hinkkanen and Martin [4, Theorem 2.1]
proved the following result.

######  Theorem 2.1.

The Fatou set $F(S)$ of $S$ is S-forward invariant and Julia set $J(S)$ of $S$
is S-backward invariant.

If $S$ is a transcendental semigroup, then K. K. Poon [11, Theorem 2.1] proved
the following result.

######  Theorem 2.2.

The Fatou set $F(S)$ of $S$ is S-forward invariant and Julia set $J(S)$ of $S$
is S-backward invariant.

Dinesh Kumar and Sanjay Kumar [7, Theorem 4.1] proved the following result
that shows escaping set $I(S)$ is also S-forward invariant. Here, we provide
another proof based on our definition 1.1 of escaping set.

######  Theorem 2.3.

The escaping set $I(S)$ of transcendental semigroup $S$ is S-forward
invariant.

Before proving this theorem 2.3, we define and discuss some special subsets of
semigrioup $S$.

######  Definition 2.2 (Subsemigroup, left ideal, right ideal and ideal).

A non-empty subset $T$ of semigroup $S$ is a subsemigroup of $S$ if $f\circ
g\in T$ for all $f,\;g\in T$. A non-empty subset $I$ of $S$ is called a left
(or right) ideal of $S$ if $f\circ g\in I$ (or $g\circ f\in I)$ for all $f\in
S$ and $g\in I$. $I$ is called an ideal of $S$ if it is both left and right
ideal of $S$.

Every (left or right) ideal is a subsemigroup but converse may not hold. The
left ideal, right ideal and an ideal of a semigroup $S$ can be constructed
easily as shown in the following proposition.

######  Proposition 2.1.

For any $f\in S$, the set $S\circ f=\\{g\circ f:g\in S\\}$ is a left ideal of
$S$, the set $f\circ S=\\{f\circ g:g\in S\\}$ is an right ideal of $S$ and the
set $S\circ f\circ S=\\{h\circ f\circ g:h,g\in S\\}$ is an ideal of $S$. For
any $n\in\mathbb{N},\;(g\circ f)^{n}=h\circ f$ for some $h\in S$ and $(h\circ
f\circ g)^{n}=p\circ f\circ q$ for some $p,q\in S$.

###### Proof of the theorem 2.1.

Let $z\in I(S)$. Then by the definition 1.1, semigroup $S$ is iteratively
divergent at $z$ and for any $g\in S$, the subsemigroup $S\circ g=\\{f\circ
g:f\in S\\}$ is also iteratively divergent at $z$. That is, $(f\circ
g)^{n}(z)\to\infty\;\text{as}\;n\to\infty$ for all $f\in S$. By the propositon
2.1, for all $n\in\mathbb{N}$, we have $(f\circ g)^{n}=h_{n_{i}}\circ g$ for
some $h_{n_{i}}\in S$ where $n_{i}$ depends on $n$. Therefore, $(f\circ
g)^{n}(z)\to\infty$ as $n\to\infty\Rightarrow(h_{n_{i}}\circ
g)(z)=h_{n_{i}}(g(z))\to\infty$. Since $h_{n_{i}}\in S$ is transcendental
entire function with $h_{n_{i}}(g(z))\to\infty$, we must have
$h_{n_{i}}=p^{m_{i}}$ for some $p\in S$ and $m_{i}\in\mathbb{N}$. Thus, from
$(f\circ g)^{n}(z)\to\infty\;\text{as}\;n\to\infty$ we get
$p^{m_{i}}(g(z))\to\infty$ as $m_{i}\to\infty$. This proves that $S$ diverges
at $g(z)$, so $g(z)\in I(S)$ for all $g\in S$. Hence $I(S)$ is S-forward
invariant. ∎

In the case of rational semigroup, Hinkkanen and Martin [4, Example1] provided
the following example that show that Fatou set $F(S)$ need not be backward
invariant and Julia set $J(S)$ need not be forward invariant.

######  Example 2.1.

For a rational semigroup $S=\langle z^{2},z^{2}/a\rangle$, where
$a\in\mathbb{C},|a|>1$, the Fatou set $F(S)=\\{z:|z|<1\;\text{or}\;|z|>|a|\\}$
is not S-backward invariant and Julia set
$J(S)=\\{z:1\leqslant|z|\leqslant|a|\\}$ is not S-forward invariant.

Fatou [2] and Julia [5] independently proved that classical Fatou set and
Julia set of rational function are completely invariant. Note that all three
sets $F(f),J(f)$ and $I(f)$ of transcendental entire function are also
completely invariant. This is also fundamental contrast between classical
complex dynamics and semigroup dynamics.

We prove under certain conditions, Fatou set is S-backward invariant and Julia
set is S-forward invariant of a rational semigroup $S$. First we define the
notion of abelian holomorphic semigroup.

######  Definition 2.3.

Let $S=\langle f_{\alpha}\rangle$ be a holomorphic semigroup. We say the $S$
is abelian if $f_{\alpha}\circ f_{\beta}=f_{\beta}\circ f_{\alpha}$ for all
generators $f_{\alpha}$ and $f_{\beta}$ of $S$.

######  Theorem 2.4.

The Fatou set $F(S)$ is S-backward invariant and Julia set $J(S)$ is S-forward
invariant if $S$ is an abelian rational semigroup.

###### Proof.

We prove that if $g(z)\in F(S)$, then $z\in F(S)$ for all $g\in S$. This
follows that $g^{-1}(F(S))\subset F(S)$ for all $g\in S$. Suppose, $g(z)\in
F(S)$. Let $U$ be a neighborhood of $g(z)$ such that $\overline{U}\subset
F(S)$. Then there is a subsequence $(f^{n_{j}})$ such that $f^{n_{j}}(g(z))\to
f(g(z))$ uniformly on $U$, where $f$ is rational or constant $\infty$. Since
$S$ is abelian, so we have $g(f^{n_{j}}(z))\to g(f(z))$ uniformly on $U$ as
well. This shows that $g\circ f^{n_{j}}\to g\circ f$ uniformly on $U$. This
proves $z\in F(S)$ for all $g\in S$. ∎

From the theorems 2.1 and 2.4, we can say that Fatou set $F(S)$ and Julia set
$J(S)$ are S-completely invariant if $S$ is an abelian rational semigroup. The
following example of Hinkkanen and Martin [4, Example 2] is best for the above
theorem 2.4.

######  Example 2.2.

The semigroup $S=\langle T_{n}(z):n=0,1,2,\ldots\rangle$ generated by
Tchebyshev polynomials $T_{n}(z)$ defined by $T_{0}(z)=1,T_{1}(z)=z$ and
$T_{n+1}(z)=2zT_{n}(z)-T_{n-1}(z)$ is abelian. Therefore, by above theorem
2.4, Fatou set $F(S)$ is S-forward invariant and Julia set $J(S)$ is
S-backward invariant.

Under the same condition of rational semigroup, we prove that Fatou set is
S-backward invariant and Julia set is S-forward invariant if $S$ is a
transcendental semigroup.

######  Theorem 2.5.

The Fatou set $F(S)$ is S-backward invariant and Julia set $J(S)$ is S-forward
invariant if $S$ is an abelian transcendental semigroup.

###### Proof.

The proof is similar to theorem 2.4. ∎

In [8, Theorem 2.1], Dinesh Kumar and Sanjay Kumar provided the following
condition for backward invariance of $I(S)$. Here, we give another proof based
on our definition 1.1 of escaping set.

######  Theorem 2.6.

The escaping set $I(S)$ of transcendental semigroup $S$ is S-backward
invariant if $S$ is an abelian transcendental semigroup.

###### Proof.

We prove that if $g(z)\in I(S)$, then $z\in I(S)$ for all $g\in S$. This
follows that $g^{-1}(I(S))\subset I(S)$ for all $g\in S$. This will be proved
if we are able to prove its contrapositive statement: if $z\notin I(S)$, then
$g(z)\notin I(S)$ for all $g\in S$. Let $z\notin I(S)$ there is some $f\in S$
which is iteratively bounded at $z$. That is, $f^{n}(z)\nrightarrow\infty$ as
$n\to\infty$. In this case, there exists a sequence $(f_{k})_{k\in\mathbb{N}}$
in $S$ containing $f$ which is iteratively bounded at $z$ and all subsequences
of this sequence containing $f$ are also iteratively bounded at $z$. Now, for
any $g\in S$, $(f_{k}\circ g)_{k\in\mathbb{N}}$ is a sequence in $S$. Since
$S$ is abelian and $g$ is a transcendental entire function, so by the
continuity of $g$ at $z\in\mathbb{C}$, we can write that $(f_{k}\circ
g)(z)=(g\circ f_{k})(z)$ for all $k\in\mathbb{N}$. From which it follows that
the sequence $(f_{k}\circ g)_{k\in\mathbb{N}}$ is iteratively bounded at $z$.
So, all subsequence of this sequences are iteratively bounded at $z$. From the
fact $(f_{k}\circ g)(z)=(g\circ f_{k})(z)$ for all $k\in\mathbb{N}$, we can
say that all subsequences of the sequence $(f_{k}\circ g)_{k\in\mathbb{N}}$
are iteratively bounded at $g(z)$. That is, $g(z)\notin I(S)$ for all $g\in
S$. Therefore, $g^{-1}(I(S))\subset I(S)$ for all $g\in S$. This proves that
$I(S)$ is backward invariant. ∎

From the result of above theorems 2.5 and 2.6, we can conclude that Fatou set
$F(S)$, Julia set $J(S)$ and escaping set $I(S)$ are S-completely invariant if
$S$ is an abelian transcendental semigroup. For example, the following
semigroups

  1. (1)

$\langle z+\gamma\sin z,\;z+\gamma\sin z+2k\pi\rangle$,

  2. (2)

$\langle z+\gamma\sin z,\;-z-\gamma\sin z+2k\pi\rangle$,

  3. (3)

$\langle z+\gamma e^{z},\;z+\gamma e^{z}+2k\pi i\rangle$,

  4. (4)

$\langle z-\sin z,\;z-\sin z+2\pi\rangle$,

  5. (5)

$\langle e^{\gamma z},\;e^{\gamma z+\frac{2\pi i}{\gamma}}\rangle$, where
$0<\gamma<e^{-1}$,

are abelian transcendental semigroups, so their Fatou set, Julia set and
escaping sets are S-completely invariant.

The theorems 2.4, 2.5 and 2.6 give a kind of connection between classical
complex dynamics and semigroup dynamics. We got completely invariant structure
of Fatou set, Julia set and escaping set in both classical complex dynamics
and semigroup dynamics because of their associated abelian semigroups. This
was our expectation that abelian semigroup (rational and transcendental) must
exhibit same dynamical features as in classical complex dynamics. Note that it
will not better to conclude that Fatou set, Julia set and escaping set can not
be completely invariant unless the semigroup is abelian.

If $S$ is a finitely generated rational semigroup, then Hiroki Sumi [14, Lemma
1.1.4 (2)] proved the following results.

######  Theorem 2.7.

If $S=\langle f_{1},f_{2},\ldots,f_{n}\rangle$ is a finitely generated
rational semigroup, then $F(S)=\bigcap_{i=1}^{n}f_{i}^{-1}(F(S))$ and
$J(S)=\bigcup_{i=1}^{n}f_{i}^{-1}(J(S))$

In the case of finitely generated transcendental semigroup, we prove following
result which is analogous to above theorem 2.7.

######  Theorem 2.8.

If $S=\langle f_{1},f_{2},\ldots,f_{n}\rangle$ is a finitely generated
transcendental semigroup, then $F(S)=\bigcap_{i=1}^{n}f_{i}^{-1}(F(S))$ and
$J(S)=\bigcup_{i=1}^{n}f_{i}^{-1}(J(S))$

###### Proof.

The Fatou set $F(S)$ is S-forward invariant in general (theorem 2.2). So,
$f_{i}(F(S))\subset F(S)$ and it follows
$F(S)\subset\bigcap_{i=1}^{n}f^{-1}_{i}(F(S))$ for all $i$.

Next, let $z_{0}\in\bigcap_{i=1}^{n}f^{-1}_{i}(F(S))$. Then (say)
$w_{i}=f_{i}(z_{0})\in F(S)$ for all $i$. The semigroup $S$ is normal at
$w_{i}$ for all $i$. In other words, every $g\in S$ is equicontinuous at
$w_{i}$ for all $i$. So, for any $\epsilon>0$, there is $\delta>0$ such that

$d(g(w),g(w_{i}))<\epsilon,\;\text{whenever}\;\;d(w,w_{i})<\delta$

for all $w\in F(S)$ and $i=1,2,3,\ldots,n$. Where $d$ represents Euclidean
metric on $\mathbb{C}$. For such $\delta$, there is $\eta>0$ such that

$d(f_{i}(z),f_{i}(z_{0}))<\delta,\;\text{whenever}\;\;d(z,z_{0})<\eta$

for all $z\in\bigcap_{i=1}^{n}f^{-1}_{i}(F(S))$ and $i=1,2,3,\ldots,n$. Thus,
ultimately, we conclude that

$d(g(f_{i}(z)),g(f_{i}(z_{0})))<\epsilon\;\;\text{whenever}\;\;d(z,z_{0})<\eta$

for all $z\in\bigcap_{i=1}^{n}f^{-1}_{i}(F(S))$ and $i=1,2,3,\ldots,n$. Here
$S=\bigcup_{i=1}^{n}(S\circ f_{i})$. So $S$ is equicontinuous at $z_{0}$. That
is, $z_{0}\in F(S)$. Hence, $F(S)=\bigcap_{i=1}^{n}f_{i}^{-1}(F(S))$. Second
part of the theorem easily follows as

$J(S)=\mathbb{C}-F(S)=\mathbb{C}-\bigcap_{i=1}^{n}f_{i}^{-1}(F(S))=\bigcup_{i=1}^{n}(\mathbb{C}-f_{i}^{-1}(F(S))=\bigcup_{i=1}^{n}f_{i}^{-1}(J(S)).$

∎

The relation $J(S)=\bigcup_{i=1}^{n}f_{i}^{-1}(J(S))$ for Julia set is called
backward self similarity. With this property, dynamics of semigroup can be
regarded as backward iterated function systems.

The theorems 2.1 and 2.6 can be used directly to express the escaping set
$I(S)$ as a finite intersection of all pre-images of itself under the
generators of the semigroup $S$ if it is finitely generated. The following
result is due to Dinesh Kumar and Sanjay Kumar [8, Theorem 2.6]. Here, we give
an alternative proof.

######  Theorem 2.9.

If $S=\langle f_{1},f_{2},\ldots,f_{n}\rangle$ is a finitely generated
transcendental semigroup, then $I(S)=\bigcap_{i=1}^{n}f_{i}^{-1}(I(S))$ if $S$
is abelian semigroup.

###### Proof.

$I(S)$ is forward invariant in general and under the assumption given in the
statement, it is also backward invariant (see for instance theorem 2.6). So,
we have $f_{i}(I(S))\subset I(S)$ and $f^{-1}_{i}(I(S))\subset I(S)$ for all
$1\leqslant i\leqslant n$. From which we get respectively
$I(S)\subset\bigcap_{i=1}^{n}f^{-1}_{i}(I(S))$ and
$I(S)\supset\bigcap_{i=1}^{n}f^{-1}_{i}(I(S))$ for all $1\leqslant i\leqslant
n$. Thus, we get $I(S)=\bigcap_{i=1}^{n}f_{i}^{-1}(I(S))$. ∎

Later, we found that the condition of abelian in semigroup $S$ is not required
to hold essence of above theorem 2.9. For
$I(S)\subset\bigcap_{i=1}^{n}f^{-1}_{i}(I(S))$ as above. Next, let
$z_{0}\in\bigcap_{i=1}^{n}f^{-1}_{i}(I(S))$. Then (say) $w_{i}=f_{i}(z_{0})\in
I(S)$ for all $i$. By the definition 1.1, semigroup $S$ is iteratively
divergent at $w_{i}$ for all $i$. By the theorem 1.1, every sequence
$(f_{j})_{j\in\mathbb{N}}$ in $S$ has a subsequence
$(f_{j_{k}})_{k\in\mathbb{N}}$ which diverges to $\infty$ at
$w_{i}=f_{i}(z_{0})$. This proves that the sequence $(f_{j_{k}}\circ
f_{i})_{k,i\in\mathbb{N}}$ diverges to $\infty$ at $z_{0}$. Hence the result.

##  3\. Further results in abelian semigroup dynamics

We have some further expectations that semigroup dynamics behave just like
classical complex dynamics if corresponding semigroup is abelian. We see more
specific results in abelian semigroup dynamics. This also exhibits a nice
connection between classical complex dynamics and semigroup dynamics.

######  Theorem 3.1.

Let $S$ be an abelian rational semigroup. Then $J(S)=J(f)$ for all $f\in S$ of
degree at least two.

We recall the following result of Fatou [3] and Julia [6] concerning commuting
rational functions.

######  Lemma 3.1.

Let $f$ and $g$ be two rational functions of degree at least two such that
$f\circ g=g\circ f$. Then $J(f)=J(g)$.

###### Proof of the theorem 3.1.

Since semigroup $S$ is abelian, so we have $f_{i}\circ f_{j}=f_{j}\circ f_{i}$
for all generators $f_{i}$ and $f_{j}$ with $i\neq j$. Then by above lemma
3.1, $J(f_{i})=J(f_{j})$ for all $i$ and $j$ with $i\neq j$. Also, every $f\in
S$ permutes with each generator $f_{i}$ for all $i$, so again by the same
lemma 3.1, $J(f)=J(f_{i})$ for all $i$. This fact together with the fact of
theorem 1.3, we can conclude that $J(S)=J(f)$ for all $f\in S$. ∎

The analogous result in transcendental semigroup may not hold in general
because of the essence of above lemma 3.1 is still unanswered for permutable
transcendental entire functions. Julia sets for two permutable entire
functions were studied in [12, 13, 15] where we found certain conditions from
which we can get the essence of above lemma 3.1. If we expose extra conditions
in the statement, then result analogous to the theorem 3.1 holds in the case
of transcendental semigroup. One of the analogous result of the theorem 3.1
was proved by K.K. Poon [11, Theorem 5.1]. Here, we only give sketch of the
proof similar to above proof of the theorem 3.1.

######  Theorem 3.2.

Let $S$ be an abelian transcendental semigroup in which each generator is of
finite type. Then $J(S)=J(f)$ for all $f\in S$.

Before proving this theorem 3.2, We recall the notion of finite type, bounded
type functions and other related objects. Recall that the set
$CV(f)=\\{w\in\mathbb{C}:w=f(z)\;\ \text{such that}\;\ f^{\prime}(z)=0\\}$ is
called the set of critical values. The set $AV(f)$ consisting of all
$w\in\mathbb{C}$ such that there exists a curve
$\Gamma:[0,\infty)\to\mathbb{C}$ so that $\Gamma(t)\to\infty$ and
$f(\Gamma(t))\to w$ as $t\to\infty$ is called the set of asymptotic values of
$f$ and the set $SV(f)=\overline{(CV(f)\cup AV(f))}$ is called the set of
singular values of $f$. If $SV(f)$ has only finitely many elements, then $f$
is said to be of finite type. If $SV(f)$ is a bounded set, then $f$ is said to
be of bounded type. The sets

$\mathscr{S}=\\{f:f\;\ \textrm{is of finite
type}\\}\;\;\text{and}\;\;\mathscr{B}=\\{f:f\;\ \textrm{is of bounded
type}\\}$

are respectively called Speiser class and Eremenko-Lyubich class. Again, we
recall the following result of K. K. Poon [11, Lemmas 5.1 and 5.2] concerning
commuting transcendental maps.

######  Lemma 3.2.

Let $f$ and $g$ be two transcendental entire functions of finite type. Then
$f\circ g$ is of finite type. Moreover, if $f$ and $g$ are permutable, then
$J(f)=J(g)$.

###### Proof of the theorem 3.2.

Since semigroup $S$ is abelian, then by above lemma 3.2, $J(f_{i})=J(f_{j})$
for all generators $f_{i}$ and $f_{j}$ with $i\neq j$. Each $f=f_{i_{1}}\circ
f_{i_{2}}\circ\ldots f_{i_{m}}$ is of bounded type by the same lemma 3.2 and
$J(f)=J(f_{i})$ for all $i$. This fact together with the fact of theorem 1.4,
we can conclude that $J(S)=J(f)$ for all $f\in S$.. ∎

We expect that the condition mentioned in the theorem 3.2 will also be enough
to hold $I(S)=I(f)$ for all $f\in S$.

######  Theorem 3.3.

Let $S$ is an abelian transcendental semigroup in which each generator is of
finite type (or bounded type). Then $I(S)=I(f)$ for all $f\in S$.

######  Lemma 3.3.

If $f$ and $g$ are permutable transcendental entire functions of finite type
(or bounded type), then $I(f)=I(g)$.

###### Proof.

As given in the statement of this lemma, Poon [11, Lemma 5.2] showed that
$F(f)=F(g)$(Theorem 3.2). Eremenko and Lyubich [1] proved that if
transcendental function $f\in\mathscr{B}$, then $I(f)\subset J(f)$, and
$J(f)=\overline{I(f)}$. For any function of finite type (or bounded type), we
must have $\overline{I(f)}=\overline{I(g)}$. This lemma will be proved if we
show $J(f)-I(f)=J(g)-I(g)$. Let $z\in J(f)-I(f)$. Then $z$ is a non-escaping
point of $J(f)$ and so the sequence $(f^{n})$ has a bounded subsequence at
$z$. $J(f)=J(g)$ implies that the sequence $(g^{n})$ has also a bounded
subsequence at $z$. So $z\in J(g)-I(g)$. Therefore, $J(f)-I(f)\subset
J(g)-I(g)$. By similar fashion, we can show that $J(g)-I(g)\subset J(f)-I(g)$.
Hence, we got our claim.

∎

###### Proof of the Theorem 3.3.

Since semigroup $S$ is abelian, so we have $f_{i}\circ f_{j}=f_{j}\circ f_{i}$
for all generator $f_{i}$ and $f_{j}$ with $i\neq j$. So by above lemma 3.3,
we have $I(f_{i})=I(f_{j})$. Any $f\in S$ can be written as $f=f_{i_{1}}\circ
f_{i_{2}}\circ f_{i_{3}}\circ\cdots\circ f_{i_{m}}$. By permutability of each
$f_{i}$, we can rearrange $f_{i_{j}}$ and ultimately represented by

$f=f_{1}^{t_{1}}\circ f_{2}^{t_{2}}\circ\ldots\circ f_{n}^{t_{n}}$

where each $t_{k}\geqslant 0$ is an integer for $k=1,2,\ldots,n$. The lemms
3.2 can be applied repeatably to show each of
$f_{1}^{t_{1}},f_{2}^{t_{2}},\ldots,f_{n}^{t_{n}}$ is of finite (or bounded)
type and so $f=f_{1}^{t_{1}}\circ f_{2}^{t_{2}}\circ\ldots\circ f_{n}^{t_{n}}$
is itself finite (or bounded) type. Since each $f_{i}$ permutes with $f$ and
hence again by above lemma 3.3, $I(f_{i})=I(f)$ for all $f\in S$. Therefore,
$I(S)=I(f)$ for each $f\in S$. ∎

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