#  Configuration determination for chiral and polar crystals
by anisotropic NMR shift

Hiroaki Kusunose1,2 and Jun Kikuchi1 1Department of Physics, Meiji University,
Kawasaki 214-8571, Japan
2Quantum Research Center for Chirality, Institute for Molecular Science,
Okazaki 444-8585, Japan

###### Abstract

We propose a method to perform a configuration determination for chiral and/or
polar crystals by utilizing anisotropy of NMR shift. The chirality
(handedness) or polarity of a crystal, that is characterized by its sign, can
be extracted from the asymmetric magnetic-field angle dependence of NMR shift
in the appropriate plane, as the configuration is reflected in the off-
diagonal components of the shift tensor. This method is applicable to the
triclinic, monoclinic, and trigonal crystal structures among 230
crystallographic space groups, and the appropriate planes to extract the
asymmetric field-angle dependence are tabulated for all relevant space groups.
We discuss how to determine the appropriate plane, and to identify each
contribution of twin domains, and so on, by using the specific examples of Te,
IrSn4 and RhSn4, and the spontaneous symmetry-breaking phases of URhSn. We
also argue that an absolute-configuration determination is also possible in
accordance with the relation between chirality or polarity of crystal and the
sign of the relevant off-diagonal component of the shift tensor, provided by
theoretical evaluation of internal magnetic field from surrounding ions,
and/or by experimental input on the shape of the asymmetric etch pit on the
cleaved face of crystal.

##  I Introduction

Inversion-symmetry breaking together with spin-orbit coupling has been
attracted much attention as they are fundamental source to bring about various
nonreciprocal phenomena and cross-correlated responses [1, 2, 3]. Among them,
polar crystals provide typical playgrounds, in which some physical properties
of materials, including advanced applications, such as pyroelectricity,
piezoelectricity, ferroelectricity, second harmonic generation, and
electrooptic effect, are only allowed, or they are strongly enhanced.
Moreover, chiral crystals add an ability to convert from axial quantities to
polar ones and vice versa, such as a conversion from the applied electric
current to the angular momentum [4, 5, 6, 7, 8, 9, 10]. It is remarkable that
the spin degree of freedom is controlled even in nonmagnetic chiral materials,
which is the so-called “Chirality-Induced Spin Selectivity” (CISS) [11, 12,
13, 14, 15, 16, 17, 18].

Polarity of a crystal is determined by the sign of the polarization.
Similarly, chirality (handedness) of a crystal is determined by the sign of
the “chiralization”, which is recently introduced quantitatively by using the
concept of the electric-toroidal monopole that corresponds to a sort of “order
parameter” of the chirality [19, 20, 21, 9, 22, 23, 24]. Since the
nonreciprocal phenomena and cross-correlated responses in polar or chiral
crystals depend essentially on their polarity or chirality, it is
fundamentally significant to make a configuration determination of samples.

Circular dichroism is a very efficient method to determine the configuration
of chirality, although there are several difficulties [25]. Single crystal
X-ray diffraction is also useful, and the Flack parameter is a standard
indicator of chirality or polarity [26, 27, 28]. Nevertheless, these methods
sometime do not work out because of sample quality, size, and resolution of
X-ray beam, and so on. Therefore, a complementary method of a configuration
determination for chirality or polarity is required.

In this paper, we propose a method to perform a configuration determination
for chiral and/or polar crystals by using the anisotropic magnetic-field angle
dependence of NMR shift. The chirality (handedness) or polarity of a crystal,
that is characterized by its sign, can be extracted from the asymmetric field-
angle dependence in the appropriate plane, as the configuration is reflected
in the off-diagonal components of the shift tensor at the nuclear site.

This paper is organized as follows. In Sec. II, we present the fundamental
expressions of the NMR shift, and decompose the components of the shift and
electric-field gradient (EFG) tensors by using the orthogonal basis set. In
the decomposed expression, the symmetry of the shift and EFG tensors become
apparent which are tabulated for all site symmetries of nuclear sites. In Sec.
III, we discuss how to determine the appropriate plane from group theoretical
point of view, and summarize all appropriate planes of the relevant space
groups (triclinic, monoclinic, and trigonal crystal systems) to perform a
configuration determination. In Sec. IV, we demonstrate the present method by
using the specific examples of Te (high-symmetry Wyckoff position in trigonal
system), IrSn4 and RhSn4 (general Wyckoff position in trigonal system), and
the spontaneous symmetry-breaking phases of URhSn (high-symmetry Wyckoff
position in ordered chiral or polar system). Section V summarizes the paper.
We have two appendices. The expressions of shift and EFG tensors and their
first-order perturbation forms are given in Appendix A. The algorithm for
fitting of experimental data is briefly explained in Appendix B.

##  II Anisotropy of NMR shift

###  II.1 Nuclear spin Hamiltonian and NMR shift

Let us begin with the nuclear spin Hamiltonian to discuss the anisotropy of
the NMR shift [29]. The Hamiltonian for the nuclear spin $\bm{I}_{j}$
(dimensionless) at $j$ site is expressed as

$\displaystyle
H^{(j)}=\hbar\omega_{0}\sum_{\alpha\beta}\left[-h^{\alpha}(\delta_{\alpha\beta}+K_{\alpha\beta}^{(j)})I_{j}^{\beta}+\widetilde{V}_{\alpha\beta}^{(j)}Q_{j}^{\alpha\beta}\right],$
(1) $\displaystyle\quad
Q_{j}^{\alpha\beta}=\frac{1}{2}(I_{j}^{\alpha}I_{j}^{\beta}+I_{j}^{\beta}I_{j}^{\alpha})-\frac{1}{3}I(I+1)\delta_{\alpha\beta},$
(2)
$\displaystyle\quad\widetilde{V}_{\alpha\beta}^{(j)}=\frac{eQ/\hbar\omega_{0}}{2I(2I-1)}V_{\alpha\beta}^{(j)},$
(3)

where the first, second, and third terms are the Zeeman coupling, hyperfine
interaction, and quadrupolar interaction with the quadrupole moment $Q$,
respectively. The second-rank operator $Q_{j}^{\alpha\beta}$ is the electric
quadrupole operator, which is finite only when the nuclear spin $I>1/2$. Here,
$\bm{h}=\bm{H}/|\bm{H}|$ is the unit vector along the applied magnetic-field
direction, and $\omega_{0}=\gamma_{\rm n}|\bm{H}|$ is the nuclear resonance
frequency with the nuclear gyromagnetic ratio $\gamma_{\rm n}$. The
coefficients $K_{\alpha\beta}^{(j)}$ and $V_{\alpha\beta}^{(j)}$ are the shift
and EFG tensors, respectively, which are both time-reversal even second-rank
polar tensor with real components, and the latter is traceless symmetric.

The eigenvalues of the Hamiltonian (3) are obtained by diagonalizing
$(2I+1)\times(2I+1)$ matrix for the basis ($m=I,I-1,\cdots,-I$), and we denote
them as $\hbar\Omega_{m}^{(j)}$ in ascending order. The label $m$ of the
eigenvalues is assigned so that they are reduced to those of the Zeeman
coupling in the noninteracting limit. The total shift of the resonance
frequency relative to the bare one in the presence of the hyperfine and
quadrupolar interactions is defined by

$\displaystyle
T_{m}^{(j)}\equiv\frac{\Omega_{m-1}^{(j)}-\Omega_{m}^{(j)}-\omega_{0}}{\omega_{0}}\quad(m=I,I-1,\cdots,-I+1).$
(4)

The contribution of the hyperfine interaction to $T_{m}^{(j)}$ is often called
the Knight shift.

When the hyperfine and quadrupolar interactions are weak enough as compared
with the Zeeman coupling, we evaluate the eigenvalues by the first-order
perturbation as

$\displaystyle\frac{\Omega_{m}^{(j)}}{\omega_{0}}=-m-\sum_{\alpha\beta}\left(K_{\alpha\beta}^{(j)}m-\frac{\widetilde{V}_{\alpha\beta}^{(j)}}{2}[3m^{2}-I(I+1)]\right)h^{\alpha}h^{\beta}.$
(5)

Using this expression, the total shift (4) is given by

$\displaystyle
T_{m}^{(j)}=\sum_{\alpha\beta}\left[\frac{1}{2}(K_{\alpha\beta}^{(j)}+K_{\beta\alpha}^{(j)})-3\widetilde{V}_{\alpha\beta}^{(j)}\left(m-\frac{1}{2}\right)\right]h^{\alpha}h^{\beta}.$
(6)

Note that in the first-order perturbation, only the symmetric components of
$K_{\alpha\beta}^{(j)}$ contribute to the total shift as $h^{\alpha}h^{\beta}$
is symmetric with respect to $\alpha\leftrightarrow\beta$. Then, the expansion
coefficients, $(k_{x}^{\prime},k_{y}^{\prime},k_{z}^{\prime})$, as defined
below can be ignored.

From this expression, it is evident that an asymmetric field-angle dependence
appears in the presence of the off-diagonal components. It is the key
ingredient to perform a configuration determination for chirality and/or
polarity of crystal as discussed later.

###  II.2 Decomposition of shift and EFG tensors

In general, $3\times 3$ time-reversal even second-rank polar tensor can be
decomposed as the linear combination of nine bases composed of the electric
monopole ($\mathbb{Q}_{0}$), electric quadupoles ($\mathbb{Q}_{U}$,
$\mathbb{Q}_{V}$, $\mathbb{Q}_{YZ}$, $\mathbb{Q}_{ZX}$, $\mathbb{Q}_{XY}$),
and the electric-toroidal dipoles ($\mathbb{G}_{X}$, $\mathbb{G}_{Y}$,
$\mathbb{G}_{Z}$), and they are defined as follows
($\mathbb{G}_{\alpha}=i\mathbb{M}_{\alpha}$),

$\displaystyle\mathbb{Q}_{0}=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\\
0&1&0\\\
0&0&1\end{pmatrix},\quad\mathbb{Q}_{U}=\frac{1}{\sqrt{6}}\begin{pmatrix}-1&0&0\\\
0&-1&0\\\ 0&0&2\end{pmatrix},$ (7)
$\displaystyle\mathbb{Q}_{V}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&0\\\
0&-1&0\\\
0&0&0\end{pmatrix}\quad\mathbb{Q}_{YZ}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&0\\\
0&0&1\\\ 0&1&0\end{pmatrix},$ (8)
$\displaystyle\mathbb{Q}_{ZX}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&1\\\
0&0&0\\\
1&0&0\end{pmatrix},\quad\mathbb{Q}_{XY}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&1&0\\\
1&0&0\\\ 0&0&0\end{pmatrix},$ (9)
$\displaystyle\mathbb{M}_{X}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&0\\\
0&0&-i\\\
0&i&0\end{pmatrix},\quad\mathbb{M}_{Y}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&i\\\
0&0&0\\\ -i&0&0\end{pmatrix},$ (10)
$\displaystyle\mathbb{M}_{Z}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&-i&0\\\
i&0&0\\\ 0&0&0\end{pmatrix},$ (11)

where $U=3Z^{2}-R^{2}$ and $V=X^{2}-Y^{2}$. They are hermitian matrices, and
satisfy the orthonormality as ${\rm
Tr}(\mathbb{X}_{i}\mathbb{X}_{j})=\delta_{ij}$
($\mathbb{X}=\mathbb{Q},\mathbb{M}$). Here we have used the capital letters
$(XYZ)$ representing a local coordinate at a nuclear site.

By using these bases, the shift tensor is decomposed as

$\displaystyle\hat{K}$
$\displaystyle=\sum_{i}^{0,U,V,YZ,ZX,XY}k_{i}\,\mathbb{Q}_{i}+\sum_{i}^{X,Y,Z}k_{i}^{\prime}\,\mathbb{G}_{i}$
(12)
$\displaystyle=\begin{pmatrix}\frac{k_{0}}{\sqrt{3}}-\frac{k_{U}}{\sqrt{6}}+\frac{k_{V}}{\sqrt{2}}&\frac{k_{Z}+k_{Z}^{\prime}}{\sqrt{2}}&\frac{k_{Y}-k_{Y}^{\prime}}{\sqrt{2}}\\\
\frac{k_{Z}-k_{Z}^{\prime}}{\sqrt{2}}&\frac{k_{0}}{\sqrt{3}}-\frac{k_{U}}{\sqrt{6}}-\frac{k_{V}}{\sqrt{2}}&\frac{k_{X}+k_{X}^{\prime}}{\sqrt{2}}\\\
\frac{k_{Y}+k_{Y}^{\prime}}{\sqrt{2}}&\frac{k_{X}-k_{X}^{\prime}}{\sqrt{2}}&\frac{k_{0}}{\sqrt{3}}+\frac{2k_{U}}{\sqrt{6}}\end{pmatrix},\quad$
(13)

where the coefficients are denoted as
$(k_{YZ},k_{ZX},k_{XY})\to(k_{X},k_{Y},k_{Z})$ for notational simplicity, and
they can be obtained by $k_{i}={\rm Tr}(\hat{K}\mathbb{Q}_{i})$ and
$k_{i}^{\prime}=-{\rm Tr}(\hat{K}\mathbb{G}_{i})$ as

$\displaystyle k_{0}=\frac{1}{\sqrt{3}}(K_{XX}+K_{YY}+K_{ZZ}),$ (14)
$\displaystyle k_{U}=\frac{1}{\sqrt{6}}(2K_{ZZ}-K_{XX}-K_{YY}),\quad
k_{V}=\frac{1}{\sqrt{2}}(K_{XX}-K_{YY}),$ (15) $\displaystyle
k_{X}=\frac{1}{\sqrt{2}}(K_{YZ}+K_{ZY}),\quad
k_{Y}=\frac{1}{\sqrt{2}}(K_{ZX}+K_{XZ}),$ (16) $\displaystyle
k_{Z}=\frac{1}{\sqrt{2}}(K_{XY}+K_{YX}),\quad
k_{X}^{\prime}=\frac{1}{\sqrt{2}}(K_{YZ}-K_{ZY}),$ (17) $\displaystyle
k_{Y}^{\prime}=\frac{1}{\sqrt{2}}(K_{ZX}-K_{XZ}),\quad
k_{Z}^{\prime}=\frac{1}{\sqrt{2}}(K_{XY}-K_{YX}).$ (18)

Table 1: Site symmetry and active component of the multipole bases. The primary axis of the local coordinate is taken as $Z$ axis. The abbreviations, $U=2Z^{2}-X^{2}-Y^{2}$ and $V=X^{2}-Y^{2}$, are used. Crystal system | Site | International | # acitve | $\mathbb{Q}_{0}$ | $\mathbb{Q}_{U}$ | $\mathbb{Q}_{V}$ | $\mathbb{Q}_{YZ}$ | $\mathbb{Q}_{ZX}$ | $\mathbb{Q}_{XY}$ | $\mathbb{G}_{X}$ | $\mathbb{G}_{Y}$ | $\mathbb{G}_{Z}$
---|---|---|---|---|---|---|---|---|---|---|---|---
| symmetry | representation | symmetric+ | $(k_{0})$ | $(k_{U})$ | $(k_{V})$ | $(k_{X})$ | $(k_{Y})$ | $(k_{Z})$ | $(k_{X}^{\prime})$ | $(k_{Y}^{\prime})$ | $(k_{Z}^{\prime})$
|  |  | anti-symmetric |  | $(v_{U})$ | $(v_{V})$ | $(v_{X})$ | $(v_{Y})$ | $(v_{Z})$ |  |  |
triclinic | C1 | $1$ | $6+3$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
| Ci | $\bar{1}$ | $6+3$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
monoclinic | C2 | $2$ | $4+1$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |  |  | $\checkmark$ |  |  | $\checkmark$
| Cs | $m$ | $4+1$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |  |  | $\checkmark$ |  |  | $\checkmark$
| C2h | $2/m$ | $4+1$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |  |  | $\checkmark$ |  |  | $\checkmark$
orthorhombic | D2 | $222$ | 3 | $\checkmark$ | $\checkmark$ | $\checkmark$ |  |  |  |  |  |
| C2v | mm2 | 3 | $\checkmark$ | $\checkmark$ | $\checkmark$ |  |  |  |  |  |
| D2h | mmm | 3 | $\checkmark$ | $\checkmark$ | $\checkmark$ |  |  |  |  |  |
tetragonal | C4 | $4$ | $2+1$ | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  | $\checkmark$
| S4 | $\bar{4}$ | $2+1$ | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  | $\checkmark$
| C4h | 4/m | $2+1$ | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  | $\checkmark$
| D4 | 422 | 2 | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  |
| C4v | 4mm | 2 | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  |
| D2d |  $\bar{4}$2m | 2 | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  |
| D4h | 4/mmm | 2 | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  |
trigonal | C3 | $3$ | $2+1$ | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  | $\checkmark$
| C3i | $\bar{3}$ | $2+1$ | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  | $\checkmark$
| D3 | 32 | 2 | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  |
| C3v | 3m | 2 | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  |
| D3d |  $\bar{3}$m | 2 | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  |
hexagonal | C6 | 6 | $2+1$ | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  | $\checkmark$
| C3h | $\bar{6}$ | $2+1$ | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  | $\checkmark$
| C6h | 6/m | $2+1$ | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  | $\checkmark$
| D6 | 622 | 2 | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  |
| C6v | 6mm | 2 | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  |
| D3h |  $\bar{6}$2m | 2 | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  |
| D6h | 6/mmm | 2 | $\checkmark$ | $\checkmark$ |  |  |  |  |  |  |
cubic | T | 23 | 1 | $\checkmark$ |  |  |  |  |  |  |  |
| Th | m$\bar{3}$ | 1 | $\checkmark$ |  |  |  |  |  |  |  |
| O | 432 | 1 | $\checkmark$ |  |  |  |  |  |  |  |
| Td |  $\bar{4}$3m | 1 | $\checkmark$ |  |  |  |  |  |  |  |
| Oh | m$\bar{3}$m | 1 | $\checkmark$ |  |  |  |  |  |  |  |

The EFG tensor $\widetilde{V}_{\alpha\beta}$ can also be decomposed in a
similar way, and we denote the coefficients as $v_{i}$. As
$\widetilde{V}_{\alpha\beta}$ is traceless symmetric,
$v_{0}=v_{X}^{\prime}=v_{Y}^{\prime}=v_{Z}^{\prime}=0$.

The number of active components in the above tensors is determined by the
local site symmetry at the nuclear site. The relation between the site
symmetry and the active components are summarized in Table 1, where the
primary axis is taken as $Z$ axis. Note that the symmetric off-diagonal
components can be eliminated by appropriate rotation of the local coordinate
frame, although the anti-symmetric components cannot be. The number of the
symmetric off-diagonal components is equivalent to that of the free angle
parameters to diagonalize the symmetric part of the tensor. Only the local
symmetries 1, $\bar{1}$, 2, m, and 2/m allow the presence of symmetric off-
diagonal components, and 1, $\bar{1}$, 2, m, 2/m, 4, $\bar{4}$, 4/m, 3,
$\bar{3}$, 6, $\bar{6}$, and 6/m without vertical mirror allow the anti-
symmetric off-diagonal components.

Once $\hat{K}_{\alpha\beta}^{(j)}$ and
$\hat{\widetilde{V}}_{\alpha\beta}^{(j)}$ are determined at the particular
site of a crystal (we assign $j=1$ for this representative site), the tensors
of the other symmetry-equivalent sites can be obtained by the symmetry
operation as $\hat{K}^{(j)}=\hat{R}_{j}\hat{K}^{(1)}\hat{R}_{j}^{-1}$, and
$\hat{\widetilde{V}}_{\alpha\beta}^{(j)}$ is transformed in a similar way.
Here, $\hat{R}_{j}$ is the representation matrix of the polar vector without
the (partial) translation, and the corresponding symmetry operation with the
(partial) translation transforms the nuclear site from #1 to #$j$. Note that
when we consider the transformation property of the shift and EFG tensors, it
is sufficient to take account of the symmetry operations without the (partial)
translation of the associated point group. The coefficient of the basis
$\mathbb{X}_{i}$ for the shift tensor at $j$ site, $\hat{K}^{(j)}$, is
explicitly obtained by

$\displaystyle k_{i}^{(j)}=\sum_{k}k_{k}^{(1)}{\rm
Tr}(\mathbb{Q}_{i}\hat{R}_{j}\mathbb{Q}_{k}\hat{R}_{j}^{-1}),$ (19)
$\displaystyle k_{i}^{\prime(j)}=-\sum_{k}k_{k}^{\prime(1)}{\rm
Tr}(\mathbb{G}_{i}\hat{R}_{j}\mathbb{G}_{k}\hat{R}_{j}^{-1}),$ (20)

where common axes should be used for $\hat{R}$ and $\mathbb{Q}$, $\mathbb{G}$.

##  III Configuration determination

Figure 1:  Pair of sites (#1 and #2) with the opposite-sign off-diagonal
components ($xy$ and $zx$) related by $C_{2}$ rotation along $x$ axis
($C_{2x}$). For $C_{2}$ rotation along $z$ axis ($C_{2z}$), $yz$ and $zx$
components of #1 and #3 sites constitute a pair.

The local information at the nuclear site can be extracted by NMR measurement.
In particular, some of the off-diagonal components change their signs
depending on the chirality or polarity of crystal, since certain component of
the coordinate vector changes its sign by reversing the chirality or polarity
of crystal. Therefore, it may be useful to determine a configuration of
chirality and/or polarity by analyzing the anisotropic NMR shift carefully,
because its magnetic-field angle dependence reflects the sign and magnitude of
the off-diagonal components. In this section, we consider the condition to
realize a configuration determination, and then we demonstrate a few example
of the determination in the next section.

Let us first consider the property of the shift tensor at a general Wyckoff
position, which has only identity symmetry operation [30]. At such a nuclear
site, its shift tensor has all of the off-diagonal components as shown in
Table 1.

In the case of cubic system that always has three 2-fold axes being orthogonal
with each other, there exists a pair of symmetry-equivalent sites having the
off-diagonal components with the opposite signs, as shown in Fig. 1. Since the
contribution of such a pair of sites to the NMR spectrum is identical to that
of the corresponding pair in the crystal with opposite chirality or polarity,
a configuration determination does not work out in cubic system.

Except for cubic system, the main crystallographic principal axis to
characterize the chirality or polarity is $c$ axis (taken as $z$ axis) for
hexagonal, trigonal, tetragonal, and orthorhombic systems, while $b$ axis
(taken as $y$ axis) for monoclinic system in the standard setting. When the
chirality or polarity of crystal is reversed, the sign of $(yz,zx)$ components
is reversed for the former systems, while that of $(yz,xy)$ components is
reversed for the latter. Meanwhile, there exists 2-fold $z$ axis for
hexagonal, tetragonal, and orthorhombic systems. Since there exists a pair of
sites having the opposite signs of $(yz,zx)$ components as in the cubic
system, it is impossible to perform a configuration determination of the
chirality or polarity.

Table 2: Sohncke space group including chiral one and the shift tensor parameters at general Wyckoff position (WP), where $z_{0}=1/3$. Note that the relations between the crystal $(abc)$ and orthogonal $(xyz)$ axes are given by $a\parallel x$, $c\parallel z$ for No.151/153 and No.152/154. The relations, $(xyz)=(YZX)$ for No.151/153 and $(xyz)=(ZXY)$ for No.152/154, are used, as $C_{2}$ rotations exist along $y$ and $x$ axes, respectively. The signs in No.144/145, No.151/153, No.152/154 are for the right/left-handed structures, respectively. The asymmetric magnetic-field angle dependence reflecting the chirality appears in the “plane”. Note that $k_{x}=k_{z}=k_{x}^{\prime}=k_{z}^{\prime}=0$ for the WPs (3a,3b) ($y_{b}=2x_{a}$, $z_{c}=-1/3,1/6$) in No.151/153, while, $k_{y}=k_{z}=k_{y}^{\prime}=k_{z}^{\prime}=0$ for (3a,3b) ($y_{b}=0$, $z_{c}=1/3,-1/6$) in No.152/154. $(x_{a},y_{b},z_{c})$ is the fractional coordinate of primitive translation vectors.  Space group | WP | # | Site | $\mathbb{Q}_{0}$ | $\mathbb{Q}_{u}$ | $\mathbb{Q}_{v}$ | $\mathbb{Q}_{yz}$ | $\mathbb{Q}_{zx}$ | $\mathbb{Q}_{xy}$ | $\mathbb{G}_{x}$ | $\mathbb{G}_{y}$ | $\mathbb{G}_{z}$ | Plane
---|---|---|---|---|---|---|---|---|---|---|---|---|---
149 (D${}_{3}^{1}$, P312) | 6l(3j,3k) |  | ($z_{0}=0$) | (=No.$151$) |  |  |  |  |  |  | $zx$
150 (D${}_{3}^{2}$, P321) | 6g(3e,3f) |  | ($z_{0}=0$) | (=No.$152$) |  |  |  |  |  |  | $yz$
151/153 | 6c(3a,3b) | 1(1) | $(x_{a},y_{b},\pm z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}$ | $\pm k_{x}$ | $\pm k_{y}$ | $k_{z}$ | $\pm k_{x}^{\prime}$ | $\pm k_{y}^{\prime}$ | $k_{z}^{\prime}$ | $zx$
(D${}_{3}^{3}$, P3112/D${}_{3}^{5}$, P3212) |  | 2(1) | $(-x_{a}+y_{b},y_{b},\mp z_{c}\pm z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}$ | $\mp k_{x}$ | $\pm k_{y}$ | $-k_{z}$ | $\mp k_{x}^{\prime}$ | $\pm k_{y}^{\prime}$ | $-k_{z}^{\prime}$ |
|  | 3(2) | $(x_{a},x_{a}-y_{b},\mp z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(+)}$ | $\mp k_{x}^{(+)}$ | $\pm k_{y}^{(-)}$ | $-k_{z}^{(-)}$ | $\mp k_{x}^{\prime(-)}$ | $\pm k_{y}^{(+)}$ | $-k_{z}^{\prime}$ |
|  | 4(3) | $(-y_{b},-x_{a},\mp z_{c}\mp z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(-)}$ | $\mp k_{x}^{(-)}$ | $\pm k_{y}^{(+)}$ | $-k_{z}^{(+)}$ | $\mp k_{x}^{\prime(+)}$ | $\pm k_{y}^{\prime(-)}$ | $-k_{z}^{\prime}$ |
|  | 5(3) | $(-y_{b},x_{a}-y_{b},\pm z_{c}\pm z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(+)}$ | $\pm k_{x}^{(+)}$ | $\pm k_{y}^{(-)}$ | $k_{z}^{(-)}$ | $\pm k_{x}^{\prime(-)}$ | $\pm k_{y}^{\prime(+)}$ | $k_{z}^{\prime}$ |
|  | 6(2) | $(-x_{a}+y_{b},-x_{a},\pm z_{c}\mp z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(-)}$ | $\pm k_{x}^{(-)}$ | $\pm k_{y}^{(+)}$ | $k_{z}^{(+)}$ | $\pm k_{x}^{\prime(+)}$ | $\pm k_{y}^{\prime(-)}$ | $k_{z}^{\prime}$ |
152/154 | 6c(3a,3b) | 1(1) | $(x_{a},y_{b},\pm z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}$ | $\pm k_{x}$ | $\pm k_{y}$ | $k_{z}$ | $\pm k_{x}^{\prime}$ | $\pm k_{y}^{\prime}$ | $k_{z}^{\prime}$ | $yz$
(D${}_{3}^{4}$, P3121/D${}_{3}^{6}$, P3221) |  | 2(1) | $(x_{a}-y_{b},-y_{b},\mp z_{c}\mp z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}$ | $\pm k_{x}$ | $\mp k_{y}$ | $-k_{z}$ | $\pm k_{x}^{\prime}$ | $\mp k_{y}^{\prime}$ | $-k_{z}^{\prime}$ |
|  | 3(2) | $(-x_{a},-x_{a}+y_{b},\mp z_{c}\pm z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(+)}$ | $\pm k_{x}^{(+)}$ | $\mp k_{y}^{(-)}$ | $-k_{z}^{(-)}$ | $\pm k_{x}^{\prime(-)}$ | $\mp k_{y}^{\prime(+)}$ | $-k_{z}^{\prime}$ |
|  | 4(3) | $(y_{b},x_{a},\mp z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(-)}$ | $\pm k_{x}^{(-)}$ | $\mp k_{y}^{(+)}$ | $-k_{z}^{(+)}$ | $\pm k_{x}^{\prime(+)}$ | $\mp k_{y}^{\prime(-)}$ | $-k_{z}^{\prime}$ |
|  | 5(3) | $(-y_{b},x_{a}-y_{b},\pm z_{c}\pm z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(+)}$ | $\pm k_{x}^{(+)}$ | $\pm k_{y}^{(-)}$ | $k_{z}^{(-)}$ | $\pm k_{x}^{\prime(-)}$ | $\pm k_{y}^{\prime(+)}$ | $k_{z}^{\prime}$ |
|  | 6(2) | $(-x_{a}+y_{b},-x_{a},\pm z_{c}\mp z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(-)}$ | $\pm k_{x}^{(-)}$ | $\pm k_{y}^{(+)}$ | $k_{z}^{(+)}$ | $\pm k_{x}^{\prime(+)}$ | $\pm k_{y}^{\prime(-)}$ | $k_{z}^{\prime}$ |
155 (D${}_{3}^{7}$, R32) | 18f(9d,9e) |  | ($z_{0}=0$) | (=No.$152$) |  |  |  |  |  |  | $yz$
Table 3: Polar space group and the shift tensor parameters at general Wyckoff position (WP), where $z_{0}=0$. Note that $b\parallel y$ and $x$, $z$ are taken arbitrary for No.6, and $a\parallel x$, $c\parallel z$ for No.156 and No.157. The relations, $y\parallel Z$ for No.6, $(xyz)=(ZXY)$ for No.156, and $(xyz)=(YZX)$ for No.157, are used. The asymmetric magnetic-field angle dependence reflecting the polarity appears in the “plane”. Note that $k_{x}=k_{z}=k_{x}^{\prime}=k_{z}^{\prime}=0$ for the WPs (1a,1b) ($y_{b}=0,1/2$) in No.6 and (3c) ($y_{b}=0$) in No.157, while, $k_{y}=k_{z}=k_{y}^{\prime}=k_{z}^{\prime}=0$ for (3d) ($y_{b}=2x_{a}$) in No.156.  Space group | WP | # | Site | $\mathbb{Q}_{0}$ | $\mathbb{Q}_{u}$ | $\mathbb{Q}_{v}$ | $\mathbb{Q}_{yz}$ | $\mathbb{Q}_{zx}$ | $\mathbb{Q}_{xy}$ | $\mathbb{G}_{x}$ | $\mathbb{G}_{y}$ | $\mathbb{G}_{z}$ | Plane
---|---|---|---|---|---|---|---|---|---|---|---|---|---
6 (C${}_{\rm s}^{1}$, Pm) | 2c(1a,1b) | 1(1) | $(x_{a},y_{b},z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}$ | $k_{x}$ | $k_{y}$ | $k_{z}$ | $k_{x}^{\prime}$ | $k_{y}^{\prime}$ | $k_{z}^{\prime}$ | $zx$
|  | 2(1) | $(x_{a},-y_{b},z_{c}+z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}$ | $-k_{x}$ | $k_{y}$ | $-k_{z}$ | $-k_{x}^{\prime}$ | $k_{y}^{\prime}$ | $-k_{z}^{\prime}$ |
7 (C${}_{\rm s}^{2}$, Pc) | 2a |  | ($z_{0}=1/2$) | (=No.$6$) |  |  |  |  |  |  | $zx$
8 (C${}_{\rm s}^{3}$, Cm) | 4b |  | ($z_{0}=0$) | (=No.$6$) |  |  |  |  |  |  | $zx$
9 (C${}_{\rm s}^{4}$, Cc) | 4a |  | ($z_{0}=1/2$) | (=No.$6$) |  |  |  |  |  |  | $zx$
156 (C${}_{\rm 3v}^{1}$, P3m1) | 6e(3d) | 1(1) | $(x_{a},y_{b},z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}$ | $k_{x}$ | $k_{y}$ | $k_{z}$ | $k_{x}^{\prime}$ | $k_{y}^{\prime}$ | $k_{z}^{\prime}$ | $yz$
|  | 2(2) | $(-y_{b},x_{a}-y_{b},z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(+)}$ | $k_{x}^{(+)}$ | $k_{y}^{(-)}$ | $k_{z}^{(-)}$ | $k_{x}^{\prime(-)}$ | $k_{y}^{\prime(+)}$ | $k_{z}^{\prime}$ |
|  | 3(3) | $(-x_{a}+y_{b},-x_{a},z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(-)}$ | $k_{x}^{(-)}$ | $k_{y}^{(+)}$ | $k_{z}^{(+)}$ | $k_{x}^{\prime(+)}$ | $k_{y}^{\prime(-)}$ | $k_{z}^{\prime}$ |
|  | 4(1) | $(-x_{a}+y_{b},y_{b},z_{c}+z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}$ | $k_{x}$ | $-k_{y}$ | $-k_{z}$ | $k_{x}^{\prime}$ | $-k_{y}^{\prime}$ | $-k_{z}^{\prime}$ |
|  | 5(3) | $(x_{a},x_{a}-y_{b},z_{c}+z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(+)}$ | $k_{x}^{(+)}$ | $-k_{y}^{(-)}$ | $-k_{z}^{(-)}$ | $k_{x}^{\prime(-)}$ | $-k_{y}^{\prime(+)}$ | $-k_{z}^{\prime}$ |
|  | 6(2) | $(-y_{b},-x_{a},z_{c}+z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(-)}$ | $k_{x}^{(-)}$ | $-k_{y}^{(+)}$ | $-k_{z}^{(+)}$ | $k_{x}^{\prime(+)}$ | $-k_{y}^{\prime(-)}$ | $-k_{z}^{\prime}$ |
157 (C${}_{\rm 3v}^{2}$, P31m) | 6d(3c) | 1(1) | $(x_{a},y_{b},z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}$ | $k_{x}$ | $k_{y}$ | $k_{z}$ | $k_{x}^{\prime}$ | $k_{y}^{\prime}$ | $k_{z}^{\prime}$ | $zx$
|  | 2(2) | $(-y_{b},x_{a}-y_{b},z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(+)}$ | $k_{x}^{(+)}$ | $k_{y}^{(-)}$ | $k_{z}^{(-)}$ | $k_{x}^{\prime(-)}$ | $k_{y}^{\prime(+)}$ | $k_{z}^{\prime}$ |
|  | 3(3) | $(-x_{a}+y_{b},-x_{a},z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(-)}$ | $k_{x}^{(-)}$ | $k_{y}^{(+)}$ | $k_{z}^{(+)}$ | $k_{x}^{\prime(+)}$ | $k_{y}^{\prime(-)}$ | $k_{z}^{\prime}$ |
|  | 4(1) | $(x_{a}-y_{b},-y_{b},z_{c}+z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}$ | $-k_{x}$ | $k_{y}$ | $-k_{z}$ | $-k_{x}^{\prime}$ | $k_{y}^{\prime}$ | $-k_{z}^{\prime}$ |
|  | 5(3) | $(-x_{a},-x_{a}+y_{b},z_{c}+z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(+)}$ | $-k_{x}^{(+)}$ | $k_{y}^{(-)}$ | $-k_{z}^{(-)}$ | $-k_{x}^{\prime(-)}$ | $k_{y}^{\prime(+)}$ | $-k_{z}^{\prime}$ |
|  | 6(2) | $(y_{b},x_{a},z_{c}+z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(-)}$ | $-k_{x}^{(-)}$ | $k_{y}^{(+)}$ | $-k_{z}^{(+)}$ | $-k_{x}^{\prime(+)}$ | $k_{y}^{\prime(-)}$ | $-k_{z}^{\prime}$ |
158 (C${}_{\rm 3v}^{3}$, P3c1) | 6d |  | ($z_{0}=1/2$) | (=No.$156$) |  |  |  |  |  |  | $yz$
159 (C${}_{\rm 3v}^{4}$, P31c) | 6c |  | ($z_{0}=1/2$) | (=No.$157$) |  |  |  |  |  |  | $zx$
160 (C${}_{\rm 3v}^{5}$, R3m) | 18c(9b) |  | ($z_{0}=0$) | (=No.$156$) |  |  |  |  |  |  | $yz$
161 (C${}_{\rm 3v}^{6}$, R3c) | 18b |  | ($z_{0}=1/2$) | (=No.$156$) |  |  |  |  |  |  | $yz$
Table 4: Sohncke and polar space group and the shift tensor parameters at general Wyckoff position (WP), where $y_{0}=0$ and $z_{0}=1/3$. Note that $b\parallel y$ and $x$, $z$ are taken arbitrary for No.3, and $a\parallel x$, $c\parallel z$ for No.144, and $x$, $y$ $z$ are taken arbitrary for No.1. The relation, $y\parallel Z$ for No.3, is used. The sign in No.144/145 are for the right/left-handed crystals, respectively. The asymmetric magnetic-field angle dependence reflecting the chirality and polarity appears in the “plane”. Note that $k_{x}=k_{z}=k_{x}^{\prime}=k_{z}^{\prime}=0$ for the WPs (1a,1b,1c,1d) ($x_{a}=0,0,1/2,1/2$, $z_{c}=0,1/2,0,1/2$) in No.3.  Space group | WP | # | Site | $\mathbb{Q}_{0}$ | $\mathbb{Q}_{u}$ | $\mathbb{Q}_{v}$ | $\mathbb{Q}_{yz}$ | $\mathbb{Q}_{zx}$ | $\mathbb{Q}_{xy}$ | $\mathbb{G}_{x}$ | $\mathbb{G}_{y}$ | $\mathbb{G}_{z}$ | Plane
---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 (C${}_{1}^{1}$, P1) | 1a | 1 | $(x_{a},y_{b},z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}$ | $k_{x}$ | $k_{y}$ | $k_{z}$ | $k_{x}^{\prime}$ | $k_{y}^{\prime}$ | $k_{z}^{\prime}$ | all
3 (C${}_{2}^{1}$, P2) | 2e(1a,1b,1c,1d) | 1(1) | $(x_{a},y_{b},z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}$ | $k_{x}$ | $k_{y}$ | $k_{z}$ | $k_{x}^{\prime}$ | $k_{y}^{\prime}$ | $k_{z}^{\prime}$ | $zx$
|  | 2(1) | $(-x_{a},y_{b}+y_{0},-z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}$ | $-k_{x}$ | $k_{y}$ | $-k_{z}$ | $-k_{x}^{\prime}$ | $k_{y}^{\prime}$ | $-k_{z}^{\prime}$ |
4 (C${}_{2}^{2}$, P21) | 2a |  | ($y_{0}=1/2$) | (=No.$3$) |  |  |  |  |  |  | $zx$
5 (C${}_{2}^{3}$, C2) | 4c(2a,2b) |  | ($y_{0}=0$) | (=No.$3$) |  |  |  |  |  |  | $zx$
143 (C${}_{3}^{1}$, P3) | 3d |  | ($z_{0}=0$) | (=No.$144$) |  |  |  |  |  |  | all
144/145 | 3a | 1 | $(x_{a},y_{b},\pm z_{c})$ | $k_{0}$ | $k_{u}$ | $k_{v}$ | $\pm k_{x}$ | $\pm k_{y}$ | $k_{z}$ | $\pm k_{x}^{\prime}$ | $\pm k_{y}^{\prime}$ | $k_{z}^{\prime}$ | all
(C${}_{3}^{2}$, P31/C${}_{3}^{3}$, P32) |  | 2 | $(-y_{b},x_{a}-y_{b},\pm z_{c}\pm z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(+)}$ | $\pm k_{x}^{(+)}$ | $\pm k_{y}^{(-)}$ | $k_{z}^{(-)}$ | $\pm k_{x}^{\prime(-)}$ | $\pm k_{y}^{\prime(+)}$ | $k_{z}^{\prime}$ |
|  | 3 | $(-x_{a}+y_{b},-x_{a},\pm z_{c}\mp z_{0})$ | $k_{0}$ | $k_{u}$ | $k_{v}^{(-)}$ | $\pm k_{x}^{(-)}$ | $\pm k_{y}^{(+)}$ | $k_{z}^{(+)}$ | $\pm k_{x}^{\prime(+)}$ | $\pm k_{y}^{\prime(-)}$ | $k_{z}^{\prime}$ |
146 (C${}_{3}^{4}$, R3) | 9b |  | ($z_{0}=0$) | (=No.$144$) |  |  |  |  |  |  | all

With these observations, a configuration determination of chirality or
polarity is possible only for trigonal (C3, D3, C3v), monoclinic (C2, Cs), and
triclinic (C1) systems. Among these groups, D3 is the chiral point group, Cs,
C3v are the polar point groups, and C1, C2, C3 are the chiral and polar point
groups. The corresponding space groups are Nos.149–155 (D3), Nos.6–9 (Cs),
Nos.156–161 (C3v), No.1 (C1), Nos.3–5 (C2), and Nos.143–146 (C3),
respectively.

The above conclusion is applicable to the Wyckoff positions with higher
symmetry within the above space groups as long as their site symmetry belongs
to 2 or m as shown in Table 1. We summarize all the relevant space groups and
the site dependences of their expansion coefficients in Table 2 for Sohncke
space group including chiral one, in Table 3 for polar space group, and in
Table 4 for Sohncke and polar space group. Note that the Tables are expressed
in terms of the crystallographic orthogonal coordinate $(xyz)$ by taking the
appropriate correspondences between $(xyz)$ and $(XYZ)$. $k_{0}$ and $k_{u}$
are common in all sites. We have introduced the following abbreviations
arising from 3-fold rotation for notational simplicity,

$\displaystyle k_{v}^{(\pm)}=\frac{-k_{v}\pm\sqrt{3}k_{z}}{2},\quad
k_{z}^{(\pm)}=\frac{-k_{z}\pm\sqrt{3}k_{v}}{2},$ (21) $\displaystyle
k_{x}^{(\pm)}=\frac{-k_{x}\pm\sqrt{3}k_{y}}{2},\quad
k_{y}^{(\pm)}=\frac{-k_{y}\pm\sqrt{3}k_{x}}{2},$ (22) $\displaystyle
k_{x}^{\prime(\pm)}=\frac{-k_{x}^{\prime}\pm\sqrt{3}k_{y}^{\prime}}{2},\quad
k_{y}^{\prime(\pm)}=\frac{-k_{y}^{\prime}\pm\sqrt{3}k_{x}^{\prime}}{2}.$ (23)

##  IV Examples

Figure 2:  Equivalent Wyckoff positions of the trigonal space groups: (a) 6c
for No.152/154, (b) 3a, 3f, 3c for No.152/154, No.150, No.157, respectively.

In this section, we present three prime examples for the configuration
determination of chirality or polarity: (A) the simple chiral system, Te, (B)
the chiral system IrSn4 and RhSn4 with general Wyckoff position, and (C) URhSn
that is considered as a chiral or polar system brought about by the
spontaneous symmetry breaking. In all these example, the nuclear spin of Te or
Sn is $I=1/2$ and the quadrupolar interaction is absent. As the hyperfine
interaction is sufficiently small or assumed to be small for URhSn as compared
to $\hbar\omega_{0}$, we treat the NMR shift in the first-order perturbation
(6), and $k_{x}^{\prime}$, $k_{y}^{\prime}$, and $k_{z}^{\prime}$ can be
omitted. The applied magnetic-field direction is expressed by two angles
($\theta$, $\phi$) as

$\displaystyle\bm{h}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta),$ (24)

in the $(xyz)$ coordinate.

###  IV.1 Te

Let us first consider the case of elemental Te [31, 32, 7, 8, 9]. The space
group of Te is No.152/154 (right/left-handed structure), and 125Te is located
at Wyckoff 3a position as shown in Fig. 2(b), and its natural abundance (NA)
is $7.07\%$ and $\gamma=13.4327$ MHz/T. As the system has $C_{2}$ rotation
along $x$ axis, $C_{2x}$, at #1 site, $k_{y}=k_{z}=0$ as shown in Table 2.
Namely, the main crystallographic principal axis $Z$ is regarded as
$x\parallel a$ in crystallographic orthogonal coordinate.

The shift tensor for the right-handed structure were determined experimentally
[31] as $\sigma_{zz}=8.860\times 10^{-4}$,
$\sigma_{xx}=\sigma_{zz}-8.572\times 10^{-4}$,
$\sigma_{yy}=\sigma_{zz}+1.146\times 10^{-4}$,
$\sigma_{yz}=\sigma_{zy}=-8.830\times 10^{-4}$. By considering the different
convention of shift tensors, i.e., $\hat{K}^{(1)}=-\hat{\sigma}$, the
expansion coefficients are

$\displaystyle k_{0}^{(1)}=-1.1059\times 10^{-3},\quad
k_{u}^{(1)}=-3.0317\times 10^{-4},$ (25) $\displaystyle
k_{v}^{(1)}=6.8717\times 10^{-4},\quad k_{x}^{(1)}=1.2488\times 10^{-3}.$ (26)

The parameters for the left-handed structure (No.154) can be obtained by the
mirror operation with respect to $xy$ plane, which gives the sign change as
$k_{x}^{(1)}\to-k_{x}^{(1)}$. Note that the left-handed structure is also
obtained by the mirror operation with respect to $zx$ or its equivalent
planes.

Figure 3:  Field-angle dependences of the NMR shift of 125Te at 3a Wykoff
position for (a) $yz$ plane, (b) $zx$ plane, and (c) $xy$ plane, where the
solid (dashed) lines represent for the right (left) handed structure. The
shift is asymmetric in (a), and the lines of the right and left handed
structure are related with each other by the reflection with respect to $y$
axis. It is impossible to determine a configuration of chirality from (b) and
(c).

The field-angle dependences of the NMR shift calculated with (26) are shown in
Fig. 3 for (a) $yz$ plane, (b) $zx$ plane, and (c) $xy$ plane. These
dependences were already discussed in the pioneering work by Koma [31]. As
expected from the column “Plane” in Table 2, the difference between the
opposite chirality appears in $yz$ plane. The solid lines for the right-handed
structure is asymmetric, and these lines are the reflection of those for the
left-handed structure (the dashed lines) with respect to $y$ axis. This comes
from the fact that there is no pairs of the sites with opposite signs of the
coefficients $k_{x}$, $k_{x}^{(+)}$, and $k_{x}^{(-)}$ for one handedness, and
chirality reverses all of their signs. See the column $\mathbb{Q}_{yz}$ of
Table 2. Indeed, the following relation holds between the NMR shift of the
right-handed (R) and left-handed (L) structures,

$\displaystyle K^{(j)}_{\rm R}(\theta,\phi=\pi/2)=K^{(j)}_{\rm
L}(\pi-\theta,\phi=\pi/2).$ (27)

Note that #2 and #3 sites are equivalent in this plane owing to the presence
of $C_{2x}$ rotation.

On the other hand, in $zx$ plane in Fig. 3(b), the contributions of the pair
of #2 and #3 sites for each chirality are symmetric with respect to $x$ axis,
even though each contribution shows asymmetric angular dependence. This is
because #2 and #3 sites are related with each other by the $C_{2x}$ rotation
as discussed in the previous section, which leads to the sign reversal of
$k_{y}^{(\pm)}$ between #2 and #3 sites as shown in the column
$\mathbb{Q}_{zx}$ of Table 2. The field-angle dependences for both handedness
coincide in this plane, and it is impossible to determine the configuration of
the chirality in this plane. Moreover, in $xy$ plane, there is no distinction
between both handedness.

The anti-symmetric part of the angle dependence in $yz$ plane can be extracted
by

$\displaystyle A(yz)$
$\displaystyle\equiv\frac{1}{2}[K^{(1)}(\theta,\phi=\pi/2)-K^{(1)}(\pi-\theta,\phi=\pi/2)]$
(28) $\displaystyle=2K_{yz}\sin(2\theta).$ (29)

The sign and magnitude of the off-diagonal components $K_{yz}$ can be
determined by the linear dependence in $A(yz)$ around $\theta=\pi/2$.

Figure 4:  Subtracted spectrum (33) for Te ($w_{\rm R}>w_{\rm L}$) in
arbitrary unit. Gaussian profiles with the second moment $\Delta K=0.015$ %
are used for each resonance line. The positive (negative) intensity represents
the contribution of the right (left) handed domain. The red and blue dotted
lines represent the angle dependence of the NMR shift in the right and left
handed domains, respectively.

When both handedness is mixed in the presence of twin domains, the
contributions of both handedness appear in the spectrum. By means of the
relation (27), the contributions of the right and left handed structures can
be decomposed by the subtraction of the observed spectrum,

$\displaystyle\rho(\omega;\theta)=w_{\rm R}\bar{\rho}_{\rm
R}(\omega;\theta)+w_{\rm L}\bar{\rho}_{\rm L}(\omega;\theta),$ (30)
$\displaystyle\quad\quad\quad\bar{\rho}_{\rm R}(\omega;\theta)=\bar{\rho}_{\rm
L}(\omega;\pi-\theta),$ (31)

as

$\displaystyle\Delta\rho(\omega;\theta)$
$\displaystyle\equiv\rho(\omega;\theta)-\rho(\omega;\pi-\theta)$ (32)
$\displaystyle=(w_{\rm R}-w_{\rm L})[\bar{\rho}_{\rm
R}(\omega;\theta)-\bar{\rho}_{\rm L}(\omega;\theta)],$ (33)

where $w_{\rm R,L}$ is the domain weight of the handedness, and
$\bar{\rho}_{\rm R,L}(\omega;\theta)$ is an ideal spectrum from pure single
domain. Note that the subtracted spectrum vanishes when $w_{\rm R}=w_{\rm L}$,
otherwise the positive (negative) intensity represents the right (left) handed
spectrum for $w_{\rm R}>w_{\rm L}$. An example plot of the subtracted spectrum
for $w_{\rm R}>w_{\rm L}$ is given in Fig. 4. It may be useful to assign the
branches arising from each handedness, and then the domain weight is
determined by (31).

As discussed above, it is possible to determine the relative ratio of the
chirality of crystal by analyzing the field-angle dependence of the NMR shift.
However, in practice, the absolute determination of the chirality is
impossible by the NMR shift alone, because we cannot know a priori the sign of
the off-diagonal components for given handedness. For the absolute-
configuration determination, one needs further input from the shape of the
asymmetric etch pit on the cleaved face of crystal [31, 8] or theoretical
estimation of the sign of the off-diagonal components based on, e.g., the
orbital current model for nonmagnetic crystals [32, 33, 34, 29] or the dipole-
dipole interaction for magnetic ones [29].

###  IV.2 IrSn4 and RhSn4

Figure 5:  Field-angle dependences of the NMR shift of 117Sn at 6c Wykoff
position in (a) $yz$ plane and (b) $zx$ plane.

Next, we consider the case of a general Wyckoff position. For such purpose, we
take IrSn4 and RhSn4 [35] whose space group are the same as Te. The 117Sn
nucleus is located at 6c site as shown in Fig. 2(a)
[$(x,y,z)\simeq(0.23,0.5,0.43)$], and its NA is $7.5\%$. Note that Ir and Rh
are at 3a site, and the similar analysis of Te can be made. Since the shift
parameters are undetermined experimentally at the moment, we use the tentative
values,
$(k_{0}^{(1)},k_{u}^{(1)},k_{v}^{(1)},k_{x}^{(1)},k_{y}^{(1)},k_{z}^{(1)})=(20,1.0,3.0,5.0,1.0,10)\times
10^{-4}$. It should be emphasized that the qualitative conclusions are not
altered by different choices of parameters.

The field-angle dependences of the NMR shift are shown in Fig. 5 for (a) $yz$
plane and (b) $zx$ plane. As similar to the results of Te, the difference
between the right and left handed structures appears only in $yz$ plane. Note
that the pairs of sites, (#1,#2), (#3,#5), and (#4,#6), are equivalent in $yz$
plane as they are related by $C_{2x}$ rotation. The same relations (27) and
(29) also hold for 6c site, and the analysis using (33) can also be applied to
decompose the contributions of twin domains.

###  IV.3 URhSn

Table 5:  Wyckoff positions and site symmetry for URhSn.  Atom | Site (No.189) | No.189 | No.150 | No.157
---|---|---|---|---
U | $(0.5925,0,0)$ | 3f (m2m) | 3e (.2.) | 3c (..m)
Rh1 | $(1/3,2/3,1/2)$ | 2d ($\bar{6}..$) | 2d (3..) | 2b (3..)
Rh2 | $(0,0,0)$ | 1a ($\bar{6}$2m) | 1a (32.) | 1a (31m)
Sn | $(1/4,0,1/2)$ | 3g (m2m) | 3f (.2.) | 3c (..m)
Figure 6:  Two candidates of quadrupole order in URhSn, (a) $Q_{yz}$ order
(No.150, Sohncke) and (b) $Q_{zx}$ order (No.157 polar).  Figure 7:  Field-
angle dependences of the NMR shift of 117Sn for (a), (c) No.150 at 3f site and
(b), (d) No.157 at 3c site. Only the angle dependences for the right-handed
structure are shown for clarity. Note that $k_{x}$ ($k_{y}$) increases with
increase of the order parameter in No.150 Sohncke (No.157 polar) phase.

The hexagonal compound URhSn belongs to the achiral nonpolar space group,
No.189 (D${}_{3h}^{3}$, P$\bar{6}$2m), and exhibits successive phase
transitions at $T_{0}=54$ K and $T_{c}=16$ K [36]. The low-temperature phase
below $T_{c}$ is ferromagnetic with the ordered moments along $c$ axis, while
the intermediate phase between $T_{0}$ and $T_{c}$ is not clearly identified
yet. At the onset of the intermediate phase, the magnetic susceptibility shows
only a weak anomaly and the structural deformation was not reported within the
experimental accuracy [37, 38]. It has been argued that the electric multipole
order with the uniform ordering vector $\bm{q}=0$ is most likely [39, 40, 41,
42, 43].

The recent resonant X-ray scattering experiment observed slight increase of
the intensity only in the $\pi$-$\sigma^{\prime}$ scattering, which suggests
that the electric charge distribution has $z$ component, i.e., the quadrupole
moment $Q_{yz}$ or $Q_{zx}$ [41]. Moreover, $C_{44}$ mode in the ultrasonic
measurements shows relatively large softening toward $T_{0}$, which indicates
that $Q_{yz}$ and $Q_{zx}$ are relevant in the intermediate phase [43]. The
longitudinal $C_{33}$ mode also exhibits almost discontinuous softening just
below $T_{0}$, which is consistent with the fact that Rh1 site position
changes from $(1/3,2/3,1/2)$ to $(1/3,2/3,z)$ with the free parameter
$z\approx 1/2$. The preliminary result of Sn-NMR measurement suggests the lack
of the $zx$ mirror plane [42].

The ordering of the cluster of $Q_{yz}$ quadrupoles with
A${}_{1}^{\prime\prime}$ irreducible representation (irrep.) as shown in Fig.
6(a) leads to symmetry lowering to No.150 (Sohncke group), while that of the
cluster of $Q_{zx}$ quadrupoles with A${}_{2}^{\prime\prime}$ irrep. (Fig.
6(b)) leads to No.157 (polar group), both of which belong to the maximal
subgroups of No.189 [39, 40]. The change of site symmetry due to these phase
transitions is summarized in Table 5.

Although the NMR shift of 117Sn is symmetric in $yz$ and $zx$ planes with the
extrema at $x$, $y$, and $z$ axes in the paramagnetic phase, it becomes
asymmetric when either $Q_{yz}$ (No.150, Sohncke) or $Q_{zx}$ (No.157, polar)
ordering is realized owing to the appearance of nonzero off-diagonal component
as shown in Tables 2 and 3. It should be emphasized that the asymmetric field-
angle dependence appears in $yz$ plane for No.150 phase, while in $zx$ plane
for No.157 phase as shown below.

At #1 site, $k_{y}^{(1)}=k_{z}^{(1)}=0$ ($k_{x}^{(1)}=k_{z}^{(1)}=0$) owing to
the presence of $C_{2x}$ rotation ($\sigma_{zx}$ mirror). By using the
tentative values, $(k_{0}^{(1)},k_{u}^{(1)},k_{v}^{(1)})=(10,1.0,1.0)\times
10^{-4}$, the field-angle dependences of the NMR shift in $yz$ and $zx$ planes
are shown in Fig. 7 (a) for No.150 with $k_{x}^{(1)}=(0,0.5,1.0)\times
10^{-4}$ and (b) for No.157 with $k_{y}^{(1)}=(0,0.5,1.0)\times 10^{-4}$. The
contributions of #2 and #3 sites in $yz$ ($zx$) plane for No.150 (No.157) are
equivalent as they are related by $C_{2x}$ rotation ($\sigma_{zx}$ mirror).
Note that the field-angle dependences of No.150 (No.157) for the left handed
(negative polarity) structure are given by $k_{x}^{(1)}\to-k_{x}^{(1)}$
($k_{y}^{(1)}\to-k_{y}^{(1)}$), and the similar relation (27) between opposite
chirality (polarity) also holds. Note that the binary nature of chirality or
polarity corresponds to the existence of opposite domain of the quadrupole
order. On the other hand, the contributions of the #2 and #3 sites are
symmetric in $zx$ plane ($yz$ plane) for No.150 (No.157) since #2 and #3 sites
are related with each other by the $C_{2x}$ rotation ($\sigma_{zx}$ mirror).

As the symmetry property at the Sn site is the same as elemental Te for
No.150, the anti-symmetric part is also characterized by (29), while that for
No.157 in $zx$ plane is given by

$\displaystyle A(zx)$
$\displaystyle\equiv\frac{1}{2}[K^{(1)}(\theta,\phi=0)-K^{(1)}(\pi-\theta,\phi=0)]$
(34) $\displaystyle=2K_{zx}\sin(2\theta).$ (35)

The anti-symmetric parts in (29) and (35) are proportional to $K_{yz}$ and
$K_{zx}$, respectively, which are expected to increase with the growth of the
order parameters, $\braket{Q_{yz}}$ and $\braket{Q_{zx}}$. Therefore, they are
expected to rapidly develop just below $T_{0}$.

By the same analysis using the subtracted spectrum as (33), each contribution
of the left or right handedness (positive or negative polarity) can be
extracted. Therefore, careful analysis of the NMR shift of 117Sn is useful to
reveal the enigmatic phase transition at $T_{0}$ in URhSn.

##  V Summary

We have investigated the condition to perform a configuration determination
for chiral and/or polar crystals by using the asymmetric magnetic-field angle
dependence of NMR shift in the appropriate plane, where the asymmetry depends
on its chirality or polarity. This method is employed for the triclinic,
monoclinic, and trigonal crystal structures among 230 crystallographic space
groups. The appropriate plane to extract chirality or polarity from the
asymmetric field-angle dependence is the plane perpendicular to the 2-fold
rotation axis or the mirror plane of the crystal if it exists, otherwise all
planes can be used.

We have discussed the asymmetric behaviors of the NMR shift in the appropriate
plane, and how to identify each contribution of twin domains, and so on, by
using the specific examples of Te, IrSn4 and RhSn4, and the spontaneous
symmetry-breaking phases of URhSn. The analysis of the anisotropic NMR shift
provides a complementary method of a configuration determination for chirality
or polarity, and quantifies the ratio of twin domains, which could stimulate
further investigation on nonreciprocal phenomena and cross-correlated
responses in the inversion-symmetry breaking systems.

###### Acknowledgements.

We acknowledge many useful discussions with Yoshihiko Togawa, Hiroshi M.
Yamamoto, Jun-ichiro Kishine, Yusuke Kato, Hidekazu Mukuda, Tetsuaki Itou,
Masashi Takigawa, Tatsuma D. Matsuda, Kazumasa Hattori, Takayuki Ishitobi,
Tatsuya Yanagisawa, Yo Tokunaga, and Chihiro Tabata. This research was
supported by JSPS KAKENHI Grants Numbers JP23K03288, JP23H00091, and the
grants of Special Project (IMS program 23IMS1101), and OML Project (NINS
program No, OML012301) by the National Institutes of Natural Sciences.

##  Appendix A On shift and EFG tensors

###  A.1 Hyperfine coupling and shift tensor

Let us start with the hyperfine interaction at the $j$ nuclear site with
electronic spin $\bm{S}_{k}$ (dimensionless) at $k$ site [29],

$\displaystyle H_{\rm
hf}^{(j)}=\sum_{k}\sum_{\alpha\beta}^{x,y,z}A_{jk}^{\alpha\beta}I_{j}^{\alpha}S_{k}^{\beta},$
(36)

where $A_{jk}^{\alpha\beta}$ is the hyperfine coupling tensor between $j$ and
$k$ sites, which is second-rank polar tensor, and in general asymmetric. The
components of $A_{jk}^{\alpha\beta}$ is real as the Hamiltonian must be
hermitian, and the condition for the presence of off-diagonal components is
similar to that for the Dzyaloshinskii-Moriya interaction between $j$ and $k$
sites [44, 45].

By using the static magnetic susceptibility of electrons between $k$ and $m$
sites, $\chi_{km}^{\alpha\beta}$, we can express the expectation value of the
magnetic dipole, $-\gamma_{\rm e}\hbar\bm{S}_{k}$, for the uniform magnetic
field $\bm{H}_{m}=\bm{H}$ as

$\displaystyle-\gamma_{\rm e}\hbar
S_{k}^{\alpha}=\sum_{m\beta}\chi_{km}^{\alpha\beta}H_{m}^{\beta}=\sum_{\beta}\chi_{k}^{\alpha\beta}H^{\beta},\quad\chi_{k}^{\alpha\beta}\equiv\sum_{m}\chi_{km}^{\alpha\beta},$
(37)

where $\gamma_{\rm e}$ is the electron gyromagnetic ratio, and the local
susceptibility tensor $\chi_{k}^{\alpha\beta}$ is symmetric. Substituting this
expression into $H_{\rm hf}^{(j)}$, we obtain

$\displaystyle H_{\rm hf}^{(j)}$ $\displaystyle=-\frac{1}{\gamma_{\rm
e}\hbar}\sum_{k}\sum_{\alpha\beta\gamma}(A_{jk}^{\alpha\gamma}\chi_{k}^{\gamma\beta})I_{j}^{\alpha}H^{\beta}$
(38) $\displaystyle=-\gamma_{\rm
n}\hbar\sum_{\alpha\beta}K_{\alpha\beta}^{(j)}I_{j}^{\alpha}H^{\beta},$ (39)

where we have introduced the shift tensor as

$\displaystyle K_{\alpha\beta}^{(j)}\equiv\frac{1}{\gamma_{\rm n}\gamma_{\rm
e}\hbar^{2}}\sum_{k}\sum_{\gamma}A_{jk}^{\alpha\gamma}\chi_{k}^{\gamma\beta}.$
(40)

$K_{\alpha\beta}^{(j)}$ is in general asymmetric, and the off-diagonal
components exist when the vertical mirror symmetry is lost, which is the same
condition as the presence of the electric-toroidal dipoles as shown in Table 1
[46].

Considering $H_{\rm hf}^{(j)}$ and the Zeeman term, the total Hamiltonian
$H^{\prime}$ is given in the form
$H^{\prime}=-\hbar\omega_{0}\sum_{\alpha}I_{j}^{\alpha}g_{\alpha}^{(j)}$ with
$g_{\alpha}^{(j)}=h^{\alpha}+\sum_{\beta}K_{\alpha\beta}^{(j)}h^{\beta}$. By
taking the direction of $\bm{g}^{(j)}$ vector as the quantization axis, the
eigenvalues are $\hbar\omega_{m}^{(j)}=-\hbar\omega_{0}|\bm{g}^{(j)}|m$, and
the NMR shift is given by

$\displaystyle
K^{(j)}=[1+\sum_{\alpha\beta}\\{2K_{\alpha\beta}^{(j)}+(\hat{K}^{(j)T}\hat{K}^{(j)})_{\alpha\beta}\\}h^{\alpha}h^{\beta}]^{1/2}-1.$
(41)

In the first-order perturbation, we obtain the first term in (6), and it is
clear that the anti-symmetric components contribute from the second order in
the perturbation.

###  A.2 On expression of nuclear quadrupolar interaction

The nuclear quadrupolar interaction is given by

$\displaystyle H_{\rm
Q}^{(j)}=\frac{eQ}{2I(2I-1)}\sum_{\alpha\beta}V_{\alpha\beta}^{(j)}I_{j}^{\alpha}I_{j}^{\beta},$
(42)

where
$V_{\alpha\beta}^{(j)}=\partial^{2}\phi_{j}/\partial\alpha\partial\beta$.
Since the electrostatic potential at the nuclear site, $\phi_{j}$, satisfies
the Laplace equation, $V_{\alpha\beta}^{(j)}$ is traceless, and it is apparent
that it is symmetric. Hereafter, we omit the label $j$ for simplicity.

Using the traceless symmetric property, we have

$\displaystyle\sum_{\alpha\beta}V_{\alpha\beta}I_{\alpha}I_{\beta}=\sum_{\alpha\beta}V_{\alpha\beta}Q^{\alpha\beta},$
(43)

where $Q^{\alpha\beta}$ is defined as (3). Moreover, using ${\rm
Tr}\,\hat{V}=0$ and $\bm{I}^{2}=I_{x}^{2}+I_{y}^{2}+I_{z}^{2}$, we have the
relation,

$\displaystyle\sum_{\alpha}V_{\alpha\alpha}I_{\alpha}^{2}=\frac{1}{2}\left[V_{zz}(3I_{z}^{2}-\bm{I}^{2})+(V_{xx}-V_{yy})(I_{x}^{2}-I_{y}^{2})\right].$
(44)

Introducing the quadrupole operators as

$\displaystyle Q_{u}=3I_{z}^{2}-\bm{I}^{2},\quad Q_{v}=I_{x}^{2}-I_{y}^{2},$
(45) $\displaystyle Q_{yz}=\frac{1}{2}(I_{y}I_{z}+I_{z}I_{y}),\quad
Q_{zx}=\frac{1}{2}(I_{z}I_{x}+I_{x}I_{z}),$ (46) $\displaystyle
Q_{xy}=\frac{1}{2}(I_{x}I_{y}+I_{y}I_{x}),$ (47)

and the coefficients,

$\displaystyle V_{u}=\frac{1}{4}V_{zz},\quad
V_{v}=\frac{1}{4}(V_{xx}-V_{yy}),$ (48)

we obtain

$\displaystyle\sum_{\alpha\beta}V_{\alpha\beta}I_{\alpha}I_{\beta}=2\left[V_{u}Q_{u}+V_{v}Q_{v}\right.$
(49) $\displaystyle\hskip
42.67912pt\left.+V_{yz}Q_{yz}+V_{zx}Q_{zx}+V_{xy}Q_{xy}\right].$ (50)

Note that in terms of the so-called Stevens’ equivalent operators $O_{mn}$
[47, 48], we have the correspondences, $Q_{u}=O_{20}$, $Q_{v}=O_{22}$,
$Q_{yz}=O_{yz}$, $Q_{zx}=O_{zx}$, $Q_{xy}=O_{xy}$. In the principal frame
where $V_{\alpha\beta}$ is diagonal, it is customary to use the field gradient
$q$ and the asymmetry parameter $\eta$, which are defined as $q=V_{zz}/e$,
$\eta=V_{v}/V_{u}=(V_{xx}-V_{yy})/V_{zz}$.

Taking the direction of $\bm{h}$ as the quantization axis (denoted by
$z^{\prime}$ axis), we treat the interaction (50) in the first-order
perturbation. Since only the operator $Q_{u}$ in (50) has diagonal elements,
$\braket{m}{Q_{u}}{m}=3m^{2}-I(I+1)$, we obtain the first-order correction as

$\displaystyle\Delta\Omega_{m}=\frac{eQ/\hbar}{4I(2I-1)}V_{z^{\prime}z^{\prime}}[3m^{2}-I(I+1)].$
(51)

Using this expression and
$V_{z^{\prime}z^{\prime}}=\sum_{\alpha\beta}V_{\alpha\beta}h^{\alpha}h^{\beta}$,
we obtain the second term in (6).

##  Appendix B On fitting of experimental data

It is often encountered that the direction of the applied magnetic field with
respect to the crystal axes is undetermined in practical experiments. In this
case, we suppose that we rotate the applied magnetic field in the plane
defined by the normal vector $\bm{n}=(n_{x},n_{y},n_{z})$ ($|\bm{n}|=1$),
which is defined in the orthogonal coordinate $(xyz)$ relating to
crystallographic axes $(abc)$.

Let us denote the starting direction of the applied magnetic field as
$\bm{s}=(s_{x},s_{y},s_{z})$ ($|\bm{s}|=1$, $\bm{s}\cdot\bm{n}=0$), and the
field is rotated by an angle $\vartheta$ in this plane. Then, the magnetic
field direction is expressed as

$\displaystyle\bm{h}$ $\displaystyle=\hat{C}_{\bm{n}}(\vartheta)\bm{s}$ (52)
$\displaystyle=\begin{pmatrix}c+n_{x}^{2}\bar{c}&n_{x}n_{y}\bar{c}-n_{z}s&n_{x}n_{z}\bar{c}+n_{y}s\\\
n_{x}n_{y}\bar{c}+n_{z}s&c+n_{y}^{2}\bar{c}&n_{y}n_{z}\bar{c}-n_{x}s\\\
n_{x}n_{z}\bar{c}-n_{y}s&n_{y}n_{z}\bar{c}+n_{x}s&c+n_{z}^{2}\bar{c}\end{pmatrix}\begin{pmatrix}s_{x}\\\
s_{y}\\\ s_{z}\end{pmatrix},$ (53)

where $\hat{C}_{\bm{n}}(\vartheta)$ is the rotation matrix with respect to
$\bm{n}$, and we have used the abbreviations, $c=\cos\vartheta$,
$s=\sin\vartheta$, and $\bar{c}=1-c$. It is easily confirmed that
$\bm{n}\cdot\bm{h}=0$ and $\bm{h}=\bm{s}$ for $\vartheta=0$.

Similarly, for the $\varphi$ rotation in the plane perpendicular to
$\bm{n}\times\bm{s}$, the magnetic field direction is expressed as

$\displaystyle\bm{h}$
$\displaystyle=\hat{C}_{\bm{n}\times\bm{s}}(\varphi)\bm{s}.$ (55)

In this observation, the total shift (4) is regarded as a function of
($\bm{n}$, $\bm{s}$, $k_{i}^{(1)}$, $k_{i}^{\prime(1)}$, $v_{i}^{(1)}$) in
addition to the angles, $\vartheta$ and $\varphi$. Thus, these variables
(fitting parameters) can be determined so as to reproduce the observed total
shift
$T_{m}^{(j)}(\vartheta)=[\omega_{m-1}^{(j)}(\vartheta)-\omega_{m}^{(j)}(\vartheta)-\omega_{0}]/\omega_{0}$
and optionally $T_{m}^{(j)}(\varphi)$, by using, for example, the multiple
nonlinear regression with the backpropagation algorithm in the field of neural
networks [49]. As a result, directions of $\bm{n}$ and $\bm{s}$ are determined
in a given coordinate system ($xyz$), in addition to the shift and EFG
tensors.

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# Internet of Behavior (IoB) and Explainable AI Systems for Influencing IoT
Behavior

Haya Elayan, Moayad Aloqaily, Senior Member, IEEE, Fakhri Karray, Fellow,
IEEE, Mohsen Guizani, H. Elayan is with xAnalytics Inc., Ottawa, ON, Canada.
E-mail<EMAIL_ADDRESS>Aloqaily, M. Guizani, and F. Karray are with
the Mohamed Bin Zayed University of Artificial Intelligence (MBZUAI), UAE.
E-mails: {maloqaily; mguizani<EMAIL_ADDRESS>
###### Abstract

Pandemics and natural disasters over the years have changed the behavior of
people, which has had a tremendous impact on all life aspects. With the
technologies available in each era, governments, organizations, and companies
have used these technologies to track, control, and influence the behavior of
individuals for a benefit. Nowadays, the use of the Internet of Things (IoT),
cloud computing, and artificial intelligence (AI) have made it easier to track
and change the behavior of users through changing IoT behavior. This article
introduces and discusses the concept of the Internet of Behavior (IoB) and its
integration with Explainable AI (XAI) techniques to provide trusted and
evident experience in the process of changing IoT behavior to ultimately
improve user behavior. Therefore, a system based on IoB and XAI has been
proposed in a use case scenario of electrical power consumption that aims to
influence user consuming behavior to reduce power consumption and cost. The
scenario results showed a decrease of 522.2 kW of active power when compared
to original consumption over a 200-hours period. It also showed a total power
cost saving of €95.04 for the same period. Moreover, decreasing the global
active power will reduce the global intensity through the positive
correlation.

###### Index Terms:

IoB, XAI, Energy Sustainability, IoT, Deep Learning.

##  1 Introduction

Natural disasters and epidemics have affected people’s lives for centuries.
Tsunami, Spanish flu, HIV/AIDS, the COVID-19 pandemic, among others, have
killed millions around the world and forced people to change their behaviours.
Its impact on increasing mortality rate, causing economic damage, changing the
behavior of individuals, causing social and economic disruption, and
increasing political pressure and tensions which compels developed countries
to analyze them carefully.

While changing behavior appears to be a low-risk effect compared to mortality
or economic disaster, it is powerful enough to cause such risks. For example,
in a study of the impact of social distancing on suicide rates apart from
epidemiological deaths, results showed that a change in the social distancing
index of ten units caused a 2.9% increase in the suicide rate [1]. Moreover,
during the COVID-19 pandemic, scientists reported that wearing a mask could
prevent approximately 130,000 deaths in the USA alone. Many behavioral aspects
have been altered by the COVID-19 pandemic, such as customer interaction with
brands, employee work procedures, and business engagement with consumers. All
these and other examples have economic, technological, and physiological
effects. Therefore, tracking people’s behavior becomes crucial to influence it
in adverse situations. For example, using machine learning for mask
recognition tasks is one way to get individuals to respect regulations and
monitor negligence.

By 2023, it is predicted that the activities of 40% of the global population
will be tracked digitally to influence human behavior [2]. It is also
projected that the IoT sensors market will reach $22.48 billion, and 29.3
billion Internet-connected devices will be available by 2023. Therefore, IoT
would play a major role in detecting people’s behavior

The Professor of Psychology, Gote Nyman, was the first to announce the concept
of the Internet of Behavior (IoB). In 2012, he said that if the human
behavioral pattern is assigned to devices (e.g. IoT devices) with specific
addresses, there will be an opportunity to benefit from the knowledge gained
by analyzing the history of patterns in many businesses, societal, health,
political, and others fields. Behavior is a psychological characteristic that
can determine a person’s willingness to cooperate or collaborate. Despite the
other characteristics: Cognition, Emotion, Personality, and Inter-
Communication, behavior is responsible for the tendency to act and is highly
dependent on the other four criteria. Therefore, focusing on the behavior will
allow knowing how to influence and treat the person.

Attempting to influence and change user behavior is sensitive, as it may
encounter resistance and other psychological factors related to comfort and
trust. Emerging technologies such as XAI will help provide the user with the
required understanding and trust of any system that uses AI models. XAI aims
to use methods and techniques to explain to the user what an AI model does and
why, thus, giving a better perception of system operations. Therefore, the
process of tracking, analyzing, and influencing user behavior will become much
simpler.

This paper integrates the IoB concept with the XAI concept to propose trusted
and understandable frameworks that work in user behavior-changing areas.
Accordingly, it proposes an IoB-based system that utilizes XAI implemented in
a use case scenario of household electrical power consumption. The system aims
to change consumer behavior into an eco-friendly behavior to reduce power
consumption and therefore reduce energy waste and cost. This framework
integrates IoT, AI, Data Analytics, Behavioral Science, and XAI techniques to
achieve user and business benefits. The contributions of this paper can be
summarized as follows:

  * •

To the best of our knowledge, this is the first attempt to investigate the
concept of IoB, its workflow and benefits, challenges, and current industrial
directions.

  * •

Propose a trusted and comprehensible IoB-XAI-based system that attempts to
influence and change IoT behavior to achieve benefits for users and
businesses.

  * •

Present a use case scenario of electrical power consumption behavior to
influence consumer behavior and raise awareness to control and reduce the
power consumption process.

  * •

Discuss and compare the main characteristics of related work in the power
consumption area.

  * •

Highlight potential developments and future directions of the proposed system.

##  2 Internet of Behavior (IoB): IoT Devices

IoB is using devices to collect a massive amount of human behavioral data and
turn it into valuable insights to improve user experience and search
experience by changing user behavior, interests, and preferences. With the
increasing number of IoT devices collecting massive amounts of data, using
these devices to track behavioral data in the IoB process will make it easier,
faster, and more efficient.

IoB attempts to appropriately understand data and use this understanding to
create new products, promote current products, redesign the value chain,
increase profits, or reduce costs from a psychology perspective. Therefore,
user behavior in the IoB workflow will be tracked first using connected
devices, as shown in Fig. 1. The data generated by IoT devices will be
collected and then analyzed using data analytics and machine learning
algorithms. The analysis phase will yield useful information that must be
properly understood from a behavioral science perspective. Finally, the
knowledge gained will be used to develop business strategies and influence the
behavior of users, thus achieving a specific goal.

Figure 1: IoB Workflow

As shown in Fig. 2, the IoB process is descriptive yet proactive. It tracks
and analyzes user behavior in order to detect and accordingly influence
important psychological variables to achieve the process objective. This
usually brings lots of benefits as well as a huge set of challenges.

Figure 2: IoB Process

###  2.1 IoB Benefits

IoB is a very new concept in today’s emerging technologies. It is being used
in many applications for several benefits, however, without much knowledge
about it in the research community. Therefore, in this sub-section, we
summarize IoB benefits:

  * •

Quality of Experience, Increased Profit: Help companies resolve issues in
targeting more sales while keeping their customers satisfied at the same time
(Win-Win strategy). For example, online fashion retailer apps can use the
clicks and search history of users to suggest personalized discounts and offer
packages.

  * •

Tasks Automation: Help replace outdated fashion tactics like time-consuming
and unfavorable customer surveys (No More Tedious Tasks).

  * •

Target Customers: Give the opportunity to know the valuable customers based on
their interest and which segment to target and invest more time with. For
instance, the smartwatch company may find through tracked data that males aged
20-30 who do not exercise regularly are more likely to purchase its smartwatch
to make themselves more committed to exercise.

  * •

Accuracy: Give the ability to track and study the unobtainable behavior of how
customers interact with products and services (Notice the Unnoticed).

  * •

Real-Time Interaction: Help provide real-time interactions via notifications
or alerts to customers about targeted offers, sales, and/or advertisements.
For example, social media apps provide real-time feeds and ads based on
viewing patterns, amount of time spent in one account, interest in specific
content, or search and chat analysis.

###  2.2 IoB Challenges

####  2.2.1 Security

Dealing with sensitive and real-time data always creates security concerns for
any service user. In IoB, it is necessary to demonstrate a high level of
security and protection to users, as their behavioral data is expected to be
prone to attacks. The data sensitivity motivates cybercriminals to access,
reveal, collect and benefit from it.

####  2.2.2 Ethical Use

Behavioral data is a sensitive and personal data type, and its collection,
storage, or analysis must be accompanied by transparency and ethical use.
Users have the right to be aware of this process as well as to know that their
privacy is preserved and protected from misuse. Another aspect of ethical use
is not following the principle ”the end justifies the means”. Companies, in
most cases, aim for more profit regardless of the impact on users of changing
their behavior in critical areas such as health. Lastly, companies must ensure
that users’ consent is obtained when their data is collected and used for any
purpose.

####  2.2.3 Ostrich Effect

The ostrich effect is a phenomenon that occurs when the rational mind believes
something is important and the emotional mind expects it to be painful. People
understand the power and benefits of using IoB, however, they may be
uncomfortable (trust issues) about the tracking process, which results in
avoiding or rejecting it. Therefore, when companies implement IoB, it is not
enough to work on process understanding by users. They also have to work on
removing the discomfort of using this technology.

###  2.3 XAI for IoB

IoB challenges may create resistance on the part of users as it tries to
influence and change their behavior and track their data. Using AI models in
IoB to analyze users’ data allows taking advantage of XAI techniques to make
the operations of IoB-based systems more intelligible to the user and thus
more trustworthy.

Describing the system process, workflow, impact, and outcomes using user-
friendly mathematical and analytical proofs will give users a better
understanding of the system and increase awareness of its impact. Thus,
mitigating the problems of trust and the impact of resistance.

##  3 Eye on Industry

In this section, we will discuss how the different industries are interacting
with IoB technology.

###  3.1 Healthcare

Nowadays, smartphones allow many software companies to build and promote
different applications that care about user’s health, where they can easily
use smartphone sensors to track bio-metrics and healthy behavior. For
instance, the Beddit health app tracks sleep patterns, heart rate, and
breathing rate. After analyzing the tracked behavior, the app sends alerts,
notifications, and tips to improve the user’s sleep and motivates him/her to
achieve daily goals to reach a more positive outcome [3].

During the COVID-19 pandemic, many countries have developed health smartphone
apps for citizens to control the spread of the virus. The China Health Code
Alipay app, for example, tracks a user’s travel history, contact history, and
body bio-metrics such as temperature. Then, a colored QR code is generated to
identify his/her health status. Accordingly, some restrictions may apply to
the user and affect his/her behavior, such as permitting travel or
quarantining at home or in a central location [4].

###  3.2 Transportation

Even in transportation, IoB can be applied for benefits for drivers and
passengers. After a long history of disputes with drivers and a high turnover
rate, Uber is trying to resolve these issues and settle disputes in its favor
by influencing driver behavior using gamification. Uber has tricks like loss
aversion, recognition, and intrinsic motivation to reward drivers or make them
fear losing their winnings [5]. Also, Ford is expanding its reach by joining
forces with Argo AI to develop autonomous vehicles that adapt and behave
differently to road infrastructure designs and driving behaviors. This
technology is being tested in many locations such as the streets of Miami,
Washington, and others [6].

##  4 IoB and XAI For Power Consumption Reduction: A Use Case Scenario

This section proposes a use case scenario of a system for household electrical
power consumption. The system aims to influence the electricity consumption
behavior of a house by automatically controlling the amount of global active
power in a way that the user understands and trusts. Global active power
control will reduce energy consumption which will, in turn, reduce waste and
cost of potential used energy.

The system workflow in Fig. 3 shows that the current will pass through an
energy monitor and controller device (EMC) to know how much electricity each
IoT device is using. After that, the data will be transmitted from the EMC to
a cloud database to be stored and normalized. Then, the normalized data will
be used by an AI model to predict the global active power that may be consumed
in the next hour. The prediction result will be passed to a decision maker and
explainer to determine how much energy the appliances should use. Then, the
results will be returned to the EMC to control the consumption in the next
hour if it is above a certain threshold in order to reduce it.

Before discussing the proposed system flow in detail, a set of related
proposals in the electrical power consumption area will be discussed. Table I
shows a summary of each related proposal and compared it to the proposed
system (last row in the table). Studies [7], [8], and [9], used deep learning
algorithms: Long short term memory (LSTM), Convolutional neural network (CNN),
or both to predict the electrical consumption. However, they focused their
contribution on the architecture of the proposed model. Likewise, [10] and
[11] used LSTM and CNN together.

The researchers in [10] have proposed to predict power consumption through
proposing a framework consisting of data cleaning and model building steps.
Whereas the authors in [11] aimed to accelerate the deployment process of deep
learning models using the proposed edge-cloud framework.

On the other hand, [12] and [13] applied Deep Reinforcement Learning
algorithms (Deep RL) on the same dataset to perform tasks related to process
optimization of power consumption. While authors of [12] focused on the
proposed RL model structure, authors of [13] proposed a stochastic game
framework that controls the interaction between the consumed power and
household.

Figure 3: System Workflow

The proposed model of this paper presents a comprehensive framework consisting
of all technology needed to form a successful IoB/XAI model which will reduce
power consumption by using IoB and XAI to influence user behavior. Firstly,
the LSTM algorithm will be used to build a power consumption prediction model,
and then the power consumption process will be controlled. Meanwhile, the
workflow will be explained to the user to ensure his understanding of a
smoother behavior change process.

TABLE I: Related Work Comparison

Ref. | Algorithm | Dataset | Goal | Proposal / Framework description
---|---|---|---|---
[7] |  | CNN
---
LSTM
| Short-term individual households
---
power consumption for Australian
customers
Predicting the individual household electrical load. | Focus on deep learning model architecture
[8] | LSTM |  | Monthly electricity consumption
---
from a Korean region
| Forecast the annual electricity consumption
---
based on the history of the last three years.
Focus on deep learning model architecture
[9] |  | CNN
---
ResNet
| Electrical meters datasets of non
---
residential building in Switzerland
| Forecasting building load using Resnet to enhance
---
the ability of learning
Focus on deep learning model architecture
[10] |  | CNN
---
LSTM
| Individual household electric power
---
consumption In France /
Appliances Energy Prediction
| Predicting electricity consumption at various
---
distribution and transmission systems’ levels.
| Framework of two steps:
---
data cleaning and model building
[11] |  | CNN
---
LSTM
| Household power consumption
---
by Korea Electric Power Corporation.
| Accelerating a deep learning model
---
deployment after power data stream training
| Framework for a power flow edge
---
cloud system
[12] | Deep RL |  | Electrical submeters recorded by
---
Pecan Street. Inc
| Optimizing home energy management by taking
---
a series of actions
Focus on deep RL architecture
[13] | Deep RL |  | Electrical submeters recorded by
---
Pecan Street. Inc
| Scheduling energy consumption for
---
household appliances
| Framework for the interaction of power
---
grid and households using a stochastic
game
| Proposed
---
Model
LSTM |  | Individual household electric power
---
consumption In France
| Influencing electricity consumption behavior by
---
automatically controlling the amount of power.
| IoB and XAI based framework of four
---
steps: Data, Database, Data Normalization,
AI model, Decision making, and XAI

###  4.1 Dataset

An individual household electric power consumption dataset available online
has been used to implement the use case [14]. It contains measurements of one
house’s electrical power consumption per minute for approximately 4 years,
collected in a house located in France. The data has been re-sampled to get
the total consumption per hour which reduced the dataset size from 2,075,259
to 34,589. The dataset was divided into 50% train samples and 50% test
samples. The dataset includes the following features: Date, Time, and Voltage,
Global active power: Household global active power, Global reactive power:
Household global reactive power, Global intensity: Household global current
intensity in ampere, Sub-metering 1: Energy of the kitchen, Sub-metering 2:
Energy of the laundry room, Sub-metering 3: Energy of the electric water
heater and an air conditioner.

###  4.2 Database and Data Normalization

Electrical power consumption data is acquired from smart energy monitoring
sensors connected to the Internet. An advanced type of smart monitoring sensor
can detect the type of each appliance by connecting the monitor to the
electricity meter only,without the need to connect multiple monitoring sensors
to each appliance, This helps centralize the system and reduce the cost of its
installation.

The acquired data is huge and in its raw format. Therefore, it requires a
proper database to handle it. For example, MongoDB is one of the NoSQL
databases that can handle this data as it is document-oriented. MongoDB can
deal with semi-structured data where most IoT data exists in semi-structured
or unstructured formats. Moreover, it has high performance in data storage and
retrieval, which is necessary for second-by-second data acquisition and
transmission.

Due to the possibility of incomplete, unstructured, and inconsistent data as
IoT data is unstructured, the data normalization phase comes to normalize and
pre-process the data to get it in a suitable format without noise and
outliers. Additionally, this phase handles data loss that may occur due to a
disconnect or dropouts, so it deals with missing values by adding data
averaging or removing these values. Moreover, it scales features using scaling
methods like Min-Max-Scaler to map data to the same range.

###  4.3 Power Consumption Prediction Model

The dataset used in the scenario is time-series-based data. Time series
prediction problems are one of the difficult types of predictive modelling
problems because they add sequence dependence complexity amidst input
features. The Long-term memory network (LSTM) is a type of Recurrent Neural
Network (RNN) used in time series problems because it can successfully train
huge structures with eliminating major problems, such as vanishing gradients.

A sequential deep learning model has been built to predict the hourly power
consumption. It allows building a layered model, with each layer having
weights corresponding to the following layer. Keras library has been used for
coding the model structure. The model has been constructed using one LSTM
layer and one Dense layer and ran for 20 epochs with Adam optimizer. The model
has been trained on a machine with the following specifications: Intel
<EMAIL_ADDRESS>2 cores, RAM 13GB, and Disk 70GB.

###  4.4 Decision Maker and Explainer

The EMC will track the power consumption of appliances per hour and pass the
tracked data to a cloud database to be stored and normalized. The data will be
used by the AI model to analyze the consumption behavior and predict the power
consumption for the next hour. Based on prediction and historical data
patterns, the decision maker part will decide the amount of power that should
be used in the next hour. To personalize the user experience, the decision
maker will make the decision based on the historical user’s behavior and usage
pattern. Therefore, the decision will be made by comparing the predicted
consumption with the historical average consumption at specific times related
to the prediction hour. Take into account influences seasonal and occasional
factors.

The explainer part in the Decision Maker and Explainer is responsible for
applying the XAI techniques to describe the AI model, its impact, biases, and
outcomes to the user. The purpose is to use a set of processes and methods to
allow the user to understand and trust the results generated by the model,
thereby raising their awareness of the workings of the system and its goal in
order to have a smooth behavior change process.

The explainer will attempt to cover the following concerns: Model prediction
process, Model success and failure, and Model confidence in decision making,
using data annotation and deep learning algorithm annotation. Data annotation
describes statistical information of the dataset, such as features
distribution and correlation. While the algorithm annotation shows the
characteristics of the model and whether it is biased towards certain features
or outliers. The results will be sent to the user whenever they are requested
using any Internet-connected device.

Using XAI frameworks such as IBM AI Explainability 360 will help provide
explanations for various user concerns by providing the dataset and the built
model for the open-source toolkit, then selecting the type of consumer to
customize the explanations [15]. Finally, form different questions that cover
the user concerns and offer explanations and insights through the answers. For
example:

  1. 1.

Why is power consumption controlled? The predicted power consumption based on
the last hour consumption is 100 kW, your historical average is 55 kW.

  2. 2.

What appliance consumes the most energy? Appliance ”A” is the most consuming
device.

  3. 3.

How was energy consumption distributed during this decision? Appliance “A”
consumed 23 Wh, appliance “B” consumed 13 Wh, and appliance “C” consumed 15
Wh.

Accordingly, the entire system works to influence the user’s consumption
behavior in his favor by saving energy from waste, saving the cost of wasted
energy, and increasing his/her awareness of better consuming behavior.
Therefore, creating ‘’collective wisdom” and more responsible habits that
affect not only the individual but also the society as a whole.

###  4.5 Use Case Results

####  4.5.1 Model Performance

Model loss indicates how weak the model’s predictions are. A loss close to
zero means that the model’s prediction is perfect and the model’s performance
is good. The proposed model loss started with a training loss of 0.0125 and a
testing loss of 0.0079. Both of them continued to decline, reaching a training
loss of 0.0095 and a testing loss of 0.0078. Fig. 4(a) illustrates the loss
performance over epochs.

Fig. 4(b) shows a snapshot of the model’s perdition results for the global
active power and the actual global active power values of 200 samples. As
shown, the prediction follows the actual values pattern, while the error rate
is low.

(a)

(b)

Figure 4: Prediction Model Performance

####  4.5.2 Energy Saving

Since the proposed system predicts the global active power for the next hour
and controls the amount of used power to a certain threshold, a certain amount
of energy will be saved. The system will control the power if the prediction
is above the average consumption of the same hour of the same month and the
same weekday for the past four years. If the prediction is lower than the
average it will not trigger the next action. The system has been tested on 200
samples from the testing data to know the amount of saved energy. As shown in
Fig. 5, the system will issue warnings (red squares) on 27 hours from 200
predicted hours where the consumption is above the historical average power
consumption. The total saved power amount for the 27 hours is 522.2 kilowatts.

Figure 5: System Warnings

####  4.5.3 Cost Saving

As the system is tested on 200 samples and saved 522.2 kilowatts of global
active power, the power cost will also be saved. In France, the price of
electricity in kilowatt-hours for households is 0.182 euros. Therefore, the
system will save €95.04 for a period of 200 hours.

####  4.5.4 Global Intensity Reduction

The energy intensity indicates the quantity of energy required per unit output
or activity to produce a product or service. A high intensity indicates high
energy used to provide a service, and low intensity indicates lower energy to
provide a service. This affects the price or cost of energy conversion as low
energy intensity indicates a low cost of energy conversion into the gross
domestic product (GDP). In the dataset, the global intensity is positively
correlated with the global active power by 99% so they both move in tandem,
which means if the global active power decreases, the global intensity will
also decrease.

To sum up, the proposed system in the use case scenario managed to reduce
522.2 kilowatts of house global active power over a 200-hours period, while
helping to reduce €95.04 from wasted power cost. Moreover, the reduction of
the global active power affected the global intensity positively, which will
benefit the community as well.

##  5 Future Directions

In the future direction, there are various open issues and areas of potential
development that need to be investigated and tested:

Incorporating a user feedback subsystem into the proposed system is an
important area for improving the learning and prediction of the AI model and
the overall system. It is also important to measure the user satisfaction
factor. This will help improve the experience, fix emerging issues, and keep
the user more engaged.

Another potential development area is building a distributed version of the
proposed system. The aim is to compare the distributed version to the current
centralized version and test its impact on both technical and user experience
aspects. Such as its effect on network efficiency, cost, data storing process,
and system failures.

Moreover, sending out instructive notifications to users to teach them about
the optimal behavior is also a considerable approach in product and system
design. Users’ responses and use of instructive notices will speed up the
process of changing behavior and make it easier, clearer, and more entrenched.

Furthermore, emphasizing the use of different AI/ML learning techniques is
another development area to reduce the complexity of the AI model and increase
its accuracy.

Finally, integrating technologies that add more control over security and data
privacy is significant. The application of different security schemes,
techniques, and technologies for different levels and processes of the system
from data transmission to its storage and use will add more value to the
system.

##  6 Conclusion

Since Internet-connected devices are increasingly being used; tracking,
understanding, and influencing users’ behavior to achieve material and moral
profits has become a reality. This paper is the first attempt to investigate
the concept of IoB, its workflow, benefits, challenges, and current industrial
directions. It also proposes a trusted and comprehensible tracking-analyzing-
influencing behavior system using IoB and XAI. Moreover, a use case of an
electrical power consumption system that works for benefit of individuals,
power companies, and society to save energy waste and cost. The collected
results demonstrated that the system was able to save 522.2 kW of wasted power
and €95.04 of its cost. Furthermore, the positive correlation between active
power and power intensity will reduce the latter if the former is reduced.

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|  Haya Elayan (M’20) received the B.S. degree in computer information systems
from The University of Jordan, Amman, Jordan, in 2018. She is currently
pursuing the MRes degree in data science and analytics for health with the
University of Leeds, Leeds, U.K. She worked as a Data Scientist in several
companies. She applied machine learning techniques for data analytics, machine
intelligence, feature selection, and recommendation engines, along with
building and optimizing classifiers for product and operational capabilities.
She also worked as a Researcher and a Data Scientist, where she was
responsible for publishing academic papers and developing business insights.
---|---
|  Moayad Aloqaily (S’12, M’17) received the Ph.D. degree in electrical and
computer engineering from the University of Ottawa, Ottawa, ON, Canada, in
2016. He was an Instructor with the SYSC Department, Carleton University,
Ottawa, in 2017. From 2018 to 2019, he was an Assistant Professor with the
American University of the Middle East (AUM), Kuwait. From 2019 to 2021, he
was the Cybersecurity Program Director and an Assistant Professor with the
Faculty of Engineering, Al Ain University, United Arab Emirates. He has been
the Managing Director of xAnalytics Inc., Ottawa, since 2019. He is currently
with the Machine Learning Department, Mohamed Bin Zayed University of
Artificial Intelligence (MBZUAI), United Arab Emirates. His current research
interests include the applications of AI and ML, connected and autonomous
vehicles, blockchain solutions, and sustainable energy and data management. He
is an ACM Member and a Professional Engineer Ontario (P.Eng.). He has chaired
and co-chaired many IEEE conferences and workshops. He started his own Special
Interest Group on blockchain and application as well as internet of unmanned
aerial networks. He has served as a Guest Editor for many journals, including
IEEE Wireless Communications Magazine, IEEE Network, and Computer Networks. He
is an Associate Editor of Ad Hoc Networks, Cluster Computing, Security and
Privacy, and IEEE ACCESS. He has also been appointed as the Co-Editor-in-Chief
of IEEE COMMSOFT TC ELETTER in 2020.
---|---
|  Mohsen Guizani (S’85–M’89–SM’99–F’09) received the BS (with distinction),
MS and PhD degrees in Electrical and Computer engineering from Syracuse
University, Syracuse, NY, USA. He is currently a Professor and the Associate
Provost at Mohamed Bin Zayed University of Artificial Intelligence (MBZUAI),
Abu Dhabi, UAE. Previously, he worked in different institutions in the USA.
His research interests include applied machine learning and artificial
intelligence, Internet of Things (IoT), intelligent systems, smart city, and
cybersecurity. He has been an IEEE Fellow since 2009 and was listed as a
Clarivate Analytics Highly Cited Researcher in Computer Science in 2019, 2020
and 2021. Dr. Guizani has won several research awards including the “2015 IEEE
Communications Society Best Survey Paper Award” as well 4 Best Paper Awards
from ICC and Globecom Conferences. He is the author of ten books and more than
800 publications. He is also the recipient of the 2017 IEEE Communications
Society Wireless Technical Committee (WTC) Recognition Award, the 2018 AdHoc
Technical Committee Recognition Award, and the 2019 IEEE Communications and
Information Security Technical Recognition (CISTC) Award. He served as the
Editor-in-Chief of IEEE Network and is currently serving on the Editorial
Boards of many IEEE Transactions and Magazines. He was the Chair of the IEEE
Communications Society Wireless Technical Committee and the Chair of the TAOS
Technical Committee. He served as the IEEE Computer Society Distinguished
Speaker and is currently the IEEE ComSoc Distinguished Lecturer.
---|---
|  Fakhri Karray received the Ph.D. degree from the University of Illinois at
Urbana-Champaign, Urbana, IL, USA. He is currently the University Research
Chair Professor in electrical and computer engineering with the University of
Waterloo, Waterloo, ON, Canada and the Co-Director of the University’s
Institute for Artificial Intelligence. His research interests include
intelligent systems design, big data analytic, soft computing, sensor fusion,
and context-aware machines with applications to intelligent transportation
systems, Internet of Things, and cognitive robotics. He has authored in these
areas and has disseminated his work in journals, conference proceedings, and
textbooks. He is a fellow of the Canadian Academy of Engineering and the
Engineering Institute of Canada. He was an Associate Editor/Guest Editor for
various journals, including the IEEE Transactions on Cybernetics, IEEE
Transactions on Neural Networks and Learning, IEEE Transactions on
Mechatrocnics, and IEEE Computational Intelligence Magazine.
---|---