#  Magnetism and topology in Tb-based icosahedral quasicrystal

Shinji Watanabe  Department of Basic Sciences, Kyushu Institute of Technology,
Kitakyushu, Fukuoka 804-8550, Japan

###### Abstract

Quasicrystal (QC) possesses a unique lattice structure with rotational
symmetry forbidden in conventional crystals. The electric property is far from
complete understanding and it has been a long-standing issue whether the
magnetic long-range order is realized in the QC. The main difficulty was lack
of microscopic theory to analyze the effect of the crystalline electric field
(CEF) at the rare-earth atom in QCs. Here we show the full microscopic
analysis of the CEF in Tb-based QCs. We find that magnetic anisotropy arising
from the CEF plays a key role in realizing unique magnetic textures on the
icosahedron whose vertices Tb atoms are located at. Our analysis of the
minimal model based on the magnetic anisotropy suggests that the long-range
order of the hedgehog characterized by the topological charge of one is
stabilized in the Tb-based QC. We also find that the whirling-moment state is
characterized by unusually large topological charge of three. The magnetic
textures as well as the topological states are shown to be switched by
controlling compositions of the non-rare-earth elements in the ternary
compounds. Our model is useful to understand the magnetism as well as the
topological property in the rare-earth-based QCs and approximant crystals.

Quasicrystal (QC), which was discovered in 1984 Shechtman , has a unique
lattice structure with the rotational symmetry forbidden in conventional
crystals Tsai ; Takakura . In the QC, it has been an open question whether the
magnetic long range order is realized Suzuki . Thus far, in the rare-earth
based approximant crystal (AC) retaining the periodicity as well as the local
atomic configuration common to the QC, the magnetic long-range orders have
been observed in Cd6R (R=Nd, Sm, Gd, Tb, Dy, Ho, Er, and Tm) Tamura2010 ; Mori
; Tamura2012 ; Das  and Au-SM-R (SM=Si, Ge, Sn, and Al, and R=Gd, Tb, Dy, and
Ho) Hiroto2013 ; Hiroto2014 . Most of these measurements were performed by
using the bulk probe such as susceptibility and the detailed magnetic
structures remain unresolved. Recently, in the Tb-based ACs, the magnetic
structures have started to be identified by neutron measurements Sato2019 ;
Hiroto . Quite recently, the ferromagnetic (FM) long-range order has been
discovered experimentally in the Tb-based QC Tamura2021 .

These experimental developments urgently require theoretical investigation of
the magnetism in the rare-earth based QC and AC. Theoretical studies to date
have been performed as the model calculations mostly by the spin model and
also by the Hubbard model in a small cluster or low-dimensional systems Okabe
; Sorensen ; Coffey ; Jan ; Axe ; Wessel ; Kons ; Jan2007 ; Hucht ; Thiem ;
Komura ; Sugimoto ; Koga2017 ; Koga2020 ; STS ; Miyazaki , where the effect of
the crystalline electric field (CEF) crucial for the rare-earth system was not
taken into account microscopically.

In the rare-earth based compounds, the effect of the CEF is important for the
electronic states, which has been well recognized in the periodic crystals
such as the heavy fermion systems. In contrast, in the rare-earth based QC and
AC, understanding about the CEF is very limited because the local symmetry
around the rare-earth site is completely different from the crystallographic
point group, e.g., 5-fold symmetry allowed in the QC and AC is forbidden in
the periodic crystals. Thus far, the CEF in the rare-earth based ACs was
analyzed experimentally Hiroto ; Jaz ; Das . Recently, the full CEF
Hamiltonian in the rare-earth based QC and AC has been formulated
theoretically WM2021 .

In this article, by applying this formulation to the Tb-based QC and AC, we
report full microscopic analysis of the CEF in the Tb-based QC and AC. We show
that the magnetic anisotropy arising from the CEF plays a key role in
realizing unique magnetic textures on the icosahedron (IC) at whose vertices
the Tb moments are located in the QC and AC. We propose the minimal model to
clarify the magnetic structures in the QCs and ACs. Our analysis suggests that
the long-range order of the hedgehog characterized by the topological charge
is realized in the Tb-based QC. This is the first theoretical discovery of the
possible long-range order of the topological magnetic texture in the QC.
Distinct from the topological spin textures intensively studied so far in the
periodic crystals Taguchi ; Nagaosa ; Kanazawa ; Fujishiro ; Ishiwata ; Tokura
; Aoyama , our discovery is based on the total angular momentum on the CEF-
anisotropy-protected IC, which provides a new concept of topological
quasicrystal.

The Tb-based QC and AC consist of building blocks named the Tsai-type cluster
composed of the concentric shell structures shown in Figs. 1a-1e, where
Au70Si17Tb13 is displayed as a representative case. Namely, a set of
polyhedrons are nested like a matryoshka doll. In Fig. 1c, Tb is located at
each vertex of the IC, which forms the Tb-12 cluster. The local atomic
configuration around Tb is shown in Fig. 1f. Here, the $z$ axis is passing
through Tb from the center of the IC, which is regarded as the pseudo 5-fold
axis, and the $y$ axis is taken so as the $yz$ plane to be the mirror plane.
Namely, the configuration of atoms is symmetric in both sides with respect to
the $yz$ plane in Fig. 1f.

Figure 1: (color online) Crystal structure of Tsai-type cluster and local
atomic configuration. Tsai-type cluster consists of a, the cluster center, b,
dodecahedron, c, icosahedron, d, icosidodecahedron, and e, defect rhombic
triacontahedron with Au (yellow), Au/Si (green), and Tb (gray). f, Local
atomic configuration around Tb. (This figure is created by using Adobe
Illustrator CS5 Version 15.1.0.)

Results

Analysis of crystalline electric field. The CEF can be analyzed on the basis
of the point charge model. Since the Tb3+ ion has the $4f^{8}$ configuration,
it is regarded to have the charge of 6$|e|$ in the hole picture ($4f^{14}$ is
the closed shell of $4f$ electrons). Then, the electrostatic energy at the
Tb3+ ion is expressed as $H_{\rm CEF}=6|e|V$, where $V$ is the Coulomb
potential $V=\sum_{i=1}^{16}q_{i}/|{\bm{r}}-{\bm{R}}_{i}|$. Here, ${\bm{r}}$
and ${\bm{R}}_{i}$ are the position vectors of the Tb3+ ion and ligand ions
respectively and $q_{i}$ is the charge at the $i=1$–16th Au or Au/Si mixed
site shown in Fig. 1f. To take into account the effect of the Au/Si mixed
sites, here we set $q_{i}=(0.63Z_{\rm Au}+0.37Z_{\rm Si})|e|$ for $i=1$, 3,
and 4 and $q_{16}=(0.19Z_{\rm Au}+0.81Z_{\rm Si})|e|$ so as to reflect the
occupancy ratio Hiroto , otherwise $q_{i}=Z_{\rm Au}|e|$, where $Z_{\rm Au}$
and $Z_{\rm Si}$ are the valences of Au and Si respectively.

In metallic crystals and alloyed ACs, the valences of Au and Si are known to
be 1 and 4 respectively Pearson ; Mizutani . However, in reality, these values
can be reduced by the screening effect of the conduction electrons. Hence, we
analyze the CEF for $Z_{\rm Si}=\alpha Z_{\rm Au}$ by setting $Z_{\rm
Au}=0.223$ as a typical value which was determined by the neutron measurement
in Au70Si17Tb13 Hiroto . It is noted here that the choice of $Z_{\rm Au}$ does
not affect the CEF eigenstate as long as $\alpha$ is the same.

On the basis of the recently developed formulation (Supplementary information)
WM2021 , we diagonalize $H_{\rm CEF}$ for the total angular momentum $J=6$ as
the ground multiplet endorsed by the Hund’s rule. Then, the CEF energies
$E_{n}$ and the eigenstates $|\psi_{n}\rangle$ are obtained, as shown in Fig.
2a. The CEF energies are split into almost 7 for $\alpha=0$ to 8 for
$\alpha=4$, some of which are nearly degenerate. The CEF ground energy $E_{0}$
is well separated from the first excited energy $E_{1}$ as seen in Fig. 2a, as
indeed observed in Au70Si17Tb13 with $E_{1}-E_{0}$ being in the order of
$10^{2}$ K. Thus, the magnetism at low temperatures is dominated by the CEF
ground state.

Figure 2: (color online) Crystalline-electric-field energies and magnetic-
moment direction of the ground state. a, The $\alpha$ dependence of the CEF
energies at Tb in $Z_{\rm Si}={\alpha}Z_{\rm Au}$. b, The $\alpha$ dependence
of the largest-moment direction of the CEF ground state shown by the
normalized vector ${\bm{J}}$. (top panel) $J_{x}$ vs $\alpha$, (middle panel)
$J_{y}$ vs $\alpha$, and (bottom panel) $J_{z}$ vs $\alpha$. c, The $\alpha$
dependence of $\theta$ defined by the angle of ${\bm{J}}$ from the $z$ axis in
Fig. 1f. Inset illustrates the ${\bm{J}}$ vector in the $yz$ plane. (This
figure is created by using Adobe Illustrator CS5 Version 15.1.0.)

Magnetic anisotropy. Then we evaluate the largest magnetic-moment direction of
the CEF ground state, i.e., the principal axis. The result, which is expressed
as the normalized vector ${\bm{J}}=(J_{x},J_{y},J_{z})$, is shown in Fig. 2b.
This reveals that the magnetic moment is lying in the $yz$ plane i.e., the
mirror plane in Fig. 1f and changes its direction depending on $\alpha$.
Namely, as $\alpha$ increases from $0$, the magnetic moment rotates to
anticlockwise direction in the mirror plane, whose angle defined from the $z$
axis, $\theta$, changes from $\theta\approx 90^{\circ}$ to approach
$\theta=0^{\circ}$, as shown in Fig. 2c.

Minimal model on icosahedron. This result indicates that the direction of the
magnetic moment at each Tb site can be changed by controlling the compositions
of Au and SM in the Au-SM-Tb ACs and QCs. To clarify how the magnetic
anisotropy affects the magnetism in the IC, we consider the minimal model

$\displaystyle H=-\sum_{\langle
i,j\rangle}J_{ij}\hat{\bm{J}}_{i}\cdot\hat{\bm{J}}_{j}.$ (1)

Here, $\hat{\bm{J}}_{i}$ is the unit Ising-spin vector operator at the $i$th
Tb site, whose direction is restricted to either parallel or antiparallel to
the principal axis shown in Fig. 2b and $J_{ij}$ is the exchange interaction.
We consider the nearest neighbor (N.N.) interaction $J_{1}$ and the next. N.N.
(N.N.N.) interaction $J_{2}$. It is noteworthy that this model has recently
been used to analyze the neutron measurement in the AC Au72Al14Tb14, which
results in $J_{2}/J_{1}\approx 2.3$ and $\theta=86^{\circ}$ that successfully
reproduces not only the measured magnetic structure but also the temperature
dependence of the magnetic susceptibility as well as the magnetic-field
dependence of the magnetization Sato2019 . Hence we adopt this model as the
effective model, which is expected to be relevant generally to the Tb-based
ACs and QCs. So far, in various Tb-based ACs, negative Weiss temperature was
observed, which indicates that the antiferromagnetic (AF) interaction is
dominant between the magnetic moments at the Tb sites Suzuki . Hence we focus
on the AF interaction $J_{1}<0$ and $J_{2}<0$ and the case of the FM
interaction will be reported elsewhere W2021 .

It is noted that the Tb-based AC and QC are metallic compounds. Hence, the
RKKY interaction via the conduction electrons is considered to contribute to
the exchange interaction between the 4f magnetic moments at Tb sites.
Furthermore, in the AC, the effect of the Fermi surface such as the nesting
can also contribute to trigger the magnetic order. To describe the essential
feature of the resultant magnetism in the itinerant electron system, the
minimal model is introduced as Eq. (1). Namely, due to the strong magnetic
anisotropy arising from the CEF, the Ising model restricted to the anisotropy
direction at each Tb site, which is regarded as the strong anisotropy limit in
the anisotropic Heisenberg model, is considered to be the effective model as
the first step of analysis.

Figure 3: (color online) Ground-state phase diagram and magnetic textures in
icosahedron. a, The ground-state phase diagram for $J_{2}/J_{1}$ $(J_{1}<0)$
and $\theta$ as well as $\alpha$ in the icosahedron. Contour plot of the
topological charge $n$ is also shown. b, Magnetic texture shown by each symbol
in a. c, Antihedgehog state for $\theta=0^{\circ}$. d, Antiwhirling state for
$\theta=70^{\circ}$. e, Whirling state for $\theta=90^{\circ}$. (This figure
is created by using Adobe Illustrator CS5 Version 15.1.0.)

Ground-state phase diagram. The ground-state phase diagram for $J_{2}/J_{1}$
and the angle $\theta$ of the magnetic anisotropy in the IC is shown in Fig.
3a. Here we also show the $\alpha$ axis as the holizontal axis. It is noted
that depending on the compositions of Au and SM in the Au-SM-Tb ACs and QCs,
the $\alpha$ dependence of $\theta$ can change slightly from that shown in
Fig. 2c. Hence, the $J_{2}/J_{1}$–$\theta$ phase diagram with the wide range
of $\theta$ is generally relevant to the Tb-based ACs and QCs.

Magnetic textures on icosahedron. We find that unique magnetic states appear
depending on $J_{2}/J_{1}$ and $\theta$, as a result of the CEF-anisotropy-
protected IC. Each symbol in Fig. 3a represents the magnetic texture shown in
Fig. 3b. Note that the magnetic texture where all the moments are inverted
also has the same ground-state energy for each symbol, i.e., energetically
degenerate. The red-square symbols denote the hedgehog state where all the
magnetic moments at 12 Tb sites on the IC are directed outward (see
$\theta=0^{\circ}$ in Fig. 3b). It is remarkable that the hedgehog state is
stabilized in the rather wide-$\theta$ region (Supplementary information). The
antihedgehog state where all the moments are inverted from the hedgehog state
is shown in Fig. 3c.

Topological charge. To reveal the topological character underlying in the
magnetic textures, we define the topological charge $n$ per an IC by
calculating the solid angle subtended by three neighboring moments
${\bm{J}}_{i}$, ${\bm{J}}_{j}$, and ${\bm{J}}_{k}$ Eriksson  located at the
vertices of each triangle surface of the IC. By calculating $n$ for each
magnetic textures on the IC, we find that the topological charge of $n=1$ is
realized in the hedgehog state and $n=-1$ is realized in the antihedgehog
state. This implies the solid angle subtended by 12 Tb moments on the IC
covers the whole sphere and the hedgehog and and antihedgehog play the roles
of source or sink of emergent field, which are transcribed as the emergent
monopole and antimonopole with a “charge” $n=1$ and $n=-1$ respectively.

Whirling-moments state with unusually large topological charge. We find that
the finite topological charge also appears in the region denoted by the
symbols of orange squares in Fig. 3a. The magnetic texture (see orange square
$\theta=90^{\circ}$ in Fig. 3b) is characterized by unusually large magnitude
of topological charge $n=-3$. We call this state the antiwhirling-moments
state since the magnetic moments are whirling when it is seen from the (111)
direction (Supplementary information). Interestingly, we find the topological
transition to $n=1$ occurs around $\theta=79^{\circ}$ (see Fig. 3d). In Fig.
3a, the contour plot of the topological charge $n$ is also shown. We see that
finite integer $n$ appears in the ${\bm{J}}_{\rm tot}={\bf 0}$ state which is
denoted by the square symbols in Fig. 3a. Here, the total magnetic moment on
the IC is given by ${\bm{J}}_{\rm tot}=\sum_{i=1}^{12}{\bm{J}}_{i}$. There
exists a finite magnetic moment per an IC ${\bm{J}}_{\rm tot}\neq{\bf 0}$ for
the magnetic textures denoted by not-square symbols in Figs. 3a and 3b.

Figure 4: (color online) Long-range order of hedgehog in quasicrystal. The
long-range order of the hedgehog for $\theta=0^{\circ}$ in the QC. Tb-12
clusters at the origin and at the vertices of the icodidodecahedron are shown.
(This figure is created by using Adobe Illustrator CS5 Version 15.1.0.)

Long-range order of hedgehog in quasicrystal. To explore the magnetism in the
QC, we consider the Cd5.7Yb-type QC where the Tb-12 cluster i.e., IC is
located at the origin and the surrounding vertices of the icosidodecahedron
and they are repeatedly arranged outward in the self-similar manner Takakura .
We have applied the model (1) to the Cd5.7Yb-type QC with the AFM N.N.
interaction $J_{1}<0$ and N.N.N. interaction $J_{2}<0$ not only for the intra
IC but also for the inter ICs located on the line connecting neighboring
vertices of the $\tau^{3}$-times enlarged icosidodecahedron from that in Fig.
1d. Here $\tau=(1+\sqrt{5})/2$ is the golden ratio. Then we evaluate the inner
product $\langle\hat{\bm{J}}_{i}\cdot\hat{\bm{J}}_{j}\rangle$ for the N.N. and
N.N.N. pairs of the Tb sites in the intra IC and inter ICs. For the hedgehog
state with $\theta=0^{\circ}$, we confirmed that
$\langle\hat{\bm{J}}_{i}\cdot\hat{\bm{J}}_{j}\rangle<0$ is realized for all
the $\langle i,j\rangle$ pairs with respect to the inter ICs in addition to
the intra IC. This indicates that the hedgehog is stabilized in the
arrangement of the 30 ICs located at each vertex of the icosidodecahedron.
Indeed we confirmed that the hedgehog has the lowest ground-state energy. We
show the magnetic structure in the real space for $\theta=0^{\circ}$ in Fig.
4. The hedgehog is arranged ferromagnetically i.e., uniformly distributed in
IC to IC, where the total magnetic moment is zero ${\bm{J}}_{\rm tot}={\bf 0}$
in each IC.

This result is intuitively understood from the magnetic moment directions
shown in Fig. 4. Namely, each magnetic moment at the Tb site on the IC are
directed outward. Hence, the magnetic moments between the neighboring ICs are
directed antiferromagnetically each other at least for the N.N. and N.N.N. Tb
sites when the hedgehog is uniformly distributed. Then, if the N.N. and N.N.N.
interactions are AFM in the model (1), i.e., $J_{1}<0$ and $J_{2}<0$, the
hedgehog in the IC is able to be arranged uniformly as shown in Fig.4. For the
outer ICs than the icosidodecahedron, the magnetic moments pointing to outward
direction from the IC earn the energy gain for the neighboring IC with the
same hedgehog when the N.N. and N.N.N. interactions are AFM. Then, the uniform
arrangement of the hedgehog state is realized in each IC in the self-similar
manner, giving rise to the magnetic long-range order in the QC.

This implies that the long-range order of the hedgehog is realized in the
ground state. By calculating
$\langle\hat{\bm{J}}_{i}\cdot\hat{\bm{J}}_{j}\rangle$ for N.N. and N.N.N.
pairs of the Tb sites for the inter ICs in 30 ICs located at vertices of the
icosidodecahedron, we confirmed that uniform distribution of the hedgehog is
stabilized as the ground state at least for the region of $J_{2}/J_{1}$ and
$\theta$ symbolized by red squares in Fig. 3a. This is, to our best knowledge,
the first discovery of the possible long-range order of topological magnetic
textures in the QC. It is noted that the antihedgehog shown in Fig. 3e also
exhibits the uniform distribution. The hedgehog QC and antihedgehog QC are
energetically degenerate within the model (1).

Discussion. Our finding of the hedgehog long-range order in the QC
demonstrates that topological textures can be stabilized in the QC. As for the
antiwhirling (whirling) state [see orange square for $\theta=90^{\circ}$ in
Fig. 3b (Fig. 3e)], the long-range order can be realized uniformly when the
AFM N.N.N. interaction is dominant for the inter ICs in the QC (Supplementary
information). In this case, the antimonopole (monopole) with unusually large
topological charge $|n|=3$ orders ferromagnetically. In the hedgehog and
whiling ordered states, the non-trivial phenomena such as the topological Hall
effect are expected to emerge. Our model (1) is generically applicable to the
rare-earth based QCs and ACs with magnetic anisotropy, which is useful to
clarify the magnetic structures as well as topological properties.

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Methods

Crystalline electric field

Recently, the CEF Hamiltonian in the rare-earth based QCs and ACs have been
formulated by the operator-equivalent methods WM2021  as

$\displaystyle H_{\rm
CEF}=\sum_{\ell=2,4,6}\left[B_{\ell}^{0}(c)O_{\ell}^{0}(c)+\sum_{\eta=c,s}\sum_{m=1}^{\ell}B_{\ell}^{m}(\eta)O_{\ell}^{m}(\eta)\right],$
(2)

where $O_{\ell}^{m}(\eta)$ is the Stevens operator Stevens  and $B_{\ell}^{m}$
is the coefficient. Since $O_{\ell}^{m}(\eta)$ is expressed as the operators
of the total angular momentum $J$, $H_{\rm CEF}$ can be obtained on the basis
of $J$ and the $z$ component $J_{z}$, i.e., $|J,J_{z}\rangle$ for
$J_{z}=J,J-1,\cdots,-J+1$, and $-J$. Then, by diagonalizing the $2J+1\times
2J+1$ matrix $H_{\rm CEF}$, we obtain the CEF energies $E_{n}$ and the
eigenstate $|\psi_{n}\rangle$. For the Tb3+ ion, $J=6$ is the ground multiplet
according to the Hund’s rule. Then, $H_{\rm CEF}$ is diagonalized by setting
$J=6$ as described in the main text (see also Supplementary information).

Principal axis of the magnetization

The largest magnetic-moment direction in the CEF ground state can be
identified by diagonalizing the $3\times 3$ matrix
$M_{\xi\delta}=\langle\psi_{0}|\hat{J}_{\xi}\hat{J}_{\delta}|\psi_{0}\rangle$
for $\xi,\delta=x,y$, and $z$. Here, $\hat{J}_{\xi}$ is the operator of the
total angular momentum. The normalized eigenvector of the largest eigenvalue
of $M$, ${\bm{J}}=(J_{x},J_{y},J_{z})$, gives the direction, i.e., the
principal axis of the magnetic moment.

Minimal model

The ground state of the model (1) in the main text is obtained numerically
among $2^{12}$ spin configurations in the IC. Namely, the expectation value of
the Hamiltonian (1), i.e., the energy
$E=\langle\sigma_{1},\sigma_{2},\cdots,\sigma_{12}|H|\sigma_{1},\sigma_{2},\cdots,\sigma_{12}\rangle$
are calculated for $\sigma_{i}=\uparrow,\downarrow$ where the quantization
axis at each $i$-th site on the IC (Fig. 1c) is directed to the anisotropy
direction $\theta$ shown in the inset of Fig. 2. Then, by searching the lowest
energy numerically, we obtain the ground state. By performing this calculation
for various $J_{2}/J_{1}$ and $\theta$, we have constructed the ground-state
phase diagram for an IC in Fig. 3.

Next, we apply the model (1) to the Cd5.7Yb-type QC, where the Tb-12 cluster,
i.e., IC, is located at the origin surrounded by the $\tau^{3}$ times enlarged
icosidodecahedron from that in Fig.1d, whose vertices ICs are located at and
they are repeatedly arranged outward in the self-similar manner. In the
icosidodecahedron, there exist 30 ICs. For the neighboring pairs of ICs, whose
number is 60, we set the N.N. interaction $J_{1}$ and N.N.N. interaction
$J_{2}$ between the magnetic moments at the N.N. and N.N.N. Tb sites
respectively.

Then, we evaluate the inner product
$\langle\hat{\bm{J}}_{i}\cdot\hat{\bm{J}}_{j}\rangle$ for four sets of
$\langle i,j\rangle$ pair as the N.N. and N.N.N. pairs per each bond among the
60 bonds. Namely, the inner product for the $4\times 60$ pairs are calculated
with respect to the inter ICs. Combining this with the calculation for the
intra IC mentioned above, we search the lowest-energy state of the model (1)
on the 30 ICs located at vertices of the icosidodecahedron with totally 360 Tb
sites numerically for various $J_{2}/J_{1}$ and $\theta$ and obtain the ground
state.

Topological charge on the IC

Recently, the scalar chirality and the topological charge have been defined
for four spins located at each vertex of the tetrahedron Aoyama . It has been
shown by this definition that the hedgehog state realized in the tetrahedron
is characterized by the topological charge of one. Extending these definitions
to the IC, here we consider the scalar chirality and the topological charge in
the IC as follows.

We define the scalar chirality of the IC with its center-of-mass position
${\bm{R}}$ by $\chi({\bm{R}})=\sum_{i,j,k\in{\rm IC}}\chi_{i,j,k}$ with
$\chi_{i,j,k}\equiv{\bm{J}}_{i}\cdot({\bm{J}}_{j}\times{\bm{J}}_{k})$ where
the order of $i$, $j$, and $k$ is defined in the anticlockwise direction with
respect to the normal vector of the triangle formed by the $i$, $j$, and $k$th
sites, $\hat{n}_{i,j,k}$, pointing outward from ${\bm{R}}$. In the same way,
we define the solid angle subtended by the twelve moments on the IC as
$\Omega({\bm{R}})=\sum_{i,j,k\in{\rm IC}}\Omega_{ijk}$. Here, $\Omega_{ijk}$
is the solid angle for three magnetic moments ${\bm{J}}_{i}$, ${\bm{J}}_{j}$,
and ${\bm{J}}_{k}$, which is given by
$\Omega_{ijk}=2\tan^{-1}[\chi_{ijk}/(1+{\bm{J}}_{i}\cdot{\bm{J}}_{j}+{\bm{J}}_{j}\cdot{\bm{J}}_{k}+{\bm{J}}_{k}\cdot{\bm{J}}_{i})]$
Eriksson . The topological charge $n$ is defined as
$n\equiv\Omega({\bm{R}})/(4\pi)$ per an IC.

Data availability

All the data supporting the findings are available from the corresponding
author upon reasonable request.

Acknowledgments

This work was supported by JSPS KAKENHI Grant Numbers JP18K03542 and
JP19H00648.

Author contributions

S.W. conceived the study and led the project. Theoretical calculation was
performed by S.W. The manuscript was written by S.W.

Competing interests

The author declares no competing interests.

Additional information

Supplementary information is available for this paper.

Correspondence and requests for materials should be addressed to S.W.


# Simultaneous real and momentum space electron diffraction from a fullerene
molecule

Aiswarya R  Department of Physics, Indian Institute of Technology Patna, Bihar
801103, India  Rasheed Shaik  School of Physical Sciences, Indian Institute of
Technology Mandi, Kamand, H.P. 175075, India  Jobin Jose<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology Patna, Bihar 801103,
India  Hari R. Varma  School of Physical Sciences, Indian Institute of
Technology Mandi, Kamand, H.P. 175075, India  Himadri S. Chakraborty
<EMAIL_ADDRESS>Department of Natural Sciences, D.L. Hubbard Center for
Innovation, Northwest Missouri State University, Maryville, Missouri 64468,
USA

###### Abstract

Plane-wave electrons undergo momentum transfer as they scatter off a target in
overlapping spherical waves. The transferred momentum leads to target
structural information to be encoded in angle and energy differential
scattering. For symmetric, periodic or structured targets this can engender
diffraction in the electron intensity both in real (angular) and in momentum
space. With the example of elastic scattering from C60 we show this
simultaneous manifestation of diffraction signatures. The simulated angle-
momentum diffractograms can be imaged in experiments with a two-dimensional
detector and an energy-tunable electron gun. The result may inspire invention
of technology to extend scopes of electron diffraction from molecules and
nanostructures, open a direction of electron crystallography using the
momentum-differential diffraction, and motivate an approach to control the
time delay between the pump laser-pulse and the probe electron-pulse by tuning
the electron impact-speed in ultrafast electron diffraction experiments.

Electron diffraction (ED) studies have flourished in accessing the structural
and dynamical information of molecules saha2022 ; amini2020 , nanostructures
ponce2021 , surfaces Kienzle2012 , and crystals gemmi2019 . In standard ED
measurements, the target is irradiated by an electron beam of a fixed energy
and the scattered intensity is measured by a two-dimensional (2D) detector.
This detector can be either a luminescent, namely phosphor, screen coupled to
a charge-coupled device or a direct hit detector that collects diffracted
signal within a large solid angle. For crystals, where the sample orientation
can also be tuned, measurements are typically performed in a transmission
electron microscope which provides access to targets otherwise inaccessible by
X-ray diffraction (X-rays are blind to the Coulomb force). Ultrafast ED (UED)
centurian2022  is a popular technique which temporally resolves the
evolutionary dynamics of an excited sample. In this, the sample is initially
excited by a laser pulse (pump) and then interrogated by a time-delayed
electron pulse (probe). The ED signal is registered by varying pump-probe
delay in a controllable manner. The fundamental notion that a diffraction
pattern encodes information about the Fourier transform of the real-space
structure of the target that produced it, and vice versa, is at the heart of
these studies.

Diffraction is the bending of an incident plane wave into effective spherical
waves, at detectable distances larger than the target size, by interacting
with the structural details of the target. The resulting interference produces
patterns in the signal. Fundamentally, therefore, the pattern is a function of
the magnitude ($q$) of transferred momentum vector, the difference between the
final and the initial momentum of the projectile. Let us consider the elastic
scattering for simplicity in which the electron energy, and therefore the
magnitude of the electron momentum ($\vec{k}$), does not alter during the
scattering. In this case, $q$ for a scattering angle $\theta$ can be written
as,

$\displaystyle q$ $\displaystyle=$ $\displaystyle 2k\sin(\theta/2).$ (1)

Eq. (1) implies that one way to vary $q$ is to keep the electron energy fixed
but capture the signal at various angles. This is routinely done in current ED
measurements by using a 2D detector. This provides the fringe pattern in the
real ($\theta$) space. An alternative way, see Eq. (1), to scan the $q$-space
for a fixed detection angle is to vary the impact energy to generate
diffraction as a function of $k$. An advantage of the latter is that with
increasing energy and therefore decreasing de Broglie wavelength the electrons
can resolve finer details of the target structure – from the larger-scale
electron density to the smaller-scale ion-core and inter-nuclear structures.
For UED, the tuning of impact energy may directly control the pump-probe delay
time, in particular to resolve dynamics from picoseconds centurian2010 ;
yang2015  to nanoseconds time-scale, the latter being prevalent in the field
of atmospheric chemistry, biology and combustion science long2023 . This can
be achieved by locking on a fixed launch-time difference of the pulses, while
controlling their arrival-time delay as a function of the electron speed. In
general, by combining both these pathways (see schematic Fig. 1), namely by
using an energy-tunable source and a 2D detector, it will be possible to
generate a 2D diffraction map, the diffractogram, in combined real-and-
momentum space (Fig. 5).

Figure 1: (Color online) A cartoon to delineate the elastic electron impact
diffraction with C60. It schematically shows diffraction fringes in scattered
intensity as a function of the scattering angle $\theta$ for a fixed incident
energy and the fringes at a given scattering direction $\theta_{\scriptsize
sc}$ as a function of the magnitude of electron momentum
$k=|\vec{k}_{i}|=|\vec{k}_{f}|$ as the incident energy varies.

In this study, we calculate the elastic electron scattering differential cross
section (DCS) of a C60 molecule target and produce such diffractograms. The
study serves as a proof-of-principle to provide impetus in accessing the
entire ED landscape by including the momentum space information. This may
motivate extensions of the ED method’s general spectroscopic scope. Since the
discovery of C60 kroto85 , this highly symmetric molecule has been drawing
extensive research attention, both for fundamental reasons as well as for its
broad applied relevance full-appl2021 . The high symmetry also makes it an
efficient diffracting target. Experiments on the electron scattering from C60
have included low-energy tanaka2021 , intermediate-energy hargreaves2010 , and
high-energy garchekov1998  measurements. It is generally accepted that the
e-C60 collision DCS exhibits diffraction effects by revealing minima and
maxima as a function of $\theta$ and this pattern differs for various
scattering energies tanaka2021 ; hargreaves2010 ; garchekov1998 . The
diffraction behavior of elastic/inelastic high energy DCS was found to connect
to the geometry of C60 electronic distribution garchekov1998 ; garchekov1997 .
However, little assertion of the $k$-space diffraction effect is made to the
best of our knowledge. Further, even though some qualitative attributes are
drawn, the quantitative relation of the diffraction in DCS to the target
geometry is not demonstrated, which we include in the current study.

There is a distinction in how the diffraction is probed in e-C60 scattering
process versus in photoionization (PI) and positronium (Ps) formation with
C60, in both of which the electron exists only in the final channel (half-
scattering for PI). PI and Ps formation are intrinsically inelastic, leading
to the variation of the momentum of the observed particle, photoelectron or
Ps, as the impact energy varies. This is true for the detection of the
particle at a certain angle, integrated over a small angular range in the
forward direction, or integrated over all angles. Consequently, the
diffraction phenomenon is accessible in the momentum space in these processes.
Indeed, the $k$-space diffraction in the PI from highest occupied molecular
orbitals of C60 were measured and calculated ruedel2002 ; mccune2008 . Also,
diffraction in the angle-integrated hervieux2017  and forward direction
hervieux2019  Ps formation in positron-C60 collisions was predicted where the
feature was captured in the Ps momentum. But for PI of predominantly spherical
systems, like the valence electron cloud in C60, the diffraction in $\theta$
is forbidden AngDistPI ; Cooper1993 . This notion is similar to the classical
Fraunhoffer type diffraction through a spherical aperture guasti93 . For the
Ps formation, even though $q$ of the incoming positron is involved, it gets
non-trivially entangled with $k$ of the captured electron hervieux2017 . On
the other hand, as pointed out above, the fundamental domain to map out
diffraction in elastic scattering is by varying $q$ of the electron. The
information, thus, automatically translates into the variation either of
$\theta$ (a direct quantum mechanical analogue of classical Young’s
diffraction) or of $k$, enabling a complete description of the process. This
is also a novelty of the current study.

The details of the computational methods are included as Supplementary
Material (SM) SM . The isolated C60 molecule is modeled by a jellium-based
local-density approximation scheme madjet2008  within density functional
theory (DFT). A spherical frame is employed that includes 240 delocalized C60
electrons. Variants of this DFT model were successfully employed in explaining
measurements of (i) photoelectron ruedel2002  and photoion scully2005
intensity, respectively, at non-plasmonic and plasmonic energies, and (ii) a
recent plasmonic PI time delay spectra biswas2022 . A simple electron-less
(static) annular square well (ASW) potential to model the C60 shell with
parametric adjustment to match the electron affinity of C60 winstead2006 ,
2.65 eV known from the experiment tosatti1994anomalous , is used in parallel
to identify robust features. These potentials mediate the e-C60 interaction,
that further includes an appropriate polarization potential SM  for a fuller
description. The e-C60 elastic scattering parameters are then obtained using
the non-relativistic partial wave technique. The radial Schrödinger equation
is numerically solved to access the scattering state of energy $E$ and,
subsequently, the scattering phase shift $\delta_{\ell}$ with corresponding
angular momentum quantum number $\ell$. The total scattering amplitude for a
scattering angle $\theta$ measured from the incident (forward) direction is
then evaluated as,

$\displaystyle f(k,\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{2ik}\sum_{\ell=0}^{\infty}(2\ell+1)P_{\ell}(\cos\theta)\left[\exp(i2\delta_{\ell})-1\right],$
(2)

where the magnitude of the projectile momentum, in atomic units (a.u.),
$k=\sqrt{2E}$ and $P_{\ell}$ is the Legendre polynomial. The DCS is then
evaluated as,

$\displaystyle\frac{d\sigma}{d\Omega}$ $\displaystyle=$
$\displaystyle|f(k,\theta)|^{2}.$ (3)

Figure 2: (Color online) Angle-differential DCS for e-C60 elastic scattering
calculated in DFT and ASW models and compared with measurements tanaka2021 ;
hargreaves2010  and SMC calculations winstead2006 . Both $q$ and $\theta$
scales are shown. Two impact energies considered are 10 eV (a) and 200 eV (b).

In Fig. 2 we compare our results of the angle-differential DCS with
measurements and other calculations for two values of the impact energy. Note
that the $q$-scale is also shown. For the lower 10-eV impact energy, panel
(a), both of our DFT and ASW results agree reasonably well with overall
features of the measurements tanaka2021 . The agreement achieved by a
successful and accurate Schwinger multi-channel (SMC) method winstead2006  is
of the same level of our calculations, as also shown. We note that SMC
includes an atomistic model of C60 and therefore has richer structural
information than a jellium C60. In spite of this, the success of our DFT
results versus measurements and SMC for a low-energy impact likely indicates
that longer de Broglie wavelength of the slower electron can not resolve
atomistic structures.

For the faster 200-eV impact, panel (b), while the ASW result fails to agree
with the experiment hargreaves2010 , DFT fairs well versus measurements at
lower $q$ values around the forward scattering. At larger scattering angles,
however, DFT not only shows more structures than what are present in the
measured data, but also weakens in the overall intensity. On the other hand,
the SMC calculation hargreaves2010  compares well with the experiment at wider
angles, while not so well toward the forward scattering. In general, these
structures in DCS are qualitatively known to originate from the ED effect of
C60. However, there are other effects, particularly for higher impact energies
and backward scattering, namely the direct collisions with individual carbon
atomic centers, which the jellium DFT and ASW models do not include. On the
other hand, this atomic contribution at larger scattering angles may wash out
diffraction effects which is likely the case for measurements and SMC. Also,
the wavelength of 200-eV electrons is not short enough to diffract from the
C-C bond ($\sim$ 0.15 nm). We remark that since the soft-edged DFT potential
can scatter electrons from its long-range “wings” on either side of C60 shell
(Fig.1 in SM), while the much harder-edged ASW potential cannot, the overall
intensity of DCS in the former is much stronger.

Figure 3: (Color online) Momentum-differential DCS for e-C60 elastic
scattering calculated in DFT and ASW models, and compared with measurements
tanaka1994  and SECP calculations gianturco1999 . Both $q$ and $k$ scales are
shown. Two scattering angles considered are 40∘ (a) and 70∘ (b).

In Fig. 3 we display our momentum-differential DCS, and compare with
measurements and another theory, for two choices of the scattering angle. For
the 40∘ scattering on panel (a), our DFT results show a very good agreement
with the measurements tanaka1994 . This is particularly remarkable given that
the static exchange correlation polarization (SECP) calculations gianturco1999
(also presented) significantly disagree with the experiment. Further, ASW also
showing a poor agreement suggests the superiority of DFT at this angle. Panel
(b) exhibits results for 70∘. Both DFT and ASW qualitatively capture
structures in the measured profile tanaka1994 , while SECP still somewhat
struggles to match. In any case, the structures in $k$-differential DCS in
Fig. 3 must also be due to the diffraction mechanism, since fundamentally ED
is a function of $q$; the $q$ variation is shown in the figure. Therefore,
Figs. 2 and 3 together embody a notion of complete manifestation of ED that we
will scrutinize further in the following.

Figure 4: (Color online) Fast Fourier transforms (FFT) of e-C60 elastic
scattering DCS, computed in ASW and DFT schemes (a). The peaks are attributed
to C60 geometric features (see text). The DFT radial interaction potential is
plotted (b) to facilitate the geometric comparison.

The Fourier transform of the diffraction signal in scattered electrons maps
out the diffractor’s structural information that can be resolved by the
wavelength of the incoming electron. In order to assess this, we first
benchmark the expectation by using the ASW potential in the first Born
approximation of elastic scattering to analytically evaluate DCS as a function
of $q$ SM . The result indicates that the diffraction fringes in DCS, or
equivalently the intensity “waves” as a function of $q$, carry the
“frequencies” corresponding to C60 shell structure. These frequencies are: the
C60 shell’s inner ($d_{\scriptsize i}$) and outer ($d_{\scriptsize o}$)
diameters, the average molecular diameter $d_{\scriptsize c}=(d_{\scriptsize
i}+d_{\scriptsize o})/2$, and the shell width $\Delta=(d_{\scriptsize
i}-d_{\scriptsize o})/2$. The core mechanism includes diffraction from the
inner and outer edge of the shell in the scattering amplitude Eq. (2) and
their interference by the coherence process (squared modulus) in DCS Eq. (3).
These frequencies must be found in the reciprocal spectra of the DCS.

Therefore, we analyze numerically computed DCS as a function of $q$ by
carrying out their Fourier transform (FT). This is done by a fast FT (FFT)
algorithm after neutralizing the steady background decay of the DCS and
appropriately windowing the data before FFT feed mccune2008 . Obtained FFT
signals are presented in Fig. 4(a) that indeed reveal peaks corresponding to
C60 geometry. The ASW result better captures the positions of the edges when
compared with panel (b). However, the DFT result shows offsets and some splits
in the frequency peaks. This is due to the fact that the shell edges are
somewhat diffused in the DFT interaction potential [Fig. 4(b)]. In effect, the
diffraction in the DFT approach maps out the delocalized electron density-
profile (around the molecular shell) instead, which is realistic.

In order to image the complete landscape of the diffraction phenomenon, we
generate iso-surface 2D polar maps, the diffractrograms, of the DCS in
combined real ($\theta$) and momentum space in Fig. 5. The $\theta$ range is
chosen between $-120^{\circ}$ and $120^{\circ}$, because the signal around the
back-scattering is found too weak. The primary fringe pattern exhibits series
of parabolas. One finds an overall impression of weaker groupings of fringes,
more obvious in ASW (lower panel), which denotes superimposed wider (beating)
oscillations connected to the width $\Delta$, which is the smallest size
feature in FT [Fig. 4(a)]. Note further that the DFT image (upper panel)
reveals additional parabolas towards back-scattering. These secondary fringe
pattern originate from the even-function symmetry,
$\cos(\theta)=\cos(\pi-\theta)$, applied in Eq. (2). Both the patterns overlap
for a range of $\theta$, as seen. However, the intensity of the patterns
depends on the sum of the Legendre polynomials in Eq. (2) and their
interference in Eq. (3). It is well known that Legendre polynomials of
increasing order $\ell$ oscillate progressively in phase toward the forward
direction producing a constructive interference, resulting significantly
stronger primary-fringe intensity. But the polynomials’ out-of-phase
oscillations inducing destructive interference to the backward direction
weaken the secondary-fringe intensity. However, since with increasing $k$ more
partial waves are included, this effect enhances causing an effective
extinction of secondary fringes for higher $k$ as noticed. Obviously, the
effect is physical in the more realistic DFT approach and may be detectable in
experiments. We reiterate, similar diffractograms can be generated in
experiments using energy-tunable electron source and a 2D detector. As an
attractive prospect for contemporary ultrafast studies, similar angle-momentum
differential time-delay diffractograms can be accessed; temporal fringe
patterns were predicted for photoelectrons magrakvelidze2015

Figure 5: (Color online) Angle-momentum combined 2D iso-surface polar-images
of e-C60 elastic scattering DCS with the DFT potential (top) and ASW potential
(bottom) showing rich diffraction features.

In conclusion, considering the electron elastic collision with C60 we
demonstrate that combined real and momentum space 2D DCS maps out the complete
diffraction topography of scattering. An advantage of accessing the momentum
space, or equivalently tuning the electron’s wavelength, is to find a knob for
increasing resolution of the structure. We used a fairly successful DFT model
of C60, in parallel with a simpler model to hone insights. The computed
results show good agreements with available measurements, often achieving
better success than existing theories. The Fourier transform of the DCS is
invoked for the first time to reveal structural information which is the
capstone goal of ED studies of molecular or crystal targets. The diffractogram
simulated on the DFT track is found rich in spectroscopic information. Such
images can be generated in complete experiments with gas phase C60. The study
may also motivate experiments for fullerenes in solutions or as thin films.
Even though the collision may also generate electrons by various inelastic
channels including the C60 plasmon channel, they can be selected out by
appropriate energy analysis.

Moreover, there is a broader implication of the study. This may encourage the
extension of the current technology in order to explore the momentum space
electron scattering from a molecule in gas phase and electron transmission
through a crystal. This will tap into structural information inaccessible in
constant irradiation-energy mode. The study can also inspire the electron-
speed controlled time delay of the probe electron pulse from the pump laser
pulse in UED approaches to molecular dynamics. It can be estimated that within
the irradiation energy range from 100 keV to 100 eV one may tune the electron
pulse’s arrival delay from picoseconds to nanoseconds ranges with 10 to 100 eV
energy increments. There are many important chemical and bio-molecular
processes occur within these time scales. In addition, the evolution from
nonequilibrium nuclear dynamics from the sub-picosecond timescale
srinivasan2003 ; dudek2001  to slower nanoseconds ground-state kinetics
through microcanonical energy distribution pathways is still not clearly
known. To this end therefore, the current study may offer a handle to the
control of the time resolution while simultaneously upgrading the structure
resolution to impact ultrafast molecular science.

###### Acknowledgements.

We thank Dr. Martin Centurian for helpful discussions. The research is
supported by CRG/2022/000191, India (JJ), DST-SERB-CRG Project No.
CRG/2022/002309, India (HRV), and US National Science Foundation Grant No.
PHY-2110318 (HSC.).

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