# Spin-3/2 nuclear magnetic resonance: Exact solutions for aligned systems and
implications for probing Fe-based superconductors

Jaafar N. Ansari<EMAIL_ADDRESS>Department of Physics and Astronomy, George
Mason University, Fairfax, VA 22030, USA  Karen L. Sauer<EMAIL_ADDRESS>Department of Physics and Astronomy, George Mason University, Fairfax, VA
22030, USA  Quantum Science and Engineering Center, George Mason University,
Fairfax, VA 22030, USA

###### Abstract

The nuclear magnetic resonance (NMR) spectrum of spin-3/2 nuclei in a static
magnetic field aligned with one of the electric field gradient (EFG) principal
axes is developed analytically, based on fictitious spin-1/2 formalism.
Compact closed-form expressions for the eigenstates and transitions
frequencies, as well as the expectation value of the magnetic moment after
resonant excitation, are derived. Emphasis is placed on defining and
interpreting the associated Rabi frequencies, as a function of excitation
direction and ellipticity. It is found transitions inherently fall into two
subsets, depending on their sensitivity to excitation direction, with the Rabi
frequency of one subset directly depending on the asymmetry of the EFG. A
natural application is the study of Fe-based superconductors, whose
antiferromagnetic ordering at low temperatures leads to a strong intrinsic
magnetic field aligned with the EFG principal axes. Zero external-field NMR
spectra, from powder samples of two such Fe-based superconductors, BaFe2As2
and CaFe2As2, are analyzed and exemplify the simplicity in extracting the
internal magnetic field, the quadrupole coupling constant, and the EFG
asymmetry parameter, which are important for studying magnetic ordering,
structural properties, phase transitions, and NMR dynamics. Results compare
favorably to conventional high-field NMR experiments done with the rotation of
single crystals. Overall, the physical insights, afforded by the exact and
concise expressions, will lead to ready interpretation of spin-3/2 spectra as
well as precipitate new experimental directions.

##  I Introduction

Nuclei with spin greater than 1/2 have both a magnetic and electric quadrupole
moment. The former couples to a magnetic field, the latter to an electric
field gradient (EFG). Such nuclei can then serve as an internal probe of both
magnetic and electric phenomena, and are addressable using the magnetic
dipole’s interaction with a radio-frequency excitation, i.e., nuclear magnetic
resonance (NMR). Conventionally, experiments are often performed in two
limits: (1) in a large magnetic field which is commonly known as high-field
NMR, and (2) in the absence of a magnetic field, often referred to as nuclear
quadrupole resonance (NQR). In this paper, we are concerned with the exact NMR
theory of spin-3/2 nuclei in the presence of a magnetic field which is aligned
with one of the EFG principal axes. The full continuum of results between the
zero-field and high-field limits are addressed.

We tackle this problem using spin-1/2 formalism, which has not been done
before. Although the solution for this system is somewhat involved because the
quadrupolar and Zeeman Hamiltonians do not commute, it has been solved exactly
for arbitrary field directions [1, 2, 3, 4] and by perturbation theory many
times over using a variety of techniques [5, 6]. In addition, software has
been written which simulates such spectra [7, 8]. Previous work tackling the
full solution is useful for computational analysis and modeling, however, they
lack physical insight and the ability to be used for straightforward data
analysis. With this in mind, a set of tractable analytical expressions in the
EFG-aligned case is notably missing and would elevate the quality of both the
quantitative and qualitative analysis of these systems.

The present work addresses this problem, with the aim of going beyond just the
eigenfrequencies, eigenstates, and transition rates, by deriving explicit
expressions for the oscillating nuclear magnetization, including its
direction, produced after a pulse of resonant radio-frequency excitation. The
Rabi frequency, dictating the efficacy of the excitation, is proportional to
the radio-frequency strength, as well as a modifying geometric term. This
geometric term $\lambda$ depends on the direction of the excitation field as
well as the relative strengths of the static field components. An excitation
pattern, specific to the transition frequency, can therefore be visualized
through $\lambda$. Such visualization has only been done in the pure NQR case
[9]. Select transitions directly depend on the asymmetry of the electric field
gradient with respect to static magnetic field. The use of the Rabi-frequency
for EFG determination, including asymmetry, has been studied in the case of
pure NQR through the use of nutation spectroscopy [10, 11]. Computational
results of nutation spectroscopy in the presence of a magnetic field have also
been performed [12, 13], but lack analytical expressions. This paper remedies
this by presenting exact expressions of the Rabi frequencies for the case of
EFG alignment with the magnetic field.

We take the Fe-based superconductors (FeSCs) as an example and apply our
expressions to demonstrate how they can be used for analysis. These
unconventional high-$T_{\text{c}}$ superconductors are perfectly suited for
the type of analysis described in this paper since they have spin-3/2 nuclei
which experience a large $\sim$1 T internally-supplied magnetic field from the
antiferromagnetic ordering of Fe magnetic moments. The orthorhombic/tetragonal
crystal symmetry dictates that the field is always oriented along one of the
EFG principal axes [14, 15]. Therefore zero external-field NMR (ZNMR) and a
powder sample can be used since both the field and EFG are tied to the crystal
symmetry. Detailed analysis of these ZNMR spectra has been limited due to the
lack of such expressions that we bring to light in this paper. These materials
are of immediate interest in superconductivity research and the broader
condensed-matter field, because of the interplay of magnetic and electrical
order [16], as well as controversies surrounding the superconductivity
mechanism. Other applications can be found in vapor cells [17] and other
antiferromagnetic materials [18, 19].

So, in this paper, we add to the body of knowledge on spin-3/2 particles in
magnetic fields by studying the case of the field oriented along one of the
EFG principal axes. The paper is organized as follows. In the Theory section,
we start by using spin 1/2 formalism to derive easy-to-use analytical
expressions of the eigenenergies, eigenfrequencies, and eigenstates, along
with the matrix elements of the spin angular momentum operators. Formulas for
the expectation value of the nuclear magnetic moment and the Rabi frequencies
are also presented for the first time. The Results section is split into three
subsections: (1) the predicted Zeeman-quadrupolar spectrum is visualized and
two case studies of existing FeSC ZNMR data are examined to demonstrate the
analysis using the derived expressions; (2) the full solution of the
continuum, that is for all possible magnetic field and EFG strengths is shown;
and (3) the geometric behavior of the Rabi frequencies is discussed.

##  II Theory

An electrically quadrupolar nucleus which has a quadrupole moment $Q$ and a
nuclear magnetic moment, in the presence of an EFG and a magnetic field, will
experience energy level splitting due to both of these interactions. The EFG
comes from the local electronic environment and is a tensor defined as
$V_{ij}=\partial^{2}V/\partial x_{i}\partial x_{j}|_{\mathbf{r}_{0}}$ where
$V$ is the electric potential, and is evaluated at the position of the nucleus
$\mathbf{r}_{0}$. A set of three principal axes can be determined upon
diagonalization of the tensor. With one axis of the EFG principal axes assumed
to align with the magnetic field in the $z$-direction, the dominant
Hamiltonian is given by the sum of the Zeeman Hamiltonian and the quadrupolar
Hamiltonian [20]:

$H=\hbar\left[I_{z}\omega_{0}+\frac{1}{6}\omega_{Q}(3I_{z}^{2}-I^{2})+\frac{1}{12}\omega_{\perp}(I_{+}^{2}+I_{-}^{2})\right],$
(1)

where $\omega_{0}$ is the Zeeman frequency and $\omega_{Q}$ is the quadrupole
coupling constant defined as

$\omega_{0}=-\gamma B_{0},$ (2) $\omega_{Q}=\frac{3eQ}{2I(2I-1)\hbar}V_{zz}.$
(3)

And $\omega_{\perp}$ is defined so that
$\omega_{\perp}/\omega_{Q}=(V_{xx}-V_{yy})/V_{zz}\equiv\eta$ (commonly
referred to as the EFG asymmetry parameter):

$\omega_{\perp}=\frac{3eQ}{2I(2I-1)\hbar}(V_{xx}-V_{yy}).$ (4)

Note that $\eta$ is unrestricted in its range since $V_{zz}$ is not
necessarily the largest EFG component. The static magnetic field is denoted by
$B_{0}$, $e$ is the elementary charge, $\hbar$ is the reduced Planck constant,
$\gamma$ is the nuclear gyromagnetic ratio, and $\mathbf{I}$ is the nuclear
spin angular momentum operator divided by $\hbar$.

The Hamiltonian in Eq. 1 can be diagonalized in a number of ways but is
perhaps more elegantly done by using fictitious spin-1/2 formalism. Doing so
results in the eigenenergies and eigenstates (in the eigenbasis of $I_{z}$) as
given in Table 1 (see Compliment B${}_{\text{IV}}$ and C${}_{\text{IV}}$ of
[21]). As shown in Table 1, they are defined in terms of the quantities
$\theta_{\eta}$, $\theta_{\eta}^{\prime}$, and $\omega_{\pm}$ which are given
by

$\tan\theta_{\eta}=\frac{|\omega_{\perp}|}{\sqrt{3}(2\omega_{0}+\omega_{Q})}$
(5)
$\tan\theta^{\prime}_{\eta}=\frac{|\omega_{\perp}|}{\sqrt{3}(2\omega_{0}-\omega_{Q})},$
(6)
$\omega_{\pm}=\sqrt{\left(2\omega_{0}\pm\omega_{Q}\right)^{2}+\frac{\omega_{\perp}^{2}}{3}}.$
(7)

The angles are limited such that
$0\leq\theta_{\eta},\theta^{\prime}_{\eta}<\pi$. A geometric visualization of
the angles is given in Fig. 1. Upon inspection, the system can be
characterized by just three parameters: $\omega_{+}$, revealing the net
strength of the combined field components; $\theta_{\eta}$, a metric of the
EFG asymmetry; and the relative ratio of the Zeeman and quadrupole
contributions,

$R=\frac{2\omega_{0}-\omega_{Q}}{2\omega_{0}+\omega_{Q}}.$ (8)

An energy level diagram calculated from these quantities is shown in Fig. 2.

Table 1: Eigenstates and eigenenergies of the combined Zeeman-quadrupolar Hamiltonian given in Eq. 1, with angles and frequencies given in Eqs. 2, 5-7. Label | Eigenenergy | Eigenstate
---|---|---
1 | $\displaystyle\frac{\hbar}{2}(\omega_{0}+\omega_{+})$ | $\displaystyle\cos\frac{\theta_{\eta}}{2}\left|\frac{3}{2}\right\rangle+\sin\frac{\theta_{\eta}}{2}\left|\frac{-1}{2}\right\rangle$
2 | $\displaystyle\frac{\hbar}{2}(\omega_{0}-\omega_{+})$ | $\displaystyle-\sin\frac{\theta_{\eta}}{2}\left|\frac{3}{2}\right\rangle+\cos\frac{\theta_{\eta}}{2}\left|\frac{-1}{2}\right\rangle$
$1^{\prime}$ | $\displaystyle\frac{\hbar}{2}(-\omega_{0}+\omega_{-})$ | $\displaystyle\cos\frac{\theta_{\eta}^{\prime}}{2}\left|\frac{1}{2}\right\rangle+\sin\frac{\theta_{\eta}^{\prime}}{2}\left|\frac{-3}{2}\right\rangle$
$2^{\prime}$ | $\displaystyle\frac{\hbar}{2}(-\omega_{0}-\omega_{-})$ | $\displaystyle-\sin\frac{\theta_{\eta}^{\prime}}{2}\left|\frac{1}{2}\right\rangle+\cos\frac{\theta_{\eta}^{\prime}}{2}\left|\frac{-3}{2}\right\rangle$
Figure 1: Visual definitions of $\theta_{\eta}$ and $\theta_{\eta}^{\prime}$,
also defined in Eqs. 5-6. These angles can serve as a metric of transverse
field asymmetry compared to longitudinal field components. Figure 2: Energy
levels of the combined Zeeman-quadrupolar Hamiltonian. The spacings are drawn
to scale, having been calculated for a system with $R=0.81$ and
$\theta_{\eta}=0.1$. The transition frequencies are also labeled and defined
in Table 2. The O-transitions are shown in red, and the P-transitions in blue,
also defined in Table 2.

From these energy levels, six possible transition frequencies can be observed
as shown in Fig. 2, and are given in Table 2. The expressions for the
eigenfrequencies are in agreement with previous work using low-field
perturbation theory and exact expressions in the EFG-aligned case [22]. As
discussed in more detail below, the observability of these transitions depends
on the radiofrequency excitation direction with respect to the magnetic field.
More specifically, four of these transitions are completely unresponsive to
excitations parallel to the magnetic field and are labelled as the “O”
transitions (for “orthogonal”). The other two transitions are unresponsive to
excitations orthogonal to the magnetic field and are labelled as the “P”
transitions (for “parallel”). The P-transitions are equal to $\omega_{\pm}$
defined in Eq. 7.

Table 2: Exact expressions for the six possible transition frequencies and their associated spin angular momentum operator matrix elements. The first four lines of the table are those transitions which are unresponsive to excitations parallel to the magnetic field, labeled as the “O” transitions, and are given analogous names familiar in the high-field NMR limit. The last two unnamed transitions are unresponsive to excitations orthogonal to the magnetic field and are labeled as the “P” transitions. The labeling of the transitions (O vs. P) is with respect to maximum responsiveness. The angles and frequencies are given in Eqs. 2, 5-7. $p\phantom{{}^{\prime}}\leftrightarrow q\phantom{{}^{\prime}}$ | Frequency | $\left\langle p\right|I_{+}\left|q\right\rangle$ | $\left\langle p\right|I_{-}\left|q\right\rangle$ | $\left\langle p\right|I_{z}\left|q\right\rangle$ | Transition Name
---|---|---|---|---|---
Orthogonal “O” Transitions
$2\phantom{{}^{\prime}}\leftrightarrow 2^{\prime}$ | $\omega_{0}-\frac{\omega_{+}}{2}+\frac{\omega_{-}}{2}$ | $\sqrt{3}\cos\left(\frac{\theta_{\eta}-\theta^{\prime}_{\eta}}{2}\right)$ | $-2\cos\frac{\theta_{\eta}}{2}\sin\frac{\theta^{\prime}_{\eta}}{2}$ | 0 |  $\omega_{\ell}$, lower satellite
$1^{\prime}\leftrightarrow 2\phantom{{}^{\prime}}$ | $-\omega_{0}+\frac{\omega_{+}}{2}+\frac{\omega_{-}}{2}$ | $2\cos\frac{\theta_{\eta}}{2}\cos\frac{\theta^{\prime}_{\eta}}{2}$ | $-\sqrt{3}\sin\left(\frac{\theta_{\eta}-\theta^{\prime}_{\eta}}{2}\right)$ | 0 |  $\omega_{\text{c}}$, central
$1\phantom{{}^{\prime}}\leftrightarrow 1^{\prime}$ | $\omega_{0}+\frac{\omega_{+}}{2}-\frac{\omega_{-}}{2}$ | $\sqrt{3}\cos\left(\frac{\theta_{\eta}-\theta^{\prime}_{\eta}}{2}\right)$ | $2\cos\frac{\theta^{\prime}_{\eta}}{2}\sin\frac{\theta_{\eta}}{2}$ | 0 |  $\omega_{\text{u}}$, upper satellite
$1\phantom{{}^{\prime}}\leftrightarrow 2^{\prime}$ | $\omega_{0}+\frac{\omega_{+}}{2}+\frac{\omega_{-}}{2}$ | $\sqrt{3}\sin\left(\frac{\theta_{\eta}-\theta^{\prime}_{\eta}}{2}\right)$ | $-2\sin\frac{\theta_{\eta}}{2}\sin\frac{\theta^{\prime}_{\eta}}{2}$ | 0 |  $\omega_{\text{f}}$, forbidden
Parallel “P” Transitions
$1\phantom{{}^{\prime}}\leftrightarrow 2\phantom{{}^{\prime}}$ | $\omega_{+}$ | 0 | 0 | $-\sin\theta_{\eta}$ |
$1^{\prime}\leftrightarrow 2^{\prime}$ | $\omega_{-}$ | 0 | 0 | $-\sin\theta^{\prime}_{\eta}$ |

A measurement is performed by sending a time-dependent perturbation, a radio-
frequency pulse with frequency $\omega$ resonant with one of the six
transitions,

$H_{1}=\hbar\omega_{1}\mathbf{I}\cdot\hat{n}\cos(\omega t),$ (9)

where $\omega_{1}=-\gamma B_{1}$ is the Rabi frequency and $B_{1}$ is the
amplitude of the signal. The excitation direction is
$\hat{n}=\sin\theta\cos\varphi\hat{x}+\sin\theta\sin\varphi\hat{y}+\cos\theta\hat{z}$,
where $\varphi$ and $\theta$ are the usual azimuth and polar angles with
respect to $\mathbf{B}_{0}$. The response is a net nuclear magnetic moment.
For a transition from $p$ to $q$, its expectation value can be calculated
using fictitious spin-1/2 formalism and is given by

$\langle\bm{\mu}\rangle_{pq}=i\gamma\hbar
f_{pq}\sin(\lambda_{pq}\Theta)\langle
p|\mathbf{I}|q\rangle\mathrm{e}^{-i(\omega_{pq}t-\xi_{pq})}+\mathrm{c.c.},$
(10)

where c.c. stands for the complex conjugate of the preceding term,
$f_{pq}=\hbar\omega_{pq}/[2(2I+1)k_{B}T]$ is the Boltzmann factor, the tip
angle is $\Theta=\omega_{1}t_{\text{p}}$, where $t_{\text{p}}$ is the pulse
time, the transition frequency is denoted by $\omega_{pq}$, and $\xi_{pq}$ is
the argument of $\langle p|\mathbf{I}\cdot\hat{n}|q\rangle$. The Rabi
frequency is modified by an efficiency coefficient

$\lambda_{pq}\equiv|\langle p|\mathbf{I}\cdot\hat{n}|q\rangle|,$ (11)

a unitless factor. In practice, when excitation is in the same direction as
detection, Eq. 10 shows that the signal is proportional to
$\lambda_{pq}\sin(\lambda_{pq}\Theta)$, and is therefore proportional to
$\lambda_{pq}^{2}$ in the small angle limit. The utility of the efficiency
coefficient is that it contains all the geometric information of the effective
excitation. Using Table 2, the efficiency coefficients are a straightforward
calculation. For the P-transitions, since $I_{+}$ and $I_{-}$ are all zero,
the coefficients are

$\displaystyle\lambda_{12}=\sin\theta_{\eta}\cos\theta,$ (12)
$\displaystyle\lambda_{1^{\prime}2^{\prime}}=\sin\theta^{\prime}_{\eta}\cos\theta.$
(13)

Note that these are directly proportional to the EFG asymmetry and therefore
the parallel transitions can be a direct metric of the EFG asymmetry. And for
any O-transitions they are

$\displaystyle\lambda^{2}_{pq}=\frac{1}{4}\sin^{2}\theta$
$\displaystyle\left[\langle p|I_{+}|q\rangle^{2}+\langle
p|I_{-}|q\rangle^{2}\right.$ $\displaystyle\left.+2\langle
p|I_{+}|q\rangle\langle p|I_{-}|q\rangle\cos(2\varphi)\right].$ (14)

Eqs. 12-14 are in agreement with expressions derived in the zero-field case
[9].

##  III Results

Here, results stemming from the theory in the previous section are presented.
We first show a simulated spectrum produced from a specific EFG and magnetic
field strength and elucidate how it relates to previous experimental ZNMR data
on FeSCs. Several relationships are highlighted that can be used to understand
such previously unanalyzed data. The analyzed data and comparisons to other
experiments are summarized in Table 3. Afterwards, we discuss the more general
continuum of results, that is for any arbitrary EFG and magnetic field
strength. We also discuss the “detectability” of the peaks, which will bring
to light which parts of the continuum are easy or difficult to detect in an
experiment. Lastly, we visualize the Rabi frequencies which are derived in
Eqs. 12-14.

###  III.1 Predicted Spectrum and Fe-based Superconductors Example

The spectrum is simulated using the quantities given in Table 2 and is shown
in Fig. 3. This spectrum was generated by choosing specific $R$ and
$\theta_{\eta}$ values which closely resemble an experimental spectrum. All
frequencies are normalized with respect to $\omega_{+}$. The peak heights are
calculated, using Eqs.12-14, by $\sqrt{(\langle p|I_{+}|q\rangle^{2}+\langle
p|I_{-}|q\rangle^{2})/4+\langle p|I_{z}|q\rangle^{2}}$ and then normalized
with respect to the largest peak. Finally, artificial broadening was applied
to enhance visibility. The O/P-transitions, defined in Table 2, are shown in
the top/bottom panels and colored red/blue. In the figure, the O-transitions
have been labeled by their first letters (e.g., $\omega_{\text{c}}$
corresponds to the “central” transition); the P-transitions are simply labeled
by $\omega_{\pm}$.

Figure 3: A simulated spectrum of the combined Zeeman-quadrupolar Hamiltonian
made with $R=0.81$ and $\theta_{\eta}=0.1$, showing all 6 possible transitions
and the Zeeman frequency $\omega_{0}$. All frequencies are normalized with
respect to $\omega_{+}$. The O-transitions are shown in red, and the
P-transitions in blue, as defined in Table 2. Peak widths were chosen
arbitrarily. The frequencies and their values relative to neighboring
transitions are illustrated by the black arrow lines.

The predicted spectrum in Fig. 3 was constructed to match that of particular
FeSCs, which we have taken as a case study. FeSCs are characterized by their
strong internal magnetic field which results from the antiferromagnetic
ordering of Fe magnetic moments. The magnetic ordering is an intrinsic feature
of these materials, and therefore both the magnetic field and the EFG are
fixed to the crystal structure. Since the crystal structure of these materials
is either orthorhombic or tetragonal, depending on the phase, symmetry
dictates that the internal magnetic field will always be aligned with one of
the EFG principal axes. For these reasons, FeSCs can be studied by ZNMR using
either a powder sample or a single crystal.

Several features in Fig. 3 can be discussed. First, the Zeeman frequency is
simply the average of the lower and upper satellites. Second, from the formula
for the central transition, it is clear that if there is no EFG asymmetry,
then $\omega_{\text{c}}=\omega_{0}$ and the central transition would be
exactly centered between the satellite peaks. Therefore asymmetry of the
central peak with respect to the satellite peaks is a clear indication of EFG
asymmetry. Lastly, the quadrupole coupling constant can be determined by
subtraction in quadrature of adjacent peak averages. With regards to the
P-transitions, it is more fruitful to discuss them in terms of half of their
frequency values, which is how they are plotted in Fig. 3. This is because the
average of the lower and central transitions is $\omega_{-}/2$. Similarly, the
average of the central and upper transitions is $\omega_{+}/2$. Finally, the
forbidden transition is the sum of the lower, central, and upper transitions,
which is more evidently displayed in Fig. 2. With these relationships, one
only needs to know three transitions in order to determine the other three.

The above analysis can be illustrated by examining two structurally similar
FeSCs with very different spectra: BaFe2As2 [23] and CaFe2As2 [24], which
represent all available 75As ZNMR data with 3 peaks on FeSCs, to date. Both of
these materials were studied using a powder sample. We fit the existing
experimental data, as shown in Fig. 4. Using both the fits to determine the
frequencies and the equations of $\omega_{\ell}$, $\omega_{\text{c}}$, and
$\omega_{\text{u}}$ found in Table 2, we are able to solve for the quadrupole
coupling constant, internal magnetic field, and EFG asymmetry parameter. These
values are shown in Table 3 along with other literature values for comparison.
The values in that table agree with some observations of Fig. 4. Namely, one
can immediately see that the internal field of CaFe2As2 is nearly double that
of BaFe2As2, as determined by the average of the satellite peaks shown as a
dashed line in the figure. Further, the quadrupole coupling constant of
CaFe2As2 is 6 times larger than that of BaFe2As2, as evidenced by the spread
of the spectra around the central peak. The EFG asymmetry, which is indicated
by the shift of the central peak from the Zeeman frequency, is clearly evident
in BaFe2As2, but not in CaFe2As2. Finally, it should be noted that due to the
small values of $\theta_{\eta}$ for these materials, the forbidden and
P-transitions would be difficult to directly detect and, as expected, were not
reported in the literature. In addition, $\eta$ values were not reported
perhaps due to the difficulty of extracting it without the use of analytic
equations. Instead, we extracted those values by fitting the data reported in
the literature, as described above.

Figure 4: The experimental spectra of BaFe2As2 [23] and CaFe2As2 [24], both performed at low-temperature and at zero external-field with powder samples. Data points were taken from literature and fitted with 3 Lorentzian curves. The vertical dashed lines represent the Zeeman frequency for each system. These two systems are from the same family of FeSCs yet they represent two extremes of the EFG, indicated by central peak deviation from the Zeeman frequency, with BaFe2As2 having high EFG asymmetry and CaFe2As2 having zero EFG asymmetry. Larger EFG-quadrupolar coupling is indicated by larger spread of the spectra. The quadrupole coupling constant, EFG asymmetry parameter, and internal field values are given in Table 3. Table 3: Extracted parameters of BaFe2As2 and CaFe2As2. Rows with an asterisk were determined by solving the frequency equations from Table 2 with the fitted peaks from Fig. 4 for the 3 parameters; otherwise they are quoted directly from literature. We use the standard notation: $\nu_{Q}$ is the quadrupole coupling constant, $H_{\mathrm{int}}$ is the internal magnetic field, and $\eta$ is the EFG asymmetry parameter defined by $|V_{aa}-V_{bb}|/|V_{cc}|$, where $a$, $b$, and $c$ are the lattice parameters of the crystal structure. SC refers to “single crystal” and $H$ is an externally applied magnetic field.  Material |  $\nu_{Q}$ (MHz) |  $H_{\mathrm{int}}$ (T) | $\eta$ | Methodology | Temperature (K)
---|---|---|---|---|---
CaFe2As2 [25] | $12.4(1)$ | 2.6(1) | - | NMR SC $H||c$ | 20
CaFe2As2 [24]⋆ | $11.4(1)$ | 2.59(1) | 0.0(1) | ZNMR powder | 5
CaFe2As2 [26] | $12.38$ | 2.628 | 0.39(5) | NMR SC rotation | 20
CaFe2As2 [27] | $12.9(2)$ | 2.6(1) | - | NMR SC $H||c$ | 4.3
BaFe2As2 [28] | $2.21(1)$ | 1.46(4) | 1.18(3) | NMR SC rotation | 7
BaFe2As2 [23]⋆ | $2.09(3)$ | 1.377(4) | 2.0(2) | ZNMR powder | 1.5

In addition to the parameters we extracted, Table 3 also contains the same
parameters, as quoted in the literature from high-field NMR experiments that
used a single crystal sample. Comparing between different methodologies for
the same material, the quadrupole coupling constant and the internal field are
in good agreement with each other. However, the EFG asymmetry parameter has
notable disagreements for CaFe2As2, with ZNMR finding no EFG asymmetry but
high-field NMR finding a sizeable value of 0.39(5). It is unclear why this is
the case, but it has been shown that the structure in this material is
sensitive to different growth methodologies [29, 30]. We suspect that
depending on how the sample is grown, strain may exist that contributes to the
EFG.

###  III.2 Full Solution Detectability

The two case studies in the previous section only represent two sets of
$(R,\theta_{\eta})$ values. Depending on the strength of the magnetic field
and the EFG, spectra could be substantially different. Therefore, we show
visualizations of the full solution set for the O-transitions in Fig. 5 and
the P-transitions in Fig. 6. From these, one can quickly gain a sense of which
Zeeman-quadrupolar strengths and EFG asymmetry give frequencies that can be
observed and from which direction.

Figure 5: Full solution of the Zeeman plus quadrupolar Hamiltonian spectrum
for the O-transitions, as defined in Table 2, as a function of $R$ and
$\theta_{\eta}$. Each row corresponds to a different transition frequency, as
labeled on the right side of the figure. Each column represents a different
quantity of interest as labeled on the top of the figure. The first two
columns are the matrix elements of the raising and lowering operators. Here,
$p$ and $q$ refer to the labels of the eigenstates involved in the transition.
The third column represents the actual transition frequency. Finally, the
fourth column is the observability, which is calculated as
$\omega_{pq}\sqrt{\langle p|I_{+}|q\rangle^{2}+\langle
p|I_{-}|q\rangle^{2}}/2$, where $\omega_{pq}$ is the transition frequency of
the row. All frequency quantities are normalized with respect to $\omega_{+}$.
Figure 6: Similar to Fig. 5, except the two rows correspond to the
P-transitions, as defined in Table 2. The observability is calculated by
$\omega_{pq}|\langle p|I_{z}|q\rangle|$, where $\omega_{pq}$ is the transition
frequency of the row. All frequency quantities are normalized with respect to
$\omega_{+}$.

In the limit of small $\theta_{\eta}$ and $R\to 1$, which corresponds to high-
field NMR, only three of the four O-transitions are observable:
$\omega_{\text{u}}$, $\omega_{\text{c}}$, and $\omega_{\ell}$. Also, as
expected, they are responsive only to circularly polarized magnetic fields,
co-rotating with the standard rotating frame, with $\langle
p|I_{-}|q\rangle\to 0$. The $\langle p|I_{+}|q\rangle$ ratio of the lower,
central, and upper signals is found to be $\sqrt{3}$:2:$\sqrt{3}$, as has been
found previously[3, 31].

On the other hand, when $\omega_{0}\to 0$, corresponding to the NQR limit,
$\theta^{\prime}_{\eta}=\pi-\theta_{\eta}$ and $R\to-1$ for finite
$\omega_{Q}$. In this limit, there is only one observable frequency, indicated
by the equal values of $\omega_{\text{f}}$, $\omega_{\text{c}}$, and
$\omega_{\pm}$, as expected. This is due to the double degeneracy of spin-3/2
nuclei in zero-field: $E_{1}=E_{1}^{\prime}$ and $E_{2}=E_{2}^{\prime}$. This
creates two energy levels, each populated by nuclei with equal energies but
mutually opposite spin states. A linearly polarized excitation will therefore
excite both spin states, one for each circularly polarized component of the
incident pulse. Hence the observed signal will also be linearly polarized,
oscillating in the same direction as the excitation [32]. This is evident in
the fact that $\omega_{\text{f}}$ and $\omega_{\text{c}}$ together have an
equal but opposite response to the different helicity excitation; that is,
$\langle 1|I_{+}|2^{\prime}\rangle=-\langle 1^{\prime}|I_{-}|2\rangle$ in this
limit. Finally, since the signals of $\omega_{\pm}$ are inherently linear, we
can conclude that the overall response is linear.

In the case of comparable Zeeman and quadrupolar strengths, $-1<R<1$. The
observability of $\omega_{\text{u}}$ can be a metric of $R$, particularly for
lower asymmetry with $\theta_{\eta}\leq\pi/4$, since it disappears as $R$
becomes increasingly negative. It could be useful for tracking an
appearing/disappearing internal magnetic field close to the phase transitions,
in the case of FeSCs for example. For general observability and tracking of
dynamics, for instance $T_{1}/T_{2}$ data, the central line would be useful
due to its strong responses for $R$ not close to zero.

For the P-transitions plot shown in Fig. 6, only two rows are shown,
corresponding to $\omega_{\pm}$. Since all frequencies are normalized with
respect to $\omega_{+}$, the frequency plot for the first P-transition which
is defined as $\omega_{+}$ is seen as constant. Main features of observation
are that for small values of $\theta_{\eta}$ and all values of $R$, the
transitions would be difficult to observe directly. Detectability increases
with EFG asymmetry, and may even serve as a metric of it as the Rabi frequency
is directly proportional to $\omega_{\perp}$.

###  III.3 Geometric Dependence of the Rabi Frequency

Finally, we examine the signal’s dependence on the geometry of excitation and
detection, by visualizing the Rabi frequency coefficient, as defined in Eq.
11. Assuming the use of a single crystal, we can use this coefficient to gain
a better understanding of how the signal will behave for different geometric
orientations. In particular, we are focused on signals such that excitation
and detection are along the same direction with linear polarization.

Starting with the four O-transitions, their Rabi coefficients are given by Eq.
14. In particular, we plot $\lambda^{2}_{pq}$ in 3D real space, as shown in
Fig. 7. Due to the cyclic nature of the spin angular momentum operators in
$\theta_{\eta}$ and $\theta_{\eta}^{\prime}$, defined in Table 2, all of the
features of the plots can be reproduced by some phase shift. Namely, all of
the Rabi coefficient plots are phase shifted from the central transition plots
in the following ways: a $\theta_{\eta}^{\prime}-\pi$ phase shift for the
lower satellite, a $\theta_{\eta}+\pi$ phase shift for the upper satellite,
and both a $\theta_{\eta}^{\prime}-\pi$ and $\theta_{\eta}+\pi$ phase shift
for the forbidden transition. For that reason, only the unique features of the
plots are shown in the figure. For example, for high EFG asymmetry values, the
signal can be highly asymmetric for some of the O-transitions. In contrast,
for low EFG asymmetry, the O-transitions are all highly symmetric about the
$z$-axis. In all four plots however, the concept of the O-transition, that is
transitions which are unaffected by excitations parallel to the magnetic
field, is captured with obvious dips right on the $z$-axis.

Figure 7: Polar plots of the square of the Rabi frequency coefficients
$\lambda^{2}$, the geometric modifier of the Rabi frequency, for the
O-transitions, as defined in Eq. 14. These plots allow for a simple geometric
representation of the observed signal, relative to the magnetic field which is
oriented along $z$. A particular value of $R=0.81$ is chosen for all plots,
but with $\theta_{\eta}=0.1$ for high EFG asymmetry (bottom row) and
$\theta_{\eta}=0.001$ for low EFG asymmetry (top row). Each column corresponds
to a different O-transition, as labeled below each column. For the low EFG
asymmetry row, all plots look nearly identical, hence only one plot is shown.
All values have been normalized with respect to the maximum of each subplot.

For the two P-transitions, the Rabi coefficients are shown in Fig. 8. Only one
plot is shown because both plots, when normalized, have identical features for
all values of $R$ and $\theta_{\eta}$. The plots are symmetric about $z$, with
the values collapsing to zero on the $x$-$y$ plane. This is sensible since the
P-transitions are insensitive to excitations orthogonal to the magnetic field.
This could be potentially useful with regards to materials with internal
magnetic fields, such as the FeSCs. In this context, in theory with a single
crystal, the geometry of the internal field and the EFG, could be determined
with just the frequencies that are detected.

Figure 8: This is also a plot of the Rabi frequency coefficient, similar to
Fig. 7, except for the P-transitions, defined in Eqs. 12-13. Only one
transition is shown and for a single pair of $(R,\theta_{\eta})$ because the
features of the plot are identical for all values. As expected, no signal is
seen orthogonal to the $z$ direction, which is parallel to the magnetic field.

##  IV Conclusions

In this paper, the spectrum from the combined Zeeman and quadrupole
Hamiltonian of spin-3/2 particles is calculated for the case where the
magnetic field is aligned with one of the EFG principal axes. We derive
succinct closed-form solutions for transition frequencies. In addition, for
the first time, we derive magnetic moment arising from resonant excitation as
function of excitation strength and direction as well as explicit expressions
for the Rabi frequencies, given in Eqs. 12-14. The results can be used not
only for conventional NMR, where the magnetic field is applied externally, but
also for systems which produce static internal magnetic fields.

An important example of materials with intrinsic fields can be found in FeSCs.
Pre-existing 75As spectra from two such materials are analyzed. Because both
the magnetic and EFG fields are intrinsic to the crystal’s structure, these
spectra, arising from powder samples, are not broadened. Therefore, the field
values obtained compare well to single-crystal goniometer measurements done
with conventional NMR. For these materials, three of the six possible
transitions dominate. With analytic expressions, the spectra are readily
interpreted–the average of the satellite frequencies is the Zeeman frequency,
subtraction in quadrature of adjacent peak averages is proportional to the
quadrupole coupling constant, and the deviation of the central peak from the
Zeeman frequency can be used to calculate the EFG asymmetry. While the first
two calculations are straightforward, the latter is more involved. Therefore,
the relative uncertainty on the asymmetry parameter is larger than on the
other two field quantities.

In contrast, the Rabi frequency of the P-transitions is found to be directly
proportional to the EFG asymmetry parameter. This dependency can potentially
be exploited to characterize the asymmetry. For instance, excitation on one of
the P-transitions will alter the populations of the corresponding energy
levels shown in Fig. 2, according to the strength of the associated Rabi
frequency. The resulting population changes in the quantum levels can then be
read-out on the more readily detectable O-transitions. Although the
P-transitions are often weak, they are easily found once the three dominant
O-transitions are known, since the P-transitions are themselves related to the
frequencies of the O-transitions by averages, as shown in Fig. 3. There is
precedence for such two-frequency schemes in other systems [33, 34, 35]. Such
indirect detection of the asymmetry parameter could be sensitive and therefore
particularly useful for studying systems with small asymmetry. An interesting
example is the use of asymmetry as a metric for spin-fluctuations in FeSCs
[16].

In conclusion, the exact predictions presented here provide a firm foundation
for future research with spin-3/2 particles immersed in a magnetic field
aligned with the EFG frame.

###### Acknowledgements.

We would like to thank Shinji Kawasaki for insightful conversations and for
supplying his data to us for analysis. In addition, we would like to thank
Igor Mazin and Nick Curro for their input. This work was supported by the
National Science Foundation (Award No. 2214194) and the Quantum Science &
Engineering Center at George Mason University.

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