# Magnetic interactions in doped silicene for spintronics

Lucca Moraes Gomes  Andreia Luisa da Rosa  Institute of Physics, Quantum
Computational Materials Science Group, Campus Samambaia, Goiania, 74690-900,
Goiás, Brazil

###### Abstract

Silicon is a material whose technological application is well established, and
obtaining this material in nanostructured form increases its possibility of
integration in current technology. Silicene is a natural compatibility with
current silicon-based electronics industry. Furthermore, doping is a technique
that can be often used to adjust the band gap of silicene and at the same time
introduce new functions. Here we investigate Several aspects of silicene
doping with chromium, such as dopant solubility limits, site preference for
adsorption and doping, and formation of magnetic domains. In this work we
carried out investigation on diffusion and doping of chromium atoms on
silicene. We use density-functional theory to identify the electronic and
magnetic properties of tchromium atoms on monolayer and bilayer silicene. We
find that magnetization depends on key parameters such as buckling, interlayer
distance and adsorption site.

##  I Introduction

The development of new experimental techniques and the emergence of powerful
computers combined with iterative methods, gave rise to a new area of
reesearch, allowing to explore low-dimensional materials with interesting
properties that only can be seen and used by quantum mechanics. Those
materials can have properties that could take humanity to next generation of
technological devices because of their unique properties like high electronic
mobility, thermal conductivity and impressive tenacity. Those properties may
have application in several areas that goes from medical applications to the
enhancement of new aircrafts (see for example Dowling (2004)).

Graphene is a low dimensional material that opened the new branch of
condensed-matter-physics. The layered materials is made of carbon which a very
reactive element of Group IV that have four half occupied orbitals. Graphene
is a single layer of graphite, it has a honeycomb structure where each carbon
atom bonds to other three atoms by $sp^{2}$ hybridization, hence leaving a
half occupied orbital that is usually a $p_{z}$ orbital. The band structure of
graphene shows the so called Dirac cones. In this unique dispersion pattern,
the electrons then behave as Dirac fermions which are massless fermions. This
behavior in the band structure renders a high mobility which could have an
infinite number of electronic applications Rao  _et al._ (2009).

Two-dimensional carbon was first proposed theoretically and had the first
studied published in the beginning of the 60’s by Hans-Peter Boehm who studied
extremely thin flakes of graphite and gave the name graphene for this flakes.
The suffix ”ene” is given because of the numerous double bond that thin layers
have Boehm  _et al._ (1962). Graphene was characterized only in 2004 by Geim
and Novoselov, that granted them in 2010 the Nobel prize. They were able to
obtain graphene through a process called Scotch tape method which consist in
simply applying the tape on thin flakes of graphite until one thin layer
remain, using atomic force microscope they were able to identify the single
layered materialGeim and Novoselov (2007). Despite of the interesting
properties of graphene, the manipulation of its electronic properties is very
challenging Kariyado and Hatsugai (2016) which led to the the search for
similar or better properties.

At the same time, silicon is the most abundant element on earth crust, which
is extremely explored in the semiconductor industry. Silicene, a two-
dimensional silicon layer, became very targeted by researchers in the last few
decades. The silicene structure resembles the one of graphene Zhao  _et al._
(2016). Silicene layers were synthesized on several substrates, such as
Ag(111), Ir(111), ZrB2(0001), ZrC(111) and MoS2 surfaces by epitaxial growth
on those surfaces Leoni  _et al._ (2021); Arafune  _et al._ (2013). This has
prompted experimental studies of the growth of multi-layer silicene, though
the nature of its monolayered silicene structure has been investigated in a
few papers Zhang  _et al._ (2013); Pawlak  _et al._ (2019). Like graphene,
synthesizing free-standing silicene represents the ultimate challenge. A first
step towards this has been reported recently through chemical exfoliation from
calcium disilicide (CaSi2).

On the other hand, the discovery of atomic monolayer magnetic materials has
triggered significant interest in the magnetism/spintronics and 2D van der
Waals materials communities. The emergence of 2D magnetic materials presents a
unique opportunity to study magnetism and spintronics devices in new regimes.
This Review surveys the basic properties of these materials, methods to read
and write their magnetic states, and emerging device concepts. There is a
large class of layered magnetic materials with unique magnetic properties,
which provides an ideal platform to study magnetism and spintronics device
concepts in the 2D limit. Because these materials are atomically thin, their
magnetic states can be effectively controlled or switched by external
perturbations other than magnetic fields, such as electric fields, doping and
strain. New materials concepts, such as magnetizing 2D semiconductors by
magnetic proximity coupling and spin tunnel field-effect transistors are
promising devices.

Here we demonstrate that silicene decorated with certain 3d transition metals
(Vanadium) can sustain a stable quantum anomalous Hall effect using both
analytical model and first-principles Wannier interpolation Zhang  _et al._
(2013). Nguyen-Ha  _et al._ (2023)

In this work we investigating silicene to Defects could be associated with
vacancies in the nanosheet, adsorption of impurities in the sample or even the
substitution of atoms. This work has the objective to study the changes on the
properties by adsorbing open shell elements on silicene. The open shell
element focused in this work is Chromium (Cr). Open shell elements adsorbed on
silicene may have several application due to the modification of electronic
and magnetic properties, which could render application in the field of
spintronics, where the spin is responsible for carrying the information Johll
_et al._ (2014); Bader and Parkin (2010); Rahman  _et al._ (2021). bits etc

##  II Methodology

We performed first-principles calculations within the GGA (generalized-
gradient approximation to a better description of the electronic properties
according to the parameterization of PBE (Perdew-Burke-Ernzerhof)Perdew  _et
al._ (1996) for the exchange an correlation potential. The electrons wave-
functions were built using the projected augmented wave PAW method Blöchl
(1994). Silicene has an hexagonal symmetry with two atoms in the unit cell.
The k-mesh used is (12x12x1), centered in $\Gamma$ point, according to
Monkhorst-PackChadi (1977). The single-electron wave-functions were expanded
in plane-waves up to the energy cutoff of $400eV$. The technique used to
calculate the diffusion barrier of Cr atoms on silicene is known as NEB
(Nudged Elastic Band) Hannes  _et al._ (1998). WE have chosen 7 images to
investigate single atom difusion on silicene.

##  III Results

###  III.1 Pristine silicene

Figure 1: Projected band structure and projected density of states for bulk
silicon.

One can obtain silicene from bulk silicon by cutting the bulk silicon along
the (111) plane Curcella  _et al._ (2017). Pristine silicene was investigated
in three different configuration: buckled, planar silicene and dumbbell
silicene. The buckled silicene represents the ground state configuration for
the two-dimensional silicon sheet. Buckled silicene as seen in Fig. 3.The
buckled state stability can be understood by orbital hybridization.

Figure 2: Relative total energy of bulk silicon and silicene as a function of
the lattice parameter.

The lattice parameter is $a$ = 3.87Å, the buckling distance is $\Delta$ 0.45Å,
and the bond length between silicon atoms is $d$ = 2.28Å.

It is possible to see the indirect energy gap at the Fermi level of bulk
silicon, the calculated energy gap is Egap =0.46 eV. The experimental result
for the energy gap is is 1.17 eV Jafari  _et al._ (2022). Although the GGA
underestimate the electronic band gap as shown in Figure 1, the use of GGA
will be later justified. All the orbitals contribute to states in the top of
the valence band and in the bottom of the conduction band. The spin up and
spin down projected density of states is semiconductor.

Group IV atoms on a monolayer configuration tend to bond to other three atoms
by $sp^{2}$ hybridization, forming $\sigma$ bonds, leaving a half occupied
$p_{z}$ orbital perpendicular to the plane. What happens to the remaining
orbital depends on which atom the structure is made of. A monolayer structure
made of carbon will have adjacent unoccupied $p_{z}$ orbitals bounding to each
other forming a strong $\pi$ bond due to the relative short distance between
C-C interaction, resulting in a planar structure. Otherwise, if the structure
is made of Silicon atoms the $\pi$ bond by adjacent $p_{z}$ orbital coupling
is weakened by the relative long distance in Si-Si interaction. Thus, the
$p_{z}$ orbital will prefer to combine with $s$ orbital forming a sp3 state,
resulting in structural modification Zhao  _et al._ (2016).

Figure 3: Side and top views of buckled monolayer silicene Figure 4: Projected
band structure (left) and projected density of states (right) on $s$ and $p$
orbitals of Si atoms for buckled silicene, each orbital is specified in the
inset.

The band structure of buckled silicene shows some similarities to the graphene
one. It is possible to see the Dirac states at the symmetric point K in the
Brillouin zone crossing the Fermi level Figure 4. The Dirac cone is a very
interesting feature of the band structure dispersion. Near the Fermi level the
states have a linear dispersion that can indicate a high electron mobility
that can lead to several electronic applications in nanotechnology Zhao  _et
al._ (2016).

The projected band structure shows that the $p_{z}$ orbital is the one which
contribute the most in the Dirac cones. It is possible to notice that their is
a low contribution, thus relevant of $s$, $p_{x}$ and $p_{y}$ orbitals at the
Fermi level. The system presents $sp^{3}$ hybridization. This result matches
with the buckled stability explanation given before. Silicene is usually
called as zero gap semiconductor. With the inclusion of spin-orbit coupling,
the energy gap of the buckled structure is in the order of 10-3meV.

Planar silicene has slightly higher energy and lattice parameter compared to
buckled silicene, as expected. Despite of not being the ground state of the
free-standing 2D silicon sheet is possible to obtain a local energy minimum
for planar silicene, as seen in Figure 5. We found a lattice parameter $a$ =
3.90Å, the buckling distance is non existent and the bond length between
silicon atoms is $d$ = 2.25Å, in agreement with previous calculations Drummond
_et al._ (2012); Padilha and Pontes (2015).

Figure 5: a)Top and b) side views of planar monolayer silicene. Figure 6:
Projected band structure (left) and projected density of states (right) on $s$
and $p$ orbitals of Si atoms for planar silicene.

The band structure of planar silicene is very similar to the buckled silicene
band structure, one can notice that the Dirac cones appear at the Fermi level
at the K point. There is a small change in the density of states obviously
because of the structural modification. The zero gap semiconductor
characteristic still can be seen Figure 6.

Dumbbell silicene is a 2D silicon structure that derives from and adatoms
adsorbed on buckled silicene as seen in Fig. 7. Dumbbell silicene was recently
synthesized on Ag(110), stabilized by adatoms Leoni  _et al._ (2021) has been
demonstrated. This structure transformation leads to a band structure changes
as shown in Figure 8. Lattice parameter $a$ = 7.42Å, the thickness is $d$ =
2.73Å, and the bond length between silicon atoms $d$ = 2.35-2.37Å. The
dumbbell projected band structure in Figure 8 shows us a gap formation of 0.25
eV. The valence band top is located at the K symmetry point in the Brillouin
zone, and the bottom of conduction band is located at the $\Gamma$ point. One
can notice that the orbital that contribute mostly at the Fermi level region
is the $p_{z}$ orbital and in the conduction band.

Figure 7: Top and side views of monolayer dumbbell silicene. Figure 8: Projected band structure (left) and projected density of states (right) on $s$ and $p$ orbitals of Si atoms for dumbbell silicene. structure |  $a$ (Å) |  $\Delta$ (Å) |  $d$ (Å) | Eg (eV)
---|---|---|---|---
Bulk silicon | 2.73 | - | 3.36 | 0.46
Buckled | 3.87 | 0.38 | 2.27 | -
Planar | 3.90 | 0.0 | 2.25 | -
Dumbbell | 3.71 | 2.73 | 2.35-2.37 | 0.25
Table 1: $\Delta$, $d$ and Eg for pristine bulk Si and silicene.

The energy comparison of bulk silicon and silicene phases are shown in Figure
2. Their structural properties and electronic band gap is shown in Table 1. We
will use bulk silicon as reference to calculate the cohesive energy of the Si
based 2D materials, and to calculate the cohesive energy of bulk silicon will
be used the energy of silicon atoms set at free atoms.

The cohesive energy to of the system is given by the equation below:

$E_{\rm c}=E_{\rm silicene}-\sum_{i}\frac{E_{\rm ref}}{n_{\rm ref}}N_{i}$ (1)

structure | Ec(eV/atom)
---|---
| ref. bulk | ref. atom
Bulk silicon | 0.0 | -4.58
Buckled | 0.64 | -3.94
Planar | 0.66 | -3.92
Dumbbell | 0.44 | -4.14
Table 2: Cohesive energy Ec using bulk silicon as reference (first column) and
a free silicon atom as reference (second column).

It is possible to see that among the three two-dimensional Si-based
structures, the one with lowest cohesive energy are dumbell and buckled
silicene with very similar formation energy, as shown in Tab. 2. A positive
(negative) cohesive energy means an endothermic (exothermic) process.

##  IV Chromium adsorption on silicene monolayers

The first we determine the most stable site adsorption site for the chromium
atom. Three sites were investigated, represented in Figure 9: bridge (between
two silicon atoms), hollow (in the middle of the hexagon) and top (on a single
silicon atom). The follow configurations were set with a smart guess for their
positions and then relaxations were performed to obtain the energy of each
configuration.

Figure 9: Adsorption sites for Cr atoms on silicene.

The most stable configuration has a chromium atom adsorbed at a hollow site.
The chromium atom does not stay on the bridge or site, but relax spontaneously
to the hollow site

The lattice structural parametres are: $a$ = 4.08 Å, the buckling distance
$\Delta$ = 1.19Å, and the bond length between silicon atoms is $d$ = 2.64Å and
magnetic moment of 3.14$\mu_{B}$. The chromium adsorption changes
significantly the structure of silicene, affecting mainly the buckling
distance that goes from 0.45Åin the pristine monolayer to 1.19Åin the adsorbed
layer.

This system has a peculiar characteristic related to the variation of the
lattice parameter. The ground state configuration is where the Cr atom lies
between the silicon atoms of silicene. At this conformation the system is
metallic and shows a total magnetic moment of $3.14\mu_{B}$. If one applies a
small strain on the unit cell,the Cr atom moves to the top of the layer This
renders the silicene a semiconductor with a small gap and zero magnetic
moment. This is an interesting finding since a small strain applied on the
structure induce a huge change making the material from metallic and magnetic
to a semiconductor with no magnetic moment. This result in in agreement with
predictions of Zheng  _et al._ (2015); Jafari  _et al._ (2022).

Figure 10 shows the relative energy of the lattice constant for three
different atomic heights of Cr on silicene. The ground-state is very sensitive
to the height, which could be useful to tune the magnetic properties os
silicene. Furthermore, the Cr atom induces a strain on silicene, making the
Si-Si distances 0.4 Ålonger, and the lattice parameter differ by 0.2 Å.

The band structure of silicene doped with Cr atom shows that the systems goes
from a zero gap semiconductor (pristine buckled silicene) with Dirac cones at
K point, to a metal with a net magnetic moment, as seen in Figure 11. Those
states at the Fermi level are mostly d orbitals from Cr atom. It is possible
to notice that the $d$ orbitals from Cr atom hybridize with the
$p_{x},p_{y},p_{z}$ orbitals from Si atoms, mostly with $p_{z}$. There is a
negligible contribution from $s$ orbitals from Si atoms, which is the orbital
that contribute mostly to the hybridization with $p_{z}$ on the pristine
layer. The Dirac cones are no longer at the Fermi level.

Figure 10: Different heights of Cr atom position on buckled silicene. Figure
11: Projected band structure (left) and project density of state (right) for
relaxed Cr-doped buckled silicene.

In order to identify better the hybridization between Cr atoms and Si atoms
and the effect of quantum confinmene we compare the two forms 2D and 3D of
CrSi2. The bulk CrSi2 Chen  _et al._ (2018) has the structure shown in Figure
LABEL:fig:bulk_CrSi, its has a hexagonal unit cell and is made of three Cr
atoms and six nearest-neighbors Si atoms. The lattice parameter is $a=4.41\AA$
and $c=6.35\AA$. Bulk CrSi2 shows a symmetric density of states, therefore is
a non-magnetic material , and has a energy gap of Eg = 0.69eV, as one can see
in Figure 11.

The DOS of Cr doped buckled silicene Figure 11 and DOS of bulk CrSi2 shown in
Figure 11 it is possible to notice that the contributions of Si $p$ and $s$ to
the hybridization with Cr $d$ orbitals are much larger in Cr doped buckled
silicene. It means that the scattering of the Cr atoms by the Si atoms is much
more intense in Cr doped buckled silicene than in bulk CrSi2. It results in
more localized states as one can see in Figure LABEL:fig:chargedensity_diff.
More localized states results in better qubits that can be used in spintronics
applications.

The concentration of Cr atoms was varied from 2% up to 50%. The variation was
made adding one Cr atom in the unit cell and then the size of the cell was
increased. For example: for 50% of concentration the unit cell used was the
(1x1) silicene unit cell (2 Si atoms and 1 Cr atom), for 12,5% of
concentration the cell used was the (2x2) silicene cell (8 Si atoms 1 Cr
atom).

The initial position for each Cr atom on silicene is the same as calculated in
the (1x1) unit cell. The Cr atom is positioned in the half height (define
this) of silicene. Since Cr atom produces a strain in the silicon honeycomb.
It is important to notice that at low concentrations the lattice parameter
does not change much compared to the pristine layer. Another aspect to be
noticed is that the Cr atom increased the buckling distance of silicon sheet.

Adsorption energies of Cr atoms were calculated with respect to bcc Cr (body-
centered cubic) which is going to be our reference material for Cr atoms. Bcc
chromium (two atoms in the atomic basis) is the most stable structure for pure
chromium lattice with a lattice parameter of 2.85Å. The metallic phase of
chromium has antiferromagnetic properties, all the Cr atoms presents
magnetization and their total angular momentum are oriented anti-parallel in a
way that the liquid magnetic moment of the crystal is zero.

The reference for the silicon chemical potential is taken as the total energy
for the pristine buckled silicene, for chromium the bcc phase. The general
form of the energy adsorption equation is given below:

$E_{\rm ads}=E_{\rm silicene/Cr}-\frac{E_{\rm silicene}}{N_{\rm
silicene}}\cdot N_{Si}-\frac{E_{\rm bulkCr}}{2}\cdot N_{\rm Cr}.$ (2)

Figure 12: Adsorption energy and magnetic moment of Cr atoms on silicene by
concentration.

One can see that the adsorption energy decreases as the increase of
concentration of Cr atom adsorbed in silicene. A possible explanation is the
lateral interaction between Cr atoms, closer they are from each other, more
they interact. Another aspect to be analysed is the fact that all adsorption
energies are positive.

Figure 12 b shows the magnetic moment as a function of the concentration. It
is easy to notice that the magnetic moment decrease as the concentration of Cr
atoms adsorbed increases.

##  V Substitutional Cr in silicene

The substitution process is to remove a Si atom from the monolayer and adsorb
in its place another type of atom. In this work the substitutional atom is the
Cr atom and the formation energy of this kind of defect was calculated. The
expression that calculates the formation energy is very similar to Equation 2
and is based on Equation 1. The expression is:

$E_{\rm f}=E_{\rm silicene/Cr}-E_{\rm silicene}-E_{\rm chromium}$ (3)

The concentration of this kind of defect was varied and the formation energy
was calculated.

Figure 13: Formation energy and magnetic moment of substitutional Cr atom in
silicene as a function of the concentration.

One can observe that all formation energies are negative, it means that the
system needs to lose energy to form the defect. For low concentrations the
formation energy is low. For higher concentrations of the substitutional
chromium, the formation energy is relatively higher. It could be due to the
Cr-Cr interaction that it becomes more difficult to substitute Cr atoms when
there is already a certain number of substituted Cr atoms in silicene layer.

Figure 13 shows the magnetic moment as a function of substituted Cr atom
concentration on silicene. After analysing the graphic one can see that the
magnetic moment is higher for lower concentrations of Cr atoms. This finding
is similar to adsorbed chromium.

The NEB method is used to determine the path which a atom on the crystal
energy surface would take when connected by two states, which are minimum
energy states of the system. NEB will seek for the less costly (in energy)
path between two states. The path through the potential energy surface show us
the energy barrier which the atom needs to overcome to go to a minimum energy
state to another minimum energy state.

NEB method was performed to calculate the energy barriers for the Cr atom on
the potential energy surface of silicene mono layer. The NEB was performed in
two cases: Cr atom goes from a hollow site to an adjacent one, along a path
parallel or perpendicular to silicene plane, as shown in Figure
LABEL:fig:barrier_geometry.

Figure 14: Diffusion barrier for parallel and perpendicular path.

The diffusion barrier energy for the parallel path is 1.60 eV. The barrier has
a significantly high energy, which means that it is unlikely to a Cr atom goes
from one hollow site to an adjacent one. The barrier presents a local minimum
energy state in the top of the barrier, represented by the step 3 showed in
Figure 14. This shape is due to the path, which passes by a minimum local
energy site which is a top site. Top site is a high symmetry point in the
hexagonal silicene lattice, usually high symmetry sites are lower energy
states which explains the formation of local minimum energy point. This shape
of barrier could be associated with buckled 2D structures. The radius of the
atom influences the adsorption position. Thus, a question is whether it is
possible to find a minimum local energy pit (as step 3 in Figure 14 of a
certain 2D buckled material adsorbed with a certain atom that could be
captured and maintained adsorbed in the top site instead of the global minimum
point. The diffusion barrier for the path perpendicular to the layer is
1.73eV. The energy of this barrier is higher compared to the path parallel to
the layer. The step 3 in Figure 14 shows the transition states is where the Cr
atom passes through the monolayer. This point can be related to the maximum
strain that the Cr atom is putting on the lattice of silicene.

Figure 15: Chromium atom pair adsorbed on silicene along the a) arm-chair direction and b) zig-zag direction. |  $\Delta$ E (eV) | a(Å) | $\mu_{\rm tot}(\mu_{B})$ | dCr-Cr (Å)
---|---|---|---|---
conc.(%) |  | AFM | FM | AFM | FM | AFM | FM
5.5 | -0.16 | 3.90 | 3.92 | 0.00 | 3.61 | 4.16 | 4.18
3.1 | -0.26 | 3.90 | 3.92 | 0.00 | 4.26 | 4.19 | 4.19
2.0 | -0.33 | 3.90 | 3.92 | 0.00 | 4.70 | 4.20 | 4.15
1.3 | -0.47 | 3.90 | 3.92 | 0.00 | 4.53 | 4.22 | 4.20
Table 3: Energy difference between AFM and FM states $\Delta$E, lattice parameter a, magnetic moment $\mu_{\rm tot}$ and Cr-Cr distance dCr-Cr for silicene monolayer doped with a pair of Cr atom along the zigzag direction. |  $\Delta$ E (eV) | a(Å) | $\mu_{\rm tot}(\mu_{B})$ | dCr-Cr (Å)
---|---|---|---|---
conc.(%) |  | AFM | FM | AFM | FM | AFM | FM
5.5 | 0.09 | 3.87 | 3.88 | 0.00 | 3.61 | 7.04 | 7.07
3.1 | 0.10 | 3.87 | 3.88 | 0.00 | 4.26 | 7.06 | 7.04
2.0 | -1.44 | 3.87 | 3.88 | 0.00 | 4.70 | 7.03 | 6.83
1.3 | -1.51 | 3.87 | 3.88 | 0.00 | 4.53 | 7.04 | 6.77
Table 4: Energy difference between AFM and FM states $\Delta$E, lattice parameter a, magnetic moment $\mu_{\rm tot}$ and Cr-Cr distance dCr-Cr for silicene monolayer doped with Cr-pair along the arm-chair direction. |  $\Delta E$ (eV)
---|---
conc. (%) | AFM | FM
5.5 | -0.92819 | -1.18623
3.1 | -0.98085 | -1.3428
2.0 | -1.01236 | 0.10065
1.3 | -1.01284 | 0.03537
Table 5: Energy comparison between arm-chair and zig-zag directions in
antiferromagnetic and ferromagnetic states.

The magnetic properties of the chromium atom on silicon monolayer has been
investigated for silicene with a pair of Cr adsorbed along two different
directions (arm-chair and zig-zag) as shown in Figure 15. The energy
difference between AFM and FM for both directions is shown in Tables 4 and 3.

###  V.1 Chromium atom-pairs adsorbed on dumbbell silicene

The pristine dumbbell silicene was doped with two Cr atoms as shown in Figure
16. A feature to be noticed is that the height of dumbbell silicene increased
around 1 Å and the lattice parameter decreased slightly when doped with Cr
atoms.Cr atoms tends to shrink the dumbbell silicene in the layers plane and
increase its height Table 6. By comparing the properties of AFM and FM states
of DB silicene it is notable that FM state presents more magnetic moment per
Cr ion in FM state. This property might be associated with the disposition of
the Cr atoms in the DB silicene structure since in AFM state has their pair of
Cr atoms closer to each other at a distance of $2.79\AA$ compared to FM state
where the Cr-Cr distance is $3.35\AA$. However, the AFM state is more stable
than FM state by 0.6 eV.

Figure 16: a)Side view b) top view of dumbbell silicene unit cell doped with two Cr atoms. | a (Å) |  $\mu$/Cr ($\mu_{B}$) | h | dCr-Cr (Å)
---|---|---|---|---
AFM | 7.12 | 1.72 | 3.70 | 2.79
FM | 7.12 | 2.69 | 3.59 | 3.35
Table 6: $a$, $\mu$/Cr, $h$ and dCr-Cr for both AFM and FM states in dumbbell
silicene upon adsorption of pair of Cr atoms.

Figure 17: Projected density of state and projected band structure of dumbbell
silicene for a) AFM and b) FM configurations.

As silicene dumbbell is doped with Cr atoms, the structure becomes metallic.
It is possible to notice the difference between AFM and FM density of states
(DOS). The contributions of Si orbitals at the Fermi level of AFM state is
very small Figure 17. Near the Fermi level the orbitals that contribute the
most are $p_{x}$ and $p_{y}$. The contributions of Si orbital at the Fermi
level of FM state are slightly larger compared to AFM state. Near the Fermi
level it is possible to notice a significant larger contribution of $p_{z}$
orbital in FM DOS, seen in Figure 17. This larger contribution of Si $p_{z}$
orbital and smaller contribution of Si $p_{x}$ and $p_{y}$ orbitals to the
hybridization with Cr $d$ orbitals might be related to the increase of the
magnetic moment of structures studied in this work.

##  VI Pristine bilayer silicene

Bilayer silicene can be obtained on a (111) Ag surface Arafune  _et al._
(2013); Curcella  _et al._ (2017). Few theoretical reports have shown that the
preferred stacking is the AB Padilha and Pontes (2015); Kharadi  _et al._
(2023); Fu  _et al._ (2014); Do  _et al._ (2019). One can stack a monolayer
right above the other, this kind of stacking called AA stacking. One can stack
the two monolayers in a way that the lattice points of one layer is shifted,
those are known as AB stacking. These two type of stacking can be branched
because of the buckling pattern. It results in the six bilayer structures that
have been studied in this work.

The relaxed structures are shown in Figure 18. The structural properties of
six stackings can be seen in Table 7. The AB, AB’ and A’B’ lattice constants
and buckling are similar. However, the interlayer distance is very different.

Figure 18: a) Top view and b) side view of various stackings in pristine bilayer silicene. Stacking |  $a$ (Å) |  $d$ (Å) $\Delta$ (Å) |
---|---|---|---
AA’ | 3.86 | 2.45 | 0.66
AAb | 3.77 | 1.81 | 0.92
AAp | 4.12 | 2.40 | 0.0
A’B’ | 3.86 | 3.35 | 0.51
AB | 3.86 | 2.51 | 0.65
AB’ | 3.85 | 2.19 | 0.68-0.77
Table 7: Structural properties of pristine bilayer stackings: lattice
parameter $a$, bond length $d$ and buckling $\Delta$ (Å). Figure 19: Energy as
a function of the lattice parameter for bilayer stackings of pristine
silicene.

Figure 19 shows the ground state energy difference among the bilayers as a
function of the lattice parameter. The AA’, AAb, AB and AB’ have very similar
energies. A’B’ stacking mode is the less stable structure among all free
standing bilayers. AAp and AB are the most stable pristine bilayers with the
largest lattice constant, in agreement with Ref. Padilha and Pontes (2015).
The band structure and density of states show that $p_{z}$ orbitals contribute
the most in Fermi level $p_{z}$ orbitals are responsible to the bonding
between the two layers, forming the bilayer. All the other stacking modes
presents the same behavior.

Figure 20: Band structure and projected density of states of a) AA (buckled)
and b) AB stacking in pristine bilayer silicene.

##  VII Intercalation of Cr in silicene bilayers

The (1x1) unit cell of the stacking modes were doped with one Cr atom between
the two layers of each kind of stacking shown in Figure 20. It is obvious that
all the structures had their properties modified by the presence of the
metallic atom Table 8. All the structures had their interlayer distance
increased. The buckling of all structures changed. Structures such as AA’ and
AB had their buckling increased, AAb was the only structure to have the
buckling decreasing after being doped. The structure which had a increasing
buckling tends to have a slightly smaller lattice constant.

Figure 21: Top view and side view of Cr doped bilayers in silicene. Stacking |  $a$ (Å) |  $d$ (Å) |  $\Delta$ (Å) |  $\mu$ ($\mu_{B}$)
---|---|---|---|---
AA’ | 3.86 | 2.72 | 0.78 | 3.68
AAb | 3.82 | 2.90 | 0.73 | 3.43
AAp | 4.22 | 2.46 | 0.0 | 3.84
BB’ | 3.61 | 2.62 | 1.25 | 1.25
AB | 3.76 | 2.80 | 0.92 - 1.07 | 3.18
AB’ | 3.70 | 2.71 | 1.05 - 1.12 | 2.35
Table 8: Structural and magnetic properties of pristine bilayer stackings.

Upon doping AAp becomes now the less stable structure. As a general behavior
the doped hcp stacking modes are energetically more favourable compared to the
remaining structures. One feature to be noticed is that the A’B’ stacking mode
have vanished and replaced by a staking mode called BB’. The ground state
configuration for doped A’B’ is probably metastable. The BB’ stacking is the
same as the AA’ but the Cr atom was doped in the top site instead of the
hollow site. Turns out that BB’ was the most stable of all stacking modes,
despite of having a flat energy ground state as can be seen in Figure 22.

Figure 22: Energy as a function of the lattice parameter for stackings of Cr
doped bilayer silicene.

The magnetic moment of doped bilayer silicene reaches the maximum in AAp (less
stable) and reaches the minimum in BB’ stacking mode (most stable). So it is
possible to see with the information of Table 8 that there is relation between
the magnetic moment and buckling of the bilayer. The larger the buckling the
smaller is the magnetic moment. So it is important to analyse the density of
states and the charge density distribution and associate this relation to the
orbitals occupation at the Fermi level.

The charge density difference allows us to visualize the charge density in the
material after including a new atom in the structure. One can obtain a image
of the charge density difference as in Figures 23.

Figure 23: Charge density difference a) top view and b) side view of bilayer
stacking modes. The yellow region is where the charge has accumulated and the
cyan region is where there is a depletion of electrons.

Figure 23 shows the charge density difference where the yellow region is where
the charge has accumulated and the cyan region is where there is a depletion
of electrons. Therefore, one can easily see that in all stacking modes the Si-
Si bond in the same layer had a charge depletion, and the charge near the Cr
atom have depleted too. It means that the silicene donated electrons to the Cr
atom as seen in Figure 23. One can observe that the charge density have
accumulated in a very specific shape between the Cr atom and the two layers.
This shape indicates a formation of a new orbital, which comes from a
hybridization of the $sp^{3}$ and $sp^{2}$ orbitals of silicene and the $d$
orbitals from Cr-$d$ orbital having a larger contribution at the Fermi level.
It is possible to notice formation of six orbitals in every stacking mode.
This happens because Cr atom electronic distribution in the valence level is
$3d^{5}4s^{1}$.

The charge density difference and the projected DOS show Si orbitals
contribution at the Fermi level compared to the Cr orbitals contribution is
very small, i.e., the potential of Si does not scatter strongly the Cr states,
hence the orbitals are more localized as one can see in Figure 23. Those
features make doped silicene bilayers good candidates for spintronics
applications.

The larger is the buckling the smaller is the magnetic moment of the
structure. In order to understand that, we take as showcases the AA’ and BB
bilayers. Their charge density distribution close to the Fermi level and the
projected density of states are shown in Figures 24 and 25, respectively. We
chose AA’ and BB’, since AA’ has a relatively low buckling a high magnetic
moment. On the other hand we have BB’ which has a high buckling and small
magnetic moment.

Figure 24: Partial charge at Fermi level for a) AA’ and b) BB’ stacking configurations. | Low buckled | High Buckled
---|---|---
Stacking | AA’ | AAb | AAp | AB | AB’ | BB’
| $\mu(\mu_{B})$
---
3.68 | 3.43 | 3.84 | 1.27 | 3.18 | 2.35
spin up | $p_{x}$ | 0.036 | 0.054 | 0.019 | 0.098 | 0.094 | 0.080
$p_{y}$ | 0.059 | 0.054 | 0.042 | 0.098 | 0.094 | 0.080
spin down | $p_{x}$ | 0.014 | 0.061 | 0.059 | 0.028 | 0.014 | 0.028
$p_{y}$ | 0.014 | 0.061 | 0.067 | 0.028 | 0.014 | 0.028
Table 9: Silicon $p_{x}$ and $p_{y}$ orbitals contribution in states/eV at the
Fermi level of stacking modes.

Figure 24 shows that the charge distribution near the Fermi level of the two
stacking modes are different.Figure 24 shows an accumulated charge in the
silicene monolayers and all around the Cr atom. So in AA’ stacking mode shown
in Figure 24 it is possible to see the charge in the outer layers of the
silicene monolayer is related to the $p_{z}$ orbital. The BB’ stacking mode
shown in Figure 24 is easy to identify that the charge is accumulating between
the Si-Si bonds, the accumulated charge are $p_{x}$ and $p_{y}$ orbital.

In Table 9 we show the silicon orbitals contribution to the hybridization with
Cr-$d$ orbitals near the Fermi level. It is possible to notice that Si $p_{x}$
and $p_{y}$ orbitals have smaller contributions in low buckled stacking modes
to the hybridization with Cr-$d$ orbitals. On the other hand the contribution
of Si $p_{x}$ and $p_{y}$ orbitals have larger contributions in the high
buckled stacking modes. Therefore, one can relate the increase of the magnetic
moment of to the density contribution of Si $p_{x}$ and $p_{y}$ in the
hybridization with Cr-$d$ orbitals. This relation can be seen by comparing the
AA’ and BB’ stacking modes. A similar behavior is found for Cr-doped dumbbell
silicene. Moreover, one can see that the charge accumulated around the Cr atom
is more localized for the AA’ compared to the BB’ stacking. Notice that the
layer distance in the two stacking modes are almost the same.

Figure 25: Projected band structure and projected density of states of Cr-
doped a) AA’ and b) BB’ stacking mode.

##  VIII Conclusions

The adsorption of Cr atom on silicene changes significantly the structural
properties and electronic properties of the monolayer. The adsorption or
substitution of Cr atom often causes a strain on silicene yielding a increase
not just in the buckling but also in the lattice constant. Moreover, magnetic
properties arise in all doped Si based structures. As the concentration of Cr
atoms adsorbed or substituted increase, the magnetic moment of the system
tends to decrease. Diffusion barrier calculations show that if an Cr atom is
adsorbed in silicene, it is very unlikely that the Cr atom moves for an
adjacent hollow site, or cross the silicene structure since the diffusion
barriers are very high. The magnetic properties study of Cr pair adsorbed on
silicene showed that arm-chair direction is usually more stable configuration
than zig-zag direction of adsorption. Low concentrations of adsorbed Cr atom
pair in arm-chair direction tends to a AFM state. But in the majority of the
systems studied their is no big energy difference between the FM and AFM
states, so strain controlled could be applied for spintronic applications.
Dumbbell silicene adsorbed with Cr atoms pair is more stable in AFM
configuration. The adsorption of the Cr pair changes strongly the pristine
dumbbell silicene structural and electronic properties. The Cr doped bilayer
showed a very interesting relation between the buckling of silicene and the
total magnetic moment of the system. The larger is the buckling the smaller is
the magnetic moment. Using the charge density analysis and projected DOS one
could relate the variation in the magnetic moment to the contribution of Si
$p_{x}$ and $p_{y}$ orbitals to the hybridization with Cr $d$ orbitals. Cr
doped bilayer silicene could be a promising material for spintronic
applications since good qubits is found in the stacking modes studied.

##  IX Acknowledgements

We acknowledge the financial support from the Brazilian funding agency CNPq
under grant numbers 313081/2017-4 and 305335/2020-0. L. M. G. thanks for a
CNPq fellowship. We thank computational resources from LaMCAD/UFG, Santos
Dumont/LNCC, CENAPAD-SP/Unicamp.

!

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# Towards Semantic Search for Community Question Answering for Mortgage
Officers

Amir Reza Rahmani  amirrezar, linweil, brianv, colinbe<EMAIL_ADDRESS>,  Linwei Li  ,  Brian Vanover  ,  Colin Bertrand  and  Shourabh Rawat  Zillow
Group Automation and AI1301 2nd AveSeattleWashingtonUSA98101

###### Abstract.

Community Question Answering (CQA) has gained increasing popularity in many
domains. Mortgage is a complex and dynamic industry, and a flexible and
efficient CQA platform can potentially enhance the quality of service for
mortgage officers significantly. We have built a dynamic CQA platform with a
state of the art semantic search engine based on recent Natural Language
Processing (NLP) techniques to dynamically and collectively capture and
transfer the maturity and tribal knowledge of the more experienced workforce
to less experienced ones. The search engine allows for both keyword and
natural language queries and is based on a fine-tuned domain-adapted Sentence-
BERT encoder linearly composed with a TF-IDF vectorizer, and reciprocal-rank
fused with a BM25 vectorizer. Domain adaptation and fine-tuning is based on
publicly available mortgage corpora. Evaluation is performed on an internally
annotated dataset using standard information retrieval metrics such as
normalized discounted cumulative gain (nDCG), precision/recall at n, mean
reciprocal rank, and mean average precision (MAP). The results indicate that
our hybrid, fine-tuned, domain-adapted search engine is a more effective
approach in responding to the information needs of our mortgage officers
compared to traditional search techniques. We aim to publish the internally-
annotated evaluation and training datasets in the near future.

semantic search, community question answering, search engine, mortgage

††copyright: none††conference: The Second Document Intelligence Workshop at
KDD; 2021; Virtual Event††ccs: Information systems Combination, fusion and
federated search

##  1\. Introduction

A CQA forum combined with an advanced search engine that allows the users to
also submit their queries in natural language could significantly boost the
efficiency of operations in a fast-paced organization. In the mortgage
industry, where operations are based on dynamic interactions with a large
population of customers, a question answering platform will help ensure the
consistency of the services offered, on-board and train new members, capture
the tribal knowledge of the team, and also offer more versatile products and
services to meet the buyers’ needs.

With these motivations in mind, we have developed a CQA system for all the
professionals who are involved in Zillow Group’s mortgage operations. Our
system consists of two major parts: a) a question-answering platform which
allows the users to post questions, answer their peers’ questions, up-vote and
approve answers, and track their peers activities, b) a state of the art
search engine that can search public mortgage resources as well as internal
contents that our mortgage team produces using the QA platform.

To build an effective system for our mortgage staff, the search engine needs
to a) accept natural language as well as keyword queries, b) return highly
relevant search results right at the top, and c) do so within a reasonable
time. Hybrid search engines, i.e. search engines that perform semantic and
keyword search, are a promising choice to satisfy these requirements. Many
efforts have been made in this direction. For example, (Esteva et al., 2020;
Mass et al., 2020) implemented retrieval systems by combining Sentence-
BERT(Reimers and Gurevych, 2019) and traditional retrieval models such as
BM25(Robertson and Zaragoza, 2009).

Our search engine uses a hybrid approach to search over all the internal and
external data. Our database consists of FAQ-answer pairs that are gathered by
using publicly available FAQ pages of business related entities (external) and
also questions and answers that our mortgage staff input into the forum
(internal). When a user inputs a query in the search box, the query is matched
against all the external FAQ-answer pairs as well as all the contents from our
internal forum. To train and tune the search engine, we fine-tune a Sentence-
BERT model and train a TF-IDF(Salton and McGill, 1986) and a BM25 model on all
the FAQ-answer pairs. At run time, for an incoming query, the Sentence-BERT
score for the question part of every FAQ-answer pair is linearly combined with
the TF-IDF score on both question and answer parts of all FAQ-answer pairs.
The resultant scores generate a ranking which is reciprocally fused with the
BM25 ranking. The linear combination between the TF-IDF and Sentence-BERT
score is controlled using a dampening factor which gives more weight on SBERT
for longer queries. Although our search algorithm is inspired by (Esteva et
al., 2020), they differ in the following aspects: a) we use FAQ-answer pairs
as our documents and we combine the contributions from both question and
answer parts during indexing and inference, b) we use a dampening factor
during inference based on the length of the query to control the degree of
Sentence-BERT and TF-IDF combination, and c) our search engine is specifically
fine-tuned to mortgage domain and is evaluated on a domain-specific dataset.

Finally, the emphasis of this work is to develop a functioning system that
translates algorithmic ideas into a novel industry context and to solve a
practical business problem. The system is currently being utilized by our
mortgage staff and with the user interface capturing feedback, we are
collecting data to further improve the system and pragmatically develop the
product road map.

Figure 1. System Architecture

System Architecture

##  2\. Model

The search engine (Figure 1) consists of two components, representation
learning and retrieval. During representation learning, a preprocessed
tokenized FAQ-answer pair $(q,a)$ is taken as the input and three embeddings
are generated through a Sentence-BERT model and a TF-IDF vectorizer. The
Sentence-BERT embedding $u$ only attends to the question part of the FAQ-
answer pair and the TF-IDF vectorizer outputs $e^{q}$ for the question and
$e^{a}$ for the answer part. During retrieval, a query $Q$ is provided and
multiple FAQ-answer pairs $(q_{1},a_{1}),\dots,(q_{n},a_{n})$ are returned as
the retrieval result. The ranking of the retrieved FAQ-answer pairs is
determined by a reciprocal rank fusion over the ranking produced by the
aforementioned embeddings’ cosine similarity scores and a BM25 ranking.

###  2.1. Representation Learning

####  2.1.1. Sentence-BERT (SBERT)

We have observed in our dataset that:

  1. (1)

The question part of an FAQ-answer pair is often a good summary of the
answer’s topic.

  2. (2)

FAQs are often grammatically complete sentences.

  3. (3)

Similar concepts can be expressed in various lexical forms, e.g. loan and
mortgage.

Transformer based language models are effective in capturing the semantics,
however, they are also known to be slow during training and inference. Based
on observations (1) and (2), SBERT model becomes a good fit for our task
because it directly generates embeddings that cluster semantically similar
sentences together without further processing needed during inference. The
SBERT model uses a siamese network to capture the semantic textual similarity
between sentence pairs and uses a triplet or a classification loss to guide
the network. Their proposed way of using cosine similarity along with the
siamese network significantly reduces the search time as each sentence is
encoded only once whereas in the original BERT model, every pair needs to be
fed in the network (separated by the ¡SEP¿ token). Furthermore, (Reimers and
Gurevych, 2019) showed that BERT embeddings (either based on the ¡CLS¿ token
or obtained by averaging the outputs) are not effective at capturing sentence
semantics and are outperformed by GloVe embeddings (Pennington et al., 2014).

The training process involves two steps:

##### BERT finetuning

We perform language model fine-tuning on the bert-base model(Wolf et al.,
2019) by applying random masks over the sentences selected from the mortgage
FAQ-answer pairs.

##### SBERT training

Based on the finetuned BERT model, we further train the SBERT model using two
different approaches. The first approach is to train the model in a
classification task as mentioned in (Reimers and Gurevych, 2019). The SBERT
model is trained on the SNLI and multi-genre NLI data (Bowman et al., 2015;
Williams et al., 2017).

The second approach is to use a triplet objective function, also discussed in
(Reimers and Gurevych, 2019). A triplet comprising an anchor sentence $s_{a}$,
a positive sentence $s_{p}$, and a negative sentence $s_{n}$ are used as the
inputs. The model generates embeddings $u_{a},u_{n},u_{p}$ for each of the
three sentences and the objective is to maximize $\lVert u_{a}-u_{n}\lVert$
and minimize $\lVert u_{a}-u_{p}\lVert$. When constructing the triplets, the
positive sentence is the sentence immediately following the anchor sentence;
and the negative sentence is sampled from a random FAQ-answer pair. During
experiments, we found that the model easily differentiates the negative
sentences from the anchor sentences, failing to learn meaningful
representations. We therefore construct another set of triplets by selecting
negative sentences from the FAQ-answer pairs of the same category as the
anchor sentence.

####  2.1.2. TF-IDF

We follow the standard TF-IDF computation to generate the embeddings of both
question and answer parts in FAQ-answer pairs.

###  2.2. Retrieval and Re-ranking

During inference, the system retrieves an ordered list of documents that best
matches the query $Q$. The initial ranking is produced by computing the
similarity scores between the query embedding and the embeddings for question
and answer parts of the FAQ-answer pair. The score is a weighted sum of cosine
similarities computed over SBERT embeddings and TF-IDF embeddings:

$\displaystyle F_{\text{initial}}(Q,(q,a))$
$\displaystyle=[\alpha+(1-\alpha)*\zeta]*F_{\text{sbert}}(Q,(q,a))$
$\displaystyle+[(1-\alpha)*(1-\zeta)]*F_{\text{tfidf}}(Q,(q,a))$
$\displaystyle F_{\text{sbert}}(Q,(q,a))$
$\displaystyle=\text{cos}(u_{Q},u_{FAQ})$ $\displaystyle
F_{\text{tfidf}}(Q,(q,a))$
$\displaystyle=w*\text{cos}(e_{Q},e^{q}_{FAQ})+(1-w)*\text{cos}(e_{Q},e^{a}_{FAQ})$
$\displaystyle\zeta$ $\displaystyle=exp(\frac{1-len(Q)}{\beta})$

where $F_{\text{sbert}}$ and $F_{\text{tfidf}}$ are the similarity scores
based on SBERT and TF-IDF, $u_{Q}$, $u_{FAQ}$ are SBERT embeddings for the
query and the question part of the FAQ-answer pair, respectively. $e_{Q}$ is
the TF-IDF embedding for the query, $e^{q}_{FAQ},e^{a}_{FAQ}$ are TF-IDF
embeddings for the question and answer parts of the FAQ-answer pair,
respectively. $\zeta$ is a damping factor which favors TF-IDF similarity for
shorter queries because they are more likely to be keywords, and $\alpha$, $w$
and $\beta$ are hyperparameters. $\alpha$ controls the combination of SBERT
and TF-IDF similarity scores, $w$ controls the combination of TF-IDF
similarity scores between the query-question and query-answer parts, and
$\beta$ controls the damping factor.

The second rankings is produced by a BM25 ranker, with the same weight $w$
combining the question and the answer parts as in TF-IDF. In the end, both
rankings are combined using reciprocal rank fusion as follows:

$\text{RRF}(Q,(q,a))=\frac{1}{k+R_{\text{SBERT+TFIDF}}}+\frac{1}{k+R_{\text{BM25}}}$

where $R_{\text{SBERT+TFIDF}}$ is the initial ranking of the FAQ-answer pairs
produced by SBERT and TF-IDF, $R_{\text{BM25}}$ is the BM25 ranking, and $k$
is a hyperparameter.

##  3\. Experimental Setup

###  3.1. Datasets

####  3.1.1. Training Dataset

The training dataset consists of around 6000 FAQ-answer pairs acquired from
publicly available mortgage data. This dataset is used for BERT domain
language model fine-tuning. The question part of the FAQ is usually less than
two sentences whereas the answer part can range from a single paragraph to a
few pages. About 80% of the FAQs have categories associated to them (e.g.
appraisal, mortgage forbearance, etc). These categories are used when creating
triplets for training SBERT (see 2.1).

####  3.1.2. Evaluation Dataset

We have picked the top 16 queries that our loan officers most frequently
search. For each query, we collect the top 50 FAQ-answer pairs based on TF-
IDF, BM25, and SBERT, respectively, keeping only the unique pairs. Therefore,
for each query, we have at most 150 FAQ-answer pairs to label. We use three
labels: 0 (irrelevant), 1 (somewhat relevant), 2 (relevant). See Appendix A
for more details on the dataset.

###  3.2. Evaluation

We evaluate our search engine against the labelled data from the previous
step. We use TREC-EVAL 111https://trec.nist.gov/trec_eval/ to evaluate our
search engine based on the following metrics:

  * •

Mean Average Precision (MAP, MAP@5, MAP@10)

  * •

Mean reciprocal rank

  * •

Normalized Discounted Cumulative Gain (nDCG@5,
nDCG@10)

  * •

P@5, P@10,

  * •

Recall@5, Recall@10

The metrics are selected based on the original system requirements, i.e.
returning the most relevant results right at the top; we need the top five
retrieved results to contain as much relevant information as possible.

We will compare the following approaches:

  * •

SBERT

  * •

TF-IDF

  * •

BM25

  * •

TF-IDF combined with SBERT

  * •

Reciprocal Rank Fusion (RRF)

The grid search is performed over $w$ and $\alpha$, which represent the linear
factor controlling the combination of FAQ and answer contributions in TF-IDF
and BM25, and the combination of TF-IDF and SBERT contributions, respectively.
They both range between 0 and 1 with a step size of $0.1$. We set $\beta$ to
$3$ for now but we intend to include it in the grid search in the future.

Table 1. Comparison of different algorithms on internal evaluation dataset Algorithm | Mean Reciprocal Rank | Recall@5 | Recall@10 | nDCG@5 | nDCG@10 | MAP | MAP@5 | MAP@10
---|---|---|---|---|---|---|---|---
TF-IDF | 0.8507 | 0.2370 | 0.3364 | 0.6160 | 0.5517 | 0.4485 | 0.2289 | 0.3014
BM25 | 0.8221 | 0.2431 | 0.3671 | 0.6579 | 0.5895 | 0.4923 | 0.2288 | 0.3283
Ours: SBERT + TF-IDF | 0.9375 | 0.2583 | 0.3597 | 0.6625 | 0.5755 | 0.4439 | 0.2518 | 0.3182
Ours: SBERT + TF-IDF + BM25 | 0.8828 | 0.2589 | 0.3586 | 0.6600 | 0.5816 | 0.4874 | 0.2308 | 0.3197
Table 2. Non-trivial queries matched by the semantic search Query | Relevant FAQ
---|---
Can we originate a loan for a home on the market? | Can a property be refinanced if it is currently listed for sale?
Minimum size for manufactured house? | What are the requirements for a living unit?
What credit counseling advice can we give borrowers? | What resources can I provide to applicants to help improve their credit score?
Table 3. Comparison of SBERT models performance on the evaluation data Model | Mean Reciprocal Rank | P@5 | P@10 | Recall@5 | Recall@10 | nDCG | MAP
---|---|---|---|---|---|---|---
SBERT-base (Base BERT + NLI) | 0.6073 | 0.3500 | 0.2250 | 0.1328 | 0.1607 | 0.2995 | 0.1580
SBERT-mor-nli (fine-tuned BERT + NLI) | 0.6646 | 0.4250 | 0.2437 | 0.1559 | 0.1790 | 0.3590 | 0.1743
SBERT-mor-triplet (fine-tuned BERT + mortgage triplets) | 0.5107 | 0.2250 | 0.1562 | 0.0572 | 0.0780 | 0.1861 | 0.0738

##  4\. System Architecture

Our system architecture (Figure 1) is inspired by (Esteva et al., 2020). The
collected FAQ-answer pairs are stored in a database table. Each document
contains an FAQ part (or title) and an answer part. We index them using SBERT,
TF-IDF, and BM25 and store the indexes in a database. The indexes are stored
as matrices containing document (title/answer) embeddings. When a new query is
inserted in the forum, it is also indexed by SBERT, TF-IDF, and BM25. All the
document indexes are loaded and the query index is compared against them.
Using cosine similarity based KNN, we find the top $N$ from each of the
indexes. We then linearly combine the SBERT and TF-IDF results according to
the equations described above and combine the resultant ranking with BM25
ranking using reciprocal rank fusion (RRF). The top $M$ results based on the
RRF are surfaced to the user. In our system, we set $N=200$ and $M=50$.

##  5\. Results

###  5.1. Search Engine Evaluation Results

We perform grid search on the hyperparameters $\alpha$ and $w$ and also on the
retrieval and ranking approaches mentioned in 3.2.

Table 1 displays our metrics for different retrieval and ranking approaches.
For each approach, the reported metric corresponds to the model with tuned
hyperparameters. The SBERT model reported in the table is trained using the
classification loss based on the finetuned BERT (SBERT-mor-nli in Table 3).
The grid search results indicate that inclusion of SBERT either in TF-
IDF_SBERT or RRF creates a more efficient ranking for the top five retrieved
results, and therefore, satisfies our requirement. This is specifically very
visible in mean reciprocal rank, nDCG@5 and MAP@5. BM25 or a combination of
BM25 and TF-IDF outperform the other models when ranking is extended beyond
the top five retrieved results. For example, BM25 outperforms when comparing
Recall@10, nDCG@10, MAP, and MAP@10.

There are some queries in our evaluation set that show clearly that inclusion
of SBERT helps in capturing the relevant documents when there is minimal
lexical overlap where the keyword based methods (BM25 and TF-IDF) fail to
capture (see Table 2).

###  5.2. Sentence-BERT Models Comparisons

We compare three SBERT models. The first one is the base model, which is used
as benchmark. This model uses regular BERT model and is trained on SNLI and
multi-genre NLI data ((Bowman et al., 2015; Williams et al., 2017)) and is
made available by the SBERT developers. The second model is built based on the
mortgage fine-tuned BERT and is trained on SNLI and NLI data. The third model
is based on the mortgage fine-tuned BERT and is trained on triplet sentences
generated based on the approach explained in Section 2.1. We compare the
performance of the three models (Table 3) on our evaluation data using the
metrics discussed in Section 3.2.

Table 3 suggests that the SBERT model based on mortgage fine-tuned BERT and
trained on NLI data outperforms both the benchmark model and the SBERT model
trained on triplets. Therefore, we use this model in the search engine. This
shows that fine-tuning BERT on the mortgage domain is effective for the
downstream task. The reason that the third model did not yield better results
is that finding difficult triplets to challenge the model into learning small
nuances between close answers is itself a challenging task and requires
further effort and a more effective strategy. Furthermore, the training
dataset is small and more training data is required for the third model to
outperform the second one.

##  6\. Discussion

We have implemented a question answering system with a state of the art hybrid
search engine for the mortgage domain. The system is customized to support and
assist the mortgage staff at Zillow Group. We are currently measuring the
impact on the business operations. Future directions mostly include but are
not limited to:

  * •

Collect more query-candidate pairs annotations.

  * •

Use larger mortgage corpora for fine-tuning the models.

  * •

Our CQA system allows for collection of other kinds of data such as votes on
questions and answers, whether an answer is approved by the original poster,
the association of user-answer and user-question, or tags assigned to each
question. The integration of such signals remains an open research question
(Radlinski et al., 2008; Xue et al., 2004).

  * •

Adopt supervised training approaches such as Learning to Rank (Joachims, 2006)
based on the collected user data.

## References

  * (1)
  * Bowman et al. (2015) Samuel R. Bowman, Gabor Angeli, Christopher Potts, and Christopher D. Manning. 2015.  A large annotated corpus for learning natural language inference.  _CoRR_ abs/1508.05326 (2015).  arXiv:1508.05326 http://arxiv.org/abs/1508.05326
  * Esteva et al. (2020) Andre Esteva, Anuprit Kale, Romain Paulus, Kazuma Hashimoto, Wenpeng Yin, Dragomir R. Radev, and Richard Socher. 2020.  CO-Search: COVID-19 Information Retrieval with Semantic Search, Question Answering, and Abstractive Summarization.  _CoRR_ abs/2006.09595 (2020).  arXiv:2006.09595 https://arxiv.org/abs/2006.09595
  * Joachims (2006) Thorsten Joachims. 2006\.  Training Linear SVMs in Linear Time.  _Proceedings of the ACM Conference on Knowledge Discovery and Data Mining(KDD)_ (2006).  https://www.cs.cornell.edu/people/tj/svm_light/svm_rank.html
  * Mass et al. (2020) Yosi Mass, Boaz Carmeli, Haggai Roitman, and David Konopnicki. 2020\.  Unsupervised FAQ Retrieval with Question Generation and BERT. In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_. Association for Computational Linguistics, Online, 807–812.  https://doi.org/10.18653/v1/2020.acl-main.74
  * Pennington et al. (2014) Jeffrey Pennington, Richard Socher, and Christopher Manning. 2014.  GloVe: Global Vectors for Word Representation. In _Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP)_. Association for Computational Linguistics, Doha, Qatar, 1532–1543.  https://doi.org/10.3115/v1/D14-1162
  * Radlinski et al. (2008) Filip Radlinski, Madhu Kurup, and Thorsten Joachims. 2008\.  How does clickthrough data reflect retrieval quality?. In _Proceedings of the 17th ACM conference on Information and knowledge management_. 43–52.
  * Reimers and Gurevych (2019) Nils Reimers and Iryna Gurevych. 2019.  Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks.  _CoRR_ abs/1908.10084 (2019).  arXiv:1908.10084 http://arxiv.org/abs/1908.10084
  * Robertson and Zaragoza (2009) Stephen Robertson and Hugo Zaragoza. 2009.  The Probabilistic Relevance Framework: BM25 and Beyond.  _Foundations and Trends® in Information Retrieval_ 3, 4 (2009), 333–389.  https://doi.org/10.1561/1500000019
  * Salton and McGill (1986) Gerard Salton and Michael J McGill. 1986.  Introduction to modern information retrieval.  (1986).
  * Williams et al. (2017) Adina Williams, Nikita Nangia, and Samuel R. Bowman. 2017\.  A Broad-Coverage Challenge Corpus for Sentence Understanding through Inference.  _CoRR_ abs/1704.05426 (2017).  arXiv:1704.05426 http://arxiv.org/abs/1704.05426
  * Wolf et al. (2019) Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, and Jamie Brew. 2019.  HuggingFace’s Transformers: State-of-the-art Natural Language Processing.  _CoRR_ abs/1910.03771 (2019).  arXiv:1910.03771 http://arxiv.org/abs/1910.03771
  * Xue et al. (2004) Gui-Rong Xue, Hua-Jun Zeng, Zheng Chen, Yong Yu, Wei-Ying Ma, WenSi Xi, and WeiGuo Fan. 2004.  Optimizing web search using web click-through data. In _Proceedings of the thirteenth ACM international conference on Information and knowledge management_. 118–126.

##  Appendix A Evaluation Dataset Statistics

The evaluation dataset contains 16 queries and 1740 FAQ-answer pairs. Table 4
lists the summarized statistics for our evaluation dataset. Table 5 shows the
16 queries that are used for evaluation. We also present an example of an FAQ-
answer pair below:

FAQ: ”How can I determine if a manufactured home is eligible for FHA
financing?”

Answer: ”Manufactured homes may be legally classified as real property or
personal property based on how they are titled in accordance with state law.
In general, manufactured homes can only be classified as real property if they
are permanently affixed to the land…” (truncated)

Table 4. Statistics of the evaluation dataset Avg. (word) length of questions | 9.38
---|---
Avg. (word) length of answers | 174.69
Avg. number of FAQ-answer pairs per query | 108.75
Avg. number of relevant FAQ-answer pairs per query | 7.00
Avg. number of partially relevant FAQ-answer pairs per query | 8.44
Avg. number of non-relevant FAQ-answer pairs per query | 93.31
Table 5. Most commonly searched mortgage queries Is there a minimum square
footage requirement for a home to be eligible for FHA financing?
---
What’s the maximum DTI ratio for a conventional loan?
Is manual underwriting allowed?
Can we originate a loan for a home on the market?
Do I need to collect reserves for a second home?
Can I give loan to a non US Citizen?
Can I give loan to a customer who has late payments?
Customer has a judgement, can I do the loan?
Can I do past-due, collection, and charge-off of non-mortgage accounts?
Customer pays alimony/child support, does that count against the DTI?
Customer only has 9 payments left on their car. Can I exclude it?
Customer has student loan, but the credit report says zero for payment?
What do I do with open accounts? Amex
He’s seasonally employed, can I use that income?
She is starting a new job, what documents do I need?
Customer has foreign income. Can we use that?


# Constructing Inverse Scattering Potentials
for $\alpha-\alpha$ System using Reference Potential Approach

O. S. K. S. Sastri, Arushi Sharma and Ayushi Awasthi

###### Abstract

Background

An accurate way to incorporate long range Coulomb interaction alongside short-
range nuclear interaction has been a challenge for theoretical physicists.

Purpose

In this paper, we propose a methodology based on the reference potential
approach for constructing inverse potentials of alpha-alpha scattering.

Methods

Two smoothly joined Morse potentials, regular for short-range nuclear
interaction and inverted for long range Coulomb, are used in tandem as a
reference potential in the phase function method to obtain the scattering
phase shifts for the S, D and G states of alpha-alpha scattering. The model
parameters are optimized by choosing to minimize the mean absolute percentage
error between the obtained and experimental scattering phase shift values.

Results

The constructed inverse potentials for S, D and G states have resulted in mean
absolute percentage errors of 0.8, 0.5, and 0.4 respectively. The obtained
resonances for D and G states closely match the experimental ones.

Conclusion

The reference potential approach using a combination of smoothly joined Morse
functions is successful in accurately accounting for the Coulomb interaction
between charged particles in nuclear scattering studies.

Introduction: One of the most difficult problems for theoretical physicists is
acquiring an accurate understanding of scattering involving a charged
projectile and a charged target due to the indefinitely long range of Coulomb
interaction[1]. Inadvertently, microscopic theories such as phenomenological
potential models used in R-matrix[2], complex scaling method[3], phase
function method[4, 5], and others face mathematical difficulties in including
Coulomb effect in a rigorous manner. As a result, they use screened Coulomb
potentials to simulate the often observed condition, in which an isolated
charge is generally surrounded by residual particles due to polarization.
Taylor[1] presented a comprehensive method of incorporating Coulomb scattering
by considering

$V_{c}^{\rho}(r)=\frac{\gamma}{r}\alpha^{\rho}(r)$ (1)

where the screening function $\alpha^{\rho}$ for a given $\rho$ must go to
zero as r tends to $\infty$ and must approach 1 as $\rho$ tends to $\infty$
with r fixed. As long as $\rho$ is very large, such a potential meets the
conditions of scattering theory and produces findings that are independent of
both properties of screened potential, their nature/shape and screening
radius.
To solve this crucial issue, we focused on $\alpha-\alpha$ scattering in our
article. Ali-Bodmer[6], Buck[7] and Odsuren[3] utilise an erf() function to
model the Coulomb interaction,

$V_{c}=\frac{4e^{2}}{r}erf(\beta
r),\text{and}~{}~{}\beta=\frac{\sqrt{3}}{2R_{\alpha}}$ (2)

The interaction resulting from this statement is referred to as an enhanced
Coulomb interaction.This results from the fact that $\alpha$-particles have a
finite size, which is determined by the radius’s rms value, $R_{\alpha}$ =
1.6782 fm.
Laha et.al. utilise screened Coulomb interaction based on Atomic Hulthen[8]
potential, given by

$V_{H}(r)=V_{o}\frac{e^{-r/a}}{(1-e^{-r/a})}$ (3)

where $V_{o}$ is strength of the potential and $a$ is the screening radius.
The two parameters $V_{o}$ and $a$ are related by sommerfeld parameter [9]

For $\alpha-\alpha$, $Z_{1}=Z_{2}=2$ , $\mu=\frac{m_{\alpha}}{2}=1864.38525$
$\frac{MeV}{c^{2}}$ , $e^{2}=1.44MeVfm$ and therefore $V_{o}a=0.2758fm^{-1}$.
This ansatz is also plotted in Fig.1. While the erf-based screening potential
has a finite value as r approaches 0 and needs to be abruptly cutoff for
bigger values of r due to its long range behaviour, the Atomic Hulthen
potential has an infinite value as r approaches 0.

Figure 1: Coulomb interaction modeled using Atomic Hulthen and erf() ansatzes

Recently, Laha et.al.[10], have incorporated a separable non-local potential
in place of the short range nuclear potential and obtained analytical solution
of Schrodinger equation. By combining Manning-Rosen and Yamaguchi potentials,
they were able to describe the scattering of two charged particles and, using
the Green’s function approach, find regular and irregular solutions. In
essence, they have replaced Hulthen function with a more generic version by
using Manning-Rosen potential as a model of interaction for screened and cut-
off Coulomb interaction.
On the other hand, we have proposed a computational approach to construct
inverse potentials[11, 12] based on phase function method. The methodology
involves utilizing Morse potential as zeroth reference [11] and optimizing the
model parameters by considering all available experimental scattering phase
shifts as in inverse scattering theory. This is called as reference potential
approach[13, 14]. This is akin to machine learning paradigm wherein one
obtains the model from available data either using a global optimisation
algorithm[15] or neural networks[16].
Even while this works quite well for determining the inverse potentials for
scattering when either of the two particles, projectile or target, is neutral,
an additional Coulomb potential is required for charged systems. Previously,
while studying $pp$ [17], $pd$ [18], $p-\alpha$ [19] and $\alpha-\alpha$ [20]
systems, we have chosen to incorporate erf() based Coulomb potential. In order
to get accurate scattering phase shifts, we cut off the erf() at certain
distance $r_{f}$ abruptly, which is a major limitation of this approach.
Recently, we have undertaken a study of $\alpha$\- $\alpha$ scattering using
various phenomenological potentials by utilising Atomic Hulthen as screened
coulomb potential [21]. It was observed that screening radius had to be varied
for different $\ell$-channels so as to obtain good match with experimental
data.
A further review of literature revealed an interesting approach suggested by
Selg[13, 14], where the molecular interaction potentials were obtained by
considering a combination of two or three Morse functions. Here’s a method for
using the experimental data at hand to construct a reference potential for a
one-dimensional quantum mechanical system. The key advantage of this
approximation over a linear fit is the low number of analytically distinct
components required to provide a decent match with the original potential
across a broad distance range.
Because a two-particle problem can always be reduced to a one-particle problem
in a spherically symmetric field, the one-dimensional inverse theory may be
applied to two body systems. As a result, the primary goal of this study is to
develop inverse potentials by examining elastic scattering of charged $\alpha$
particles with ${}^{4}He$ having orbital angular momentum $l=0,2,4$ at
energies up to $28MeV$, utilizing a combination of two Morse potentials across
two distinct regions. The conventional Morse function is the initial potential
at short distances, while the second portion is an inverted Morse function to
capture the barrier height owing to Coulomb interaction.
Methodology: Theoretically, constructing inverse potentials [22] requires not
only information regarding all the bound state energies
$E_{n}(n=0,1,\ldots,N)$ along with their related normalization constants
$C_{n}$ but also the scattering phase shifts for all scattering energies $E>0$
ranging to infinity. Most often, phase shift data is available for only
certain energies to within a limited range and hence, a rigorous solution of
quantum mechanical inverse problem is extremely difficult to compute.
Reference Potential Approach(RPA): Selg[23] suggests reference potential
approach for solving 1D quantum systems wherein a single Morse function[24] or
a combination of smoothly joined Morse potentials of the form

$V_{k}^{RPA}(r)=V_{k}+D_{k}[e^{-2\alpha_{k}(r-r_{k})}-2e^{-\alpha_{k}(r-r_{k})}]~{}~{}~{}~{}~{}k=1,2,\ldots$
(4)

can be chosen as starting point to solve time-independent Schrodinger equation
for its energy eigenvalues, scattering phase shifts, and also Jost function,
from which one can obtain the inverse potential[14].
Here $D_{k}$’s are potential depths at equilibrium distances $r_{k}$’s, and
$\alpha_{k}$’s reflect shape parameter of Morse functions. $V_{k}$’s are
constants added to total potential, whose importance shall be made clear
later. These functions are smoothly joined at various boundary points
$X_{k+1}$.
The number of distinct Morse-type components that may be added is almost
unlimited. Naturally, the more components one includes, the better the match
with the experimental data, but also the more challenging the analytical
solution to the problem gets.
Only two Morse components are utilized in this entire letter to prepare the
reference potential. The first of these is a regular Morse component
$(k=1,D_{1}>0)$, while the second is a reversed Morse component
$(k=2,D_{2}<0)$. The potential has a minimum $R_{1}$ in the range of
$0<R_{1}<X$, which uses regular Morse component and a maximum $R_{2}>X$ due to
reversed Morse potential with a negative $D_{2}$ value. Despite the fact that
this may seem unreasonable and irrational, the objective is that the height of
this artificial hump approaches zero as the parameter $r_{2}$ goes to
infinity. As a result, with a large enough $r_{2}$, the hump almost completely
vanishes, yet the analytical strategy remains simple and flexible. For
ensuring smoothness of potential at the boundary point $X$, in-between the
two, the functions and their derivatives must be continuous at $X$. That is,

$\displaystyle U_{1}(r)|_{X}=U_{2}(r)|_{X}$
$\displaystyle\frac{dU_{1}(r)}{dr}\Big{|}_{X}=\frac{dU_{2}(r)}{dr}\Big{|}_{X}$
(5)

Using these equations, two of the six parameters were determined as

$D_{1}=\frac{(V_{2}-V_{1})\alpha_{2}f_{2}}{\alpha_{2}f_{3}f_{2}-\alpha_{1}f_{4}f_{1}}\\\
$ (6)
$D_{2}=\frac{(V_{1}-V_{2})\alpha_{1}f_{1}}{\alpha_{2}f_{3}f_{2}-\alpha_{1}f_{4}f_{1}}$
(7)

where

$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle
e^{-2\alpha_{1}(r-r_{1})}-e^{-\alpha_{1}(r-r_{1})}$ $\displaystyle f_{2}$
$\displaystyle=$ $\displaystyle
e^{-2\alpha_{2}(r-r_{2})}-e^{-\alpha_{2}(r-r_{2})}$ $\displaystyle f_{3}$
$\displaystyle=$ $\displaystyle
e^{-2\alpha_{1}(r-r_{1})}-2e^{-\alpha_{1}(r-r_{1})}$ $\displaystyle f_{4}$
$\displaystyle=$ $\displaystyle
e^{-2\alpha_{2}(r-r_{2})}-2e^{-\alpha_{2}(r-r_{2})}$

Now, observing Eqs. 6 and 7, one can see that not choosing $V_{1}$ and $V_{2}$
would have led to two homogeneous equations and determination of $D_{1}$ and
$D_{2}$ wouldn’t have been feasible. It is the difference $V_{1}-V_{2}$ that
is important, and in fact, our results show that $V_{2}$ is always zero.

While Selg[25] obtains analytical expressions for energies and phase shifts by
solving radial Schrodinger equation for molecular potentials, we take an
alternative computational approach by employing phase function method.
Phase Function Method: The second-order time-independent Schrodinger equation
for a potential $V(r)$,

$\frac{\partial^{2}\psi(r)}{\partial
r^{2}}+\frac{2m}{\hbar^{2}}[E-V(r)]\psi(r)=0$ (8)

is recast into a first order non-linear differential equation of Riccati type

$\frac{d\delta_{l}(k,r)}{dr}=-\frac{U(r)}{k}[\hat{j_{l}}(kr)cos(\delta_{l}(k,r))-\hat{\eta_{l}}(kr)sin(\delta_{l}(k,r))]^{2}$
(9)

where $k=\sqrt{2\mu E/{\hbar^{2}}}$, $U(r)=2\mu V(r)/\hbar^{2}$,
$\hat{j_{l}}(kr)$ and $\hat{\eta_{l}}(kr)$ are the Riccati–Bessel and Riccati-
Neumann functions, respectively. This gives phase shift information at various
energies by directly taking potential as input.
For S($\ell=0$), D($\ell=2$) and G($\ell=4$) states, the corresponding phase
equations [20] can be obtained by substituting the appropriate $\ell$th order
Riccati-Bessel and Riccati-Neumann functions as follows:

Case 1: $\ell=0$

$\delta_{0}^{\prime}(k,r)=-\frac{U(r)}{k}\sin^{2}(\delta_{0}+kr)$ (10)

Case 2: $\ell$ = 2

$\displaystyle\delta_{2}^{\prime}(k,r)$ $\displaystyle=$
$\displaystyle-\frac{U(r)}{k}\bigg{[}-\sin{\left(\delta_{2}+kr\right)}$ (11)
$\displaystyle-\frac{3\cos{\left(\delta_{2}+kr\right)}}{kr}+\frac{3\sin{\left(\delta_{2}+kr\right)}}{(kr)^{2}}\bigg{]}^{2}$

Case 3: $\ell$ = 4

$\displaystyle\delta_{4}^{\prime}(k,r)$ $\displaystyle=$
$\displaystyle-\frac{U(r)}{k}\bigg{[}\sin{\left(\delta_{4}+kr\right)}$ (12)
$\displaystyle+\frac{10\cos{\left(\delta_{4}+kr\right)}}{kr}-\frac{45\sin{\left(\delta_{4}+kr\right)}}{(kr)^{2}}$
$\displaystyle+\frac{105\cos{\left(\delta_{4}+kr\right)}}{(kr)^{3}}+\frac{105\sin{\left(\delta_{4}+kr\right)}}{(kr)^{4}}\bigg{]}^{2}$

These equations are solved using 5th order Runge-Kutta method by choosing the
initial condition as $\delta_{\ell}(0,k)=0$ and integrating to a large
distance. So, one can optimize the model parameters of chosen potential by
iteratively solving the phase equations for various energies and minimizing a
cost function. Typically, one can employ mean squared error, mean absolute
percentage error(MAPE), $\chi^{2}$-error, etc., between computed and
experimental scattering phase shifts, as cost function.

Results and Discussion:The updated experimental data for $\alpha-\alpha$
scattering by Anil et. al.,[20] had included the Chein and Brown data [26] for
energies up to 25.5 MeV only. This has limited the available data points for
$\ell=4$ channel to only 4 energy values out of the 6 energies at which
scattering phase shifts were given by [26]. In our current analysis, we have
opted to use a combination of two Morse potentials. The first part for a
region, 0 $\leq$ r $\leq$ X, is a regular Morse with 4 parameters and the
second part for larger distances, X $\leq$ r $\leq$ rf, has an inverse Morse
function with another 4 parameters. These two have to produce a smooth shaped
potential that accounts for the observed scattering phase shifts. So,
employing continuity conditions as given in Eqs. 5, one obtains the two
parameters $D_{1}$ and $D_{2}$ in terms of the rest of the parameters as given
by Eqs.6 and 7. This reduces the number of parameters required to 6 but one
needs to fix point $X$ as well. So, observing that the relative difference
$V_{1}-V_{2}$ appears in both Eqs.6 and 7, we have chosen to set $V_{2}=0$. In
fact, we have initially run the optimization routine by fixing different
values of $X$ and observed that $V_{2}$ always comes out to be of the order of
$10^{-8}$. So, in this way, we have ensured that only 6 parameters need to be
determined by utilizing the 6 experimental data points that are available for
$\ell=4$. Hence, we have chosen experimental data all the way up to 28 MeV,
just below the threshold binding energy of alpha particles, for all the $\ell$
channels.
Other important physical considerations while optimizing the model parameters
to obtain inverse potentials are

  1. 1.

The Coulomb barrier for the $\ell=0$ potential needs to be high enough to
ensure it has a pseudo-bound state with energy 0.0918 MeV.

  2. 2.

After accounting for centrifugal potential, the depth of the $\ell=2$
potential must be lower than the $\ell=0$ potential, and the Coulomb barrier
height must be on the order of resonance energy, which has been measured to be
approximately 3 MeV.

  3. 3.

Similarly, depth of $\ell=4$ potential must be less than that of $\ell=2$ and
Coulomb barrier height must be of the order of 12 MeV, which is close to the
observed resonance energy for this channel.

As a first step, we have reconstructed the inverse potentials for
$\ell=0,~{}2,~{}4$ channels of $\alpha-\alpha$ scattering by choosing single
Morse potential with $V_{1}=0$ in Eq. 4 as a model for nuclear interaction and
considering Coulomb interaction as erf() ansatz in Eqs.2 and Atomic Hulthen
form in Eqs.3, one at a time. The Mean Absolute Percentage Error(MAPE),
between obtained and experimental values, has been minimized for optimizing
the model parameters. The best model parameters obtained, along with MAPE, are
given in Table 1.

Table 1: Parameters for nuclear interaction modeled using Morse function for various $\ell$-channels of $\alpha-\alpha$ scattering using two different ansatzes for Coulomb interaction Model | Morse + erf() | Morse + Atomic Hulthen
---|---|---
Parameters | $\ell=0$ | $\ell=2$ | $\ell=4$ | $\ell=0$ | $\ell=2$ | $\ell=4$
$D_{1}$ | 11.1792 | 27.6925 | 182.7336 | 9.8096 | 26.4894 | 240.2966
$\alpha_{1}$ | 0.5661 | 1.4539 | 1.2393 | 0.5318 | 1.5241 | 1.3572
$r_{1}$ | 3.4684 | 1.8507 | 0.4768 | 3.6399 | 1.8745 | 0.3695
$V_{d}$ | -9.5506 | -8.6002 | 3.0134 | -9.0924 | -8.8917 | 0.9819
$V_{CB}$ | 0.3543 | 3.2087 | 10.1654 | 0.0915 | 2.6126 | 10.7846
MAPE | 1.6324 | 1.6332 | 1.1988 | 1.9601 | 4.0609 | 0.5675

It should be mentioned that $r_{f}$ for erf() function has been taken as 10
fm, 6.2 fm and 5 fm and screening radius $a$ for Atomic Hulthen has been
chosen as 20 fm, 8 fm and 3.5 fm for $\ell$ = 0, 2 and 4 channels
respectively. The scattering phase shifts obtained using these two ansatzes
are shown in Fig 2. One can observe that both these ansatzes are unable to
correctly explain experimental data for $\ell$ = 2.

Figure 2: $\alpha-\alpha$ scattering phase shifts for $\ell$ = 0. 2 and 4
channels obtained using various ansatze in comparison with experimental data

Then, we have optimized the model parameters for combination of two Morse
functions as suggested by reference potential approach, by choosing to vary
all 7 of them including $X$ value simultaneously, for $\ell=0$ and $\ell=2$.
It was observed that value of $V_{2}$ is 0 in both the cases. So, our choice
of $V_{2}=0$ for $\ell=4$ is well justified. These parameters were obtained by
choosing final integration distance $r_{f}$ to be 40 fm. One can, in
principle, take $r_{f}$ to be as large as required to ensure that potential
goes to zero with distance. The final obtained parameters for inverse
potentials are given in Table 2.

Table 2: Parameters for $\alpha-\alpha$ scattering using combination of two Morse functions as reference potential Model Parameters | $\ell=0$ | $\ell=2$ | $\ell=4$
---|---|---|---
$V_{1}$ | 0.7915 | 7.0575 | 39.1351
$\alpha_{1}$ | 0.5675 | 0.8019 | 0.6877
$r_{1}$ | 3.5362 | 1.5679 | 0.8934
$\alpha_{2}$ | 0.3629 | 2.3229 | 0.8301
$r_{2}$ | 12.0073 | 4.6009 | 7.8639
X | 8.0000 | 4.4723 | 2.7694
$D_{1}$ | 10.9153 | 29.1528 | 125.4702
$D_{2}$ | 0.0932 | 1.8835 | 0.0045
$V_{d}$ | -10.3173 | -4.5497 | 0.8501
$V_{CB}$ | 0.0969 | 4.8802 | 11.1349
MAPE | 0.8 | 0.5 | 0.4
Figure 3: Inverse potentials for $\ell=0,~{}2~{}\&~{}4$ using various ansatzes
(a) without and (b) with centrifugal potential term added.

The obtained scattering phase shifts, plotted in Fig. 2, are observed to be
matching very well with the experimental data. One can also compare the MAPE
obtained using each of the ansatzes to appreciate the excellent performance of
RPA as compared to screened Coulomb ansatzes. Especially, in case of $\ell$ =
2 channel, depth of the potential $V_{d}$ after adding the centrifugal
potential seems to reasonable lesser than the depth for $\ell=0$ using RPA
which is not the case for Morse + erf() or Morse + Atomic Hulthen ansatzes.
The constructed inverse potentials for all channels are plotted without
considering centrifugal term $V_{cf}=\ell(\ell+1)/r^{2}$ in Fig. 3(a) and by
including it in 3(b). The Coulomb barrier $V_{CB}$ heights using RPA can be
observed to be always a little larger than what are obtained using the other
two models.
The scattering phase shifts for $\ell=0$ become negative beyond 20 MeV
reflecting that a repulsive core is present. While Morse with erf and Atomic
Hulthen models demonstrate that repulsion has a very hard core, RPA with two
Morse potentials reveals that the height of the repulsive component is
limited. The inset of Fig. 3(a) depicts the centrifugal barrier height of
$\ell=0$ for RPA, and it is evident that the s-state is a bound state with
resonance energy of $E_{res}=0.09686$. make the S-state to be a pseudo-bound
state. In case of $\ell=2$, the scattering phase shifts initially increase
very quickly until about 10 MeV and then show a slow decreasing trend for
higher energies. Once again, a similar observation as for potentials of
$\ell=0$ can be seen, with RPA having finite height in its repulsive part.
Finally, in case of $\ell=4$, even though the scattering phase shifts seems to
be having an increasing trend, on close observation, one can see a small shift
at higher energies towards peaking. In fact, at even higher energies than
those considered in this work, one would find scattering phase shifts to peak
and have a decreasing trend, as observed in case of $\ell=2$ channel. So, one
would expect that there would be a relatively small repulsive behavior that
would set in even for $\ell=4$ and is observed in case of RPA.
It is interesting to note that after adding centrifugal term, all the inverse
potentials show a repulsion that is just beyond the radius of alpha particle.
Further, the potential depths $V_{d}$ for various $\ell$-channels are
distinctly decreasing in RPA as compared to the other two ansatzes for which
$\ell=0$ and $\ell=2$ channels have depths that are very close. Even though
all three models show Coulomb barrier heights $V_{CB}$ to be closer to
resonances observed experimentally, it is only in RPA that they appear
naturally without the imposition of any constraints. Remember that in the case
of erf() based Coulomb interaction, we had to limit the integration distance
$r_{f}$ for various channels. Since the Coulomb potential is a long-range
potential, the interaction potential obtained by using the error function is
highly dependent on final integration point. Similarly, in the case of the
Hulthen model, we had to adjust the screening radius for various channels to
obtain good results. Both of these limitations have been overcome in RPA, in
which the results obtained are independent of the integration distance.
Partial cross sections: The computed phase shifts are used to determine
partial cross sections using

$\sigma_{l}=\frac{4\pi}{k^{2}}(2l+1)sin^{2}\delta_{l}(E)$ (13)

In case of $\ell$ = 0, there is a sharp resonance behavior very close to 0 and
it appears almost like a delta function. As a result, we skipped plotting the
partial cross section for this case. In Figure 4, the partial cross sections
for $\ell=2$ and $4$ channels are displayed and are seen to be in Breit-Wigner
form. We obtain the resonance peaks(FWHM) for $\ell=2$ and $4$ at 3.25(2.46)
MeV and 12.52(5.91) MeV respectively, as compared to experimental values of
3.03(1.51) and 11.35(3.50) [27].

Conclusion: The reference potential approach is ideally suited for accurately
describing the long range Coulomb interaction alongside the short range
nuclear interaction for scattering involving charged particles. This procedure
could overcome the limitations of previously employed ansatzes for Coulomb
interaction, such as erf() and Atomic Hulthen. It will find applications in
scattering studies of atomic, molecular, and nuclear domains. It would also be
effective in studying nuclear and astrophysical reactions.

Figure 4: Partial cross sections for $\ell=2$ and $4$ for different center of
mass energies

Acknowledgement:: A. Awasthi acknowledges financial support provided by
Department of Science and Technology (DST), Government of India vide Grant No.
DST/INSPIRE Fellowship/2020/IF200538.

Author Declaration The authors declare that they have no conflict of interest.

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