# Anisotropy of the anomalous Hall effect in the altermagnet candidate Mn5Si3
films

Miina Leiviskä  Univ. Grenoble Alpes, CNRS, CEA, Grenoble INP, IRIG-SPINTEC,
F-38000 Grenoble  Javier Rial  Univ. Grenoble Alpes, CNRS, CEA, Grenoble INP,
IRIG-SPINTEC, F-38000 Grenoble  Antonín Badura  Department of Chemical Physics
and Optics, Faculty of Mathematics and Physics, Charles University, Prague,
Czechia  Institute of Physics, Czech Academy of Sciences, Prague, Czechia
Rafael Lopes Seeger  Univ. Grenoble Alpes, CNRS, CEA, Grenoble INP, IRIG-
SPINTEC, F-38000 Grenoble  Ismaïla Kounta  Aix-Marseille University, CNRS,
CINaM, Marseille, France  Sebastian Beckert  Institute of Solid State and
Materials Physics, TU Dresden, Dresden, Germany  Dominik Kriegner  Institute
of Solid State and Materials Physics, TU Dresden, Dresden, Germany  Institute
of Physics, Czech Academy of Sciences, Prague, Czechia  Isabelle Joumard
Univ. Grenoble Alpes, CNRS, CEA, Grenoble INP, IRIG-SPINTEC, F-38000 Grenoble
Eva Schmoranzerová  Department of Chemical Physics and Optics, Faculty of
Mathematics and Physics, Charles University, Prague, Czechia  Jairo Sinova
Institute for Physics, Johannes Gutenberg University Mainz, Mainz, Germany
Olena Gomonay  Institute for Physics, Johannes Gutenberg University Mainz,
Mainz, Germany  Andy Thomas  Institute of Solid State and Materials Physics,
TU Dresden, Dresden, Germany  Leibniz Institute of Solid State and Materials
Science (IFW Dresden), Helmholtzstr. 20, 01069 Dresden, Germany  Sebastian T.
B. Goennenwein  Department of Physics, University of Konstanz, Konstanz,
Germany  Helena Reichlová  Institute of Solid State and Materials Physics, TU
Dresden, Dresden, Germany  Libor Šmejkal  Institute for Physics, Johannes
Gutenberg University Mainz, Mainz, Germany  Institute of Physics, Czech
Academy of Sciences, Prague, Czechia  Lisa Michez  Aix-Marseille University,
CNRS, CINaM, Marseille, France  Tomáš Jungwirth  Institute of Physics, Czech
Academy of Sciences, Prague, Czechia  Vincent Baltz  Univ. Grenoble Alpes,
CNRS, CEA, Grenoble INP, IRIG-SPINTEC, F-38000 Grenoble

###### Abstract

Altermagnets are compensated magnets belonging to spin symmetry groups that
allow alternating spin polarizations both in the coordinate space of the
crystal and in the momentum space of the electronic structure. In these
materials the anisotropic local crystal environment of the different
sublattices lowers the symmetry of the system so that the opposite-spin
sublattices are connected only by rotations, which results in an
unconventional spin-polarized band structure in the momentum space. This low
symmetry of the crystal structure is expected to be reflected in the
anisotropy of the anomalous Hall effect. In this work, we study the anisotropy
of the anomalous Hall effect in epitaxial thin films of Mn5Si3, an
altermagnetic candidate material. We first demonstrate a change in the
relative Néel vector orientation when rotating the external field orientation
through systematic changes in both the anomalous Hall effect and the
anisotropic longitudinal magnetoresistance. We then show that the anomalous
Hall effect in this material is anisotropic with the Néel vector orientation
relative to the crystal structure and that this anisotropy requires high
crystal quality and unlikely correlates with the magnetocrystalline
anisotropy. Our results provide further systematic support to the case for
considering epitaxial thin films of Mn5Si3 as an altermagnetic candidate
material.

##  I Introduction

The physical properties of magnetic materials are governed by the intrinsic
symmetries of their crystal and spin structures, which are represented by
symmetry groups. A good example in the realm of magnetotransport properties is
the anomalous Hall effect (AHE) [1], where a longitudinal electric field
$\mathbf{E}$ generates a transversal Hall current $\mathbf{j}_{H}$ that is
perpendicular to the Hall vector
($\mathbf{h}=(\sigma_{zy},\sigma_{xz},\sigma_{yx})^{T}$, where $\sigma_{ij}$
are the off-diagonal terms of the conductivity matrix) i.e.
$\mathbf{j}_{H}=\mathbf{h}\times\mathbf{E}$. As $\mathbf{h}$ is an axial
vector and odd upon time-reversal ($\mathcal{T}$), non-vanishing AHE is only
observed in systems where the symmetries allow for such a vector. In
ferromagnetic materials the net magnetization is also a $\mathcal{T}$-odd
axial vector and thus $\mathbf{h}$ is allowed [1] while in magnetic materials
with compensated order, $\mathbf{h}$ is allowed upon sufficient lowering of
the crystal symmetries [2, 3, 4, 5, 6, 7]. In antiferromagnetic materials [8,
9] where the opposite-spin sublattices are connected by translation or
inversion, $\mathbf{h}$ is forbidden by symmetry, while in altermagnetic
materials [10] this connecting symmetry is rotation, which allows for
$\mathbf{h}$ [11, 12, 13, 14, 15, 16, 17]. This lowered symmetry in
altermagnets stems from the two sublattices being inequivalent in terms of the
local crystal environment. Altermagnets are characterised by alternating spin-
splitting of the energy bands [18, 19, 20, 21, 22] and they show various core
spintronic properties such as the AHE presented above, the Nernst effect
(thermal counterpart of the AHE) [23], spin-polarized current generation [24,
25, 26], as well as giant and tunnel magnetoresistances [27, 28, 29] in
heterostructures.

In polycrystalline soft ferromagnets, $\mathbf{h}$ is allowed by symmetry for
any magnetization orientation. Therefore, the AHE is proportional to the
component of the magnetization that is parallel to $\mathbf{h}$, i.e. to
$M_{s}$cos$\theta$, where $M_{S}$ is the saturation magnetization and $\theta$
is the angle between the external field and the film normal, assuming that the
field is strong enough to saturate the magnetization. Exceptions to this
behavior can arise in crystalline ferromagnets [30, 31] or ferromagnets with
strong magnetocrystalline anisotropy, which causes a lag between the external
field and the magnetization. In altermagnets, specific Néel vector
($\mathbf{L}$) orientations can either allow or exclude the Hall vector
$\mathbf{h(\mathbf{L})}$ by symmetry [7, 11, 13] so that anisotropic AHE
beyond $M_{s}$cos$\theta$ can arise.

Anisotropic AHE beyond $M_{s}$cos$\theta$ in compensated magnets constitutes a
hallmark of altermagnetism, as has been recently demonstrated in RuO2 [11]
(intrinsic d-wave altermagnet, i.e. the electronic structure displays four
lobes with alternating spin-polarization) and MnTe showing spontaneous AHE
[13] (intrinsic g-wave, which acquires a d-wave character in the presence of
spin-orbit coupling). Both RuO2 and MnTe owe their altermagnetism to the
presence of non-magnetic atoms that lower the symmetry of the system such that
the opposite-spin sublattices are not connected by either inversion or
translation [10]. The saturation of $\mathbf{L}$ in RuO2 and MnTe requires
high fields of 68 [11, 12] and 4 T [13], respectively, so for further studies
on the anisotropy of AHE in altermagnets a new material candidate where
$\mathbf{L}$-saturation is realized at smaller fields would be ideal.

Figure 1: (Color Online) (a) The hexagonal unit cell of epitaxial thin films
of Mn5Si3. The two magnetically ordered opposite-spin sublattices of Mn-atoms
are highlighted in red and blue. Note, that there are two other options
(’quadrupole domains’) for the checkerboard distribution of the magnetic Mn
atoms among the Mn2 sites. (b) Optical image of a typical Hall bar device. (c)
Illustration of a Hall bar with the angle $\theta$ defining the external field
$H$ orientation relative to the film normal z, which coincides with the [0001]
crystal axis of Mn5Si3. (d) Top view of a Hall bar showing the angle $\alpha$,
which refers to the orientation of the current channel $I_{x}$ relative to the
crystal axis $[0\bar{1}10]$. Varying $\alpha$ is achieved through the
fabrication of a set of Hall bars oriented along several crystal directions.

Recently, epitaxial thin films of Mn5Si3 have emerged as such candidate [15]
as the collinear compensated phase of Mn5Si3 with a hexagonal unit cell and
four magnetic Mn atoms on the Mn2 sites in a checkerboard arrangement of the
sublattices shown in Figure 1(a) fulfills the symmetry criteria for
altermagnetism [15]. Note, that in the bulk phase of Mn5Si3 the hexagonal unit
cell is not preserved during the magnetic ordering and the opposite-spin
sublattices become connected by translation making bulk Mn5Si3
antiferromagnetic [32, 33, 34, 35]. However, when grown as epitaxial thin
films [16] the substrate stabilises the hexagonal unit cell of Mn5Si3 over the
entire temperature range [15], fulfilling the symmetry requirements for making
epitaxial Mn5Si3 a potential candidate for the altermagnetic phase with a
d-wave symmetry. In good accordance with the theoretical predictions and the
structural properties epitaxial Mn5Si3 exhibits a sizable spontaneous AHE
despite a vanishingly small net magnetization [15].

In this work we investigate the anisotropy of AHE in epitaxial Mn5Si3 in terms
of the external field orientation relative to the crystal structure. We show
that a systematic rotation of the field in different crystal planes inflicts
systematic changes on the magnetotransport properties, indicating that we are
able to control the relative $\mathbf{L}$-orientation with experimentally
reasonable external field strengths of the order of 2 T. Moreover, we show
that changing the relative $\mathbf{L}$-orientation yields anisotropic
behavior of the AHE, beyond cos$\theta$, that requires a high crystal quality
of the epitaxial films and is unlikely related to the magnetocrystalline
anisotropy, which further demonstrates that the properties of epitaxial Mn5Si3
closely match those of altermagnetic materials.

Figure 2: (Color Online) (a) Disentangling the different contributions -
symmetric-in-field $\rho_{xy,sym}$, antisymmetric-in-field $\rho_{xy,asym}$,
and ordinary Hall effect $\rho_{OHE}$ \- to the field-dependence of transverse
resistivity $\rho_{xy}$. The signal of interest here is the antisymmetric
contribution, relating to the anomalous Hall effect (AHE) resistivity
$\rho_{AHE}$. (b) The hysteresis loops of the corresponding AHE conductivity
$\sigma_{AHE}={\rho_{AHE}}/{\rho_{xx}^{2}}$ at selected external field
orientations $\theta$ at 110 K. The sweep direction is indicated by the small
arrows. The $\sigma_{AHE}$ is at maximum at $\theta=$ 0 deg (field out-of-
plane) and decreases when the field rotates away from it, almost vanishing at
$\theta=$ 0 deg (field in-plane). This is in contrast with the weak
magnetization $M$, which is isotropic for $\theta=0$ and 90 deg. Note that we
have removed the linear diamagnetic background in the hysteresis loops of $M$.

##  II Methods

The epitaxial thin films of Mn5Si3 are grown on Si(111) substrates by
molecular beam epitaxy using optimized growth parameters for realizing films
with high crystalline quality and maximum Mn5Si3 content [16]. A few nm-thick
MnSi phase acts as a thin seed layer aiding the nucleation of Mn5Si3 on
Si(111) [16]. Unless otherwise specified, the films used in this work contain
96 % Mn5Si3 and 4 % MnSi and they have a thickness of $\sim$17 nm. The high
crystallinity and the crystal orientation of the films have been verified
using transmission electron microscopy (TEM) in our previous works [15, 16].
The epitaxial relationship between the Si substrate, MnSi seed layer and
Mn5Si3 film is
Si(111)$[1\bar{1}0]$//MnSi(111)$[\bar{2}11]$//Mn5Si3(0001)$[01\bar{1}0]$. The
films were patterned into microscopic Hall bars shown in Figure 1(b) using
optical UV lithography and subsequent ion beam etching. The orientations of
the Hall bar current channels relative to the crystal axis
$[2\bar{1}\bar{1}0]$ are defined by the angle $\alpha$ as illustrated in
Figure 1(d). We measure simultaneously the longitudinal ($V_{xx}$) and
transversal ($V_{xy}$) voltages (denoted in Figure 1(b,c)) as a function of
the external field strength and orientation to obtain the corresponding field-
and angular-dependences of the longitudinal ($\rho_{xx}$) and transversal
($\rho_{xy}$) resistivities. We apply the external field always in the (yz)
plane, i.e. the plane perpendicular to the current channel. Its exact
orientation relative to the [0001] crystal axis (film normal) is given by the
angle $\theta$, as illustrated in Figure 1(c). Note that we will demonstrate
later on in the text that the important parameter causing the anisotropic AHE
is the crystal plane in which the external field rotates rather than the exact
crystal orientation of the current channel - we alter the Hall bar orientation
simply to ensure that the external field and the current channel are always
perpendicular to each other unless otherwise stated. The current density is
$3\times 10^{5}$ A cm-2 throughout this work. For all the measurements
presented in this work, the temperature was kept at 110 K, which is between
the two critical temperatures of our films [15, 16]. At this temperature the
films exhibit characteristic features of an altermagnetic phase, i.e. a
sizable spontaneous AHE despite a vanishingly small net magnetization [15].

##  III Results and discussions

First, we study how the AHE resistivity ($\rho_{AHE}$) evolves when the
external field orientation, parameterized by $\theta$, is changed. We consider
a Hall bar where the external field rotates in the (0$\bar{1}$10) plane
($\alpha$ = 90 deg). We have first measured the hysteresis loops of
$\rho_{xy}$ for selected $\theta$, from which a linear $\rho_{OHE}$ background
(slope $\sim$ 0.045 $\mu\Omega$ cm $T^{-1}$), originating from the ordinary
Hall effect has been removed and only the antisymmetric-in-field part of the
signal is used: $\rho_{AHE}=\rho_{xy,asym}=(\rho_{xy}(H)-\rho_{xy}(-H))/2$.
The symmetric part $\rho_{xy,sym}=(\rho_{xy}(H)+\rho_{xy}(-H))/2$, is
negligible in our sample at the specific measurement temperature supporting
the collinear spin configuration [17]. The different contributions to
$\rho_{xy}$ are summarized in Figure 2(a). For the remaining of the paper, we
will convert $\rho_{AHE}$ to AHE conductivity
$\sigma_{AHE}={\rho_{AHE}}/{\rho_{xx}^{2}}$, which is used to characterize the
intrinsic effects ($\rho_{xx}$ at 110 K is $\sim 180$ $\mu\Omega$ cm).

The hysteresis loops of $\sigma_{AHE}$ for various $\theta$ are shown in
Figure 2(b). We observe that the $\sigma_{AHE}$ at saturation
($\sigma_{AHE,max}$) is maximum when the field is close to the out-of-plane
[0001] axis ($\theta=0$ deg) and vanishes when the field is close to the in-
plane [2$\bar{1}\bar{1}0$] axis ($\theta=90$ deg). As expected from the
previous AHE study on the epitaxial Mn5Si3 [15], 180 degree rotation of this
field changes the polarity of $\sigma_{AHE}$. The evolution of $\sigma_{AHE}$
for the field orientations between [0001] and [000$\bar{1}$] is better visible
in Figure 3 where we directly show $\sigma_{AHE}$ as a function of $\theta$
for a 4 T external field. Between [0001] and [000$\bar{1}$] $\sigma_{AHE}$
shows step-like behavior: when the field is oriented close to either [0001] or
[2$\bar{1}\bar{1}0$] (in-plane) crystal axes the slope is considerably more
flat than in-between these directions. We note that the backward sweep does
not trace the forward sweep because the saturation is not reached at 4 T at
all field orientations as seen in Figure 2(b).

Figure 3: (Color Online) The AHE conductivity $\sigma_{AHE}$ as a function of
the orientation $\theta$ of a 4 T external field, at 110 K (full line), for a
current applied along the $[01\bar{1}0]$ direction ( $\alpha$ = 90 deg). The
forward sweep is indicated by the full line and the backward sweep by the
dotted line. The AHE signal does not correlate with the $\theta$-dependence of
the weak saturation magnetization $M_{s}$ of the hysteretic component
(circles), which is isotropic at $\theta=0$ and 90 deg. Inset: measurement
geometry recalled and crystal structure of the (01$\bar{1}$0) plane in which
the external field $H$ is rotated. Figure 4: (Color Online) The field strength
(a) and $\theta$-orientation (b) dependence (at 2 T) of $\sigma_{AHE}$ at 110
K for epitaxial films containing 96:4 and 75:25 ratios of Mn5Si3:MnSi. The
former is an altermagnetic candidate and shows sizable AHE with large
coercivity and unconventional step-like $\theta$-dependence. The latter is
expected not altermagnetic and it shows smaller AHE with vanishing coercivity
and conventional isotropic $\theta$-dependence. Figure 5: (Color Online) The
normalized AHE conductivity $\sigma_{AHE}/\sigma_{AHE,max}$ and the
longitudinal anisotropic magnetoresistance (AMR∥) when the external field is
rotated in different crystal planes. In both cases, two distinct behaviors can
be observed depending on whether the crystal plane is equivalent to
(0$\bar{1}$10) or (2$\bar{1}\bar{1}$0). Here, the field strength is 4 T and
the temperature is 110 K. The inserts recall the geometry used and the planes
in which the field is rotated.

To unravel the possible origin of this anisotropic behavior of the AHE, we
will first study the option of weak magnetization giving a contribution.
Analogously to the AHE experiments, we have measured the sample magnetization
$M$ along the external field for two field orientations $\theta=0$ and 90 deg
as shown in Figures 2b and 3. After removing a linear diamagnetic background,
we observe that the magnetization loops contain two components: a non-
hysteretic part that relates to the substrate and the sample holder, and a
hysteretic part that we attribute to the Mn5Si3 film as the magnitude of its
coercivity agrees with that of the AHE-signal. This hysteretic component
suggests a slight spontaneous canting of the magnetic sublattice moments and
can play a role in the reversal mechanism of the Néel vector. We observe that
the weak magnetization of the hysteretic component is small, i.e. $\sim 0.1$
$\mu_{B}/$u.c., and this value for $\theta=0$ and 90 deg is similar. We note
that the behavior shown in Figure 2(b) would also be expected for a
ferromagnet as $M_{z}\sim 0$ at $\theta=90$ deg, while the wide plateau in
Figure 3 could be a consequence of the magnetocrystalline anisotropy. We will
now discuss further experiments to show that the anisotropic AHE is more
likely to originate from the altermagnetic phase of our films.

First, we have measured the AHE in Mn5Si3 films with a larger proportion of
MnSi, namely 75 % Mn5Si3 and 25 % MnSi. Increasing the MnSi content
deteriorates the crystallinity of the Mn5Si3 [16], leading to a more textured
film. We expect this to either prohibit altermagnetism as the strain necessary
for stabilizing the hexagonal unit cell can relax at the grain boundaries, or
even if the hexagonal unit cell is retained in the individual grains their
different crystal orientations result in overall cancelling out of the
altermagnetic effects. Figure 4 confirms that the magnetotransport properties
of this film are different from those of the 96:4 Mn5Si3:MnSi) Mn5Si3
epitaxial films with higher crystal quality. The field dependence of
$\sigma_{AHE}$ of the 75:25 control sample has a reduced amplitude and a
vanishing coercivity, while the $\theta$-dependence of the $\sigma_{AHE}$
shows roughly a $\cos\theta$ behavior, pointing to a ferromagnetic character.
These results are in stark contrast with the behavior of the 96:4 sample and
demonstrate that the AHE with anisotropic $\theta$-dependence and non-zero
coercivity requires a high crystal quality of the Mn5Si3 epitaxial films,
which is consistent with the altermagnetic phase being at the origin of the
AHE requiring the system to have a specific symmetry. Moreover, this shows
that the spurious MnSi phase with $T_{c}\sim$ 40 K is also unlikely origin of
the AHE signal as the $\sigma_{AHE}$ should in that case be stronger for the
75:25 sample.

Next, to further study the anisotropy of AHE, we vary systematically the
crystal plane in which the external field and in effect the Néel vector
$\mathbf{L}$ (confirmed later using the anisotropic magnetoresistance
measurements) is rotated and again observe how the $\sigma_{AHE}$ is affected.
This was realized by patterning multiple Hall bars with their current channels
oriented along different crystal axes. We recall that these orientations are
denoted by $\alpha$, which is the angle between the current channel and the
crystal axis $[2\bar{1}\bar{1}0]$ as was shown in Figure 1(d). For each Hall
bar we rotate the field in a plane perpendicular to the current channel and
thus effectively we are changing the crystal plane in which the field rotates.
We have selected $\alpha$’s that reflect the hexagonal crystal structure of
our material and can be divided into two sets: for $\alpha=0,60,120$ deg the
current channel is along [$2\bar{1}\bar{1}0$] or equivalent so the field
rotates in the crystal plane ($2\bar{1}\bar{1}0$) or equivalent;

Figure 6: (Color Online) Two perpendicular Hall bars measured simultaneously
at 110 K with 2 T field. The anisotropic $\theta$-dependence of $\sigma_{AHE}$
relates to the crystal plane in which the field rotates instead of the crystal
axis along which the current channel orients.

for $\alpha=30,90,150$ deg the current channel is along [$0\bar{1}10$] or
equivalent and the field rotates in the crystal plane ($0\bar{1}10$) or
equivalent. In Figure 5 we show the $\theta$-dependence of the $\sigma_{AHE}$
(with 4 T field) for the two sets of $\alpha$s. For each set, $\sigma_{AHE}$
is maximum when the field is close to [0001] axis but significant differences
arise when the field orients towards the sample plane: for $\alpha=30,90,150$
deg, when the field orients in the [$2\bar{1}\bar{1}0$] direction or
equivalent, we observe a wide plateau around $\theta$ = 90 deg, familiar from
Figure 3, while for $\alpha$ = 0, 60, 120 deg when the field orients in the
[$0\bar{1}10$] direction or equivalent we observe a steeper linear slope
instead. We have verified that this anisotropy persists also at 2 and 7 T
fields. The hexagonal symmetry of the AHE signals for the different $\alpha$
suggests that the three possibilities for the distribution of the magnetic Mn2
atoms (Figure 1) are likely present in equal populations: if each Mn2 site is
on average magnetic, the $\alpha$-dependence of $\sigma_{AHE}$ should reflect
the hexagonal symmetry of the crystal structure. If one of these ’quadrupole
domains’ dominated, lower symmetry for the $\alpha$-dependence of
$\sigma_{AHE}$ would be expected. We have verified that the key factor
responsible for the change of behavior between the different $\alpha$s is the
crystal plane in which the field rotates rather than the crystal axis along
which the current channel is oriented, as observed in Figure 6. We note that
for Figure 6 we have used a different sample of the same material, which
explains some of the differences in the $\theta$-dependence of the
$\sigma_{AHE}$ compared to the other figures. However, the overall behavior is
the same, which confirms the reproducibility of the results discussed here. In
summary, the results in Figures 5 and 6 allow us to conclude that the
anisotropic behavior depends on the $\mathbf{L}$-orientation relative to the
hexagonal crystal symmetry of the Mn5Si3 epitaxial films.

We also consider magnetocrystalline anisotropy as the origin of the
anisotropic AHE unlikely: if we interpret the results shown in Figures 3 and 5
considering that the different crystal planes have strongly different
magnetocrystalline energy landscapes, then for $\alpha$ = 30, 90, and 150 deg
it is more energetically favorable for $\mathbf{L}$ to remain aligned along a
crystal axis yielding $\sigma_{AHE}=0$ at $\theta=90$ deg compared to $\alpha$
= 0, 60, and 120 deg. However, the hysteresis loops measured at $\theta=$ 90
deg for both $\alpha=$ 0 (not shown) and 90 deg (Figure 2(b)) superimpose at 4
T, indicating that it is unlikely that at this field $\mathbf{L}$ is lagging
behind the external field around $\theta=$ 90 deg for $\alpha$ = 30, 90, and
150 deg more than for $\alpha$ = 0, 60, and 120 deg. Moreover, we observe the
anisotropy in the $\alpha$-dependence of $\sigma_{AHE}$ persisting up to 7 T,
which is well above the saturation field of $\sigma_{AHE}$ at any $\theta$, as
shown in Figure 2(b). Along with our demonstration that the anisotropic AHE is
crystal quality-dependent and depends on the $\mathbf{L}$-orientation relative
to the hexagonal crystal symmetry, these observations provide further support
to the main AHE contribution in our Mn5Si3 films being from altermagnetism.
These results constitute the main conclusion of our work and they
systematically support the previous calculations and experiments on the same
system [15].

Finally, we confirm the ability to manipulate $\mathbf{L}$ with the external
field by measuring the anisotropic magnetoresistance (AMR), which is known to
depend on the $\mathbf{L}$-orientation. As shown in Figure 5, we have measured
the $\alpha$\- and $\theta$-dependence of the longitudinal AMR (AMR||), which
relates to $\rho_{xx}$ as
AMR${}_{||}(\theta)={(\rho_{xx}(\theta)-\langle\rho_{xx}\rangle)}/{\langle\rho_{xx}\rangle}$,
where $\langle\rho_{xx}\rangle$ indicates $\rho_{xx}$ averaged over all
angles. AMR is an even function of the field and known to have two components:
a non-crystalline one scaling with the angle between the order parameter and
the current directions, and a crystalline one relating to the order parameter
orientation relative to the crystal axes [36]. We show the $\theta$-dependence
for $\alpha$ = 0, 60, 120 deg and $\alpha$ = 30, 90, 150 deg Hall bars in
Figure 5 at conditions identical to the AHE measurements. Again we observe
different behaviors for the two sets of $\alpha$, as expected from the
crystalline AMR depending on the crystal plane in which the order parameter
rotates: for $\alpha$ = 0, 60, 120 deg the dependence follows a
$\cos{2\theta}$ behavior while for $\alpha$ = 30, 90, 150 deg higher order
even cosines appear in the signal.

Although the exact Néel vector orientation and rotation under the action of
the external magnetic field cannot be deduced from the results presented, in
this final paragraph we wish to open a discussion on the interpretation of the
anisotropic AHE in the light of the symmetries of the epitaxial Mn5Si3
considering this material to be a strong candidate for the altermagnetic
phase. First, it should be noted that in our measurement geometry (Figure 1),
the changes in $\sigma_{AHE}$ correspond to changes in the component of
$\mathbf{h}$ along the [0001] direction (out-of-plane). In altermagnets, the
presence and orientation of $\mathbf{h}$ depend on the $\mathbf{L}$-vector
orientation because the latter influences the symmetries of the system. While
accurate calculation of the entire $\theta$-dependence of
$\mathbf{h(\mathbf{L})}$ is inhibited by the unknown relevant micromagnetic
parameters, the value of $\sigma_{AHE}$ for some specific orientations can be
inferred from symmetry-considerations. The symmetry analysis on the expected
spin structure of the epitaxial Mn5Si3 shown in Figure 1(a) indicates that for
non-zero $\sigma_{AHE}$ the $\mathbf{L}$-vector cannot point along the [0001]
axis ($\mathbf{h}$ excluded), along the main in-plane axes (only $\mathbf{h}$
in-plane allowed), or in the planes of the high-symmetry in-plane axes
highlighted in Figure 1(a) (only $\mathbf{h}$ in-plane allowed). Here, the
crystal symmetries of the planes corresponding to $\alpha=$ 0, 60, and 120 deg
and $\alpha=$ 30, 90, and 150 deg differ, as seen in Figure 5 and therefore
the $\sigma_{AHE}$ likely differ, too.

##  IV Conclusion

In conclusion, the implications of our measurements on the external field
orientation dependence of $\sigma_{AHE}$ in epitaxial Mn5Si3 films are two-
fold. First, the systematic changes in $\sigma_{AHE}$ supported by the
systematic changes in AMR|| upon the field rotation underlines our ability to
manipulate the $\mathbf{L}$-orientation with an external field of the order of
2 T or more. This is an important premise for further experimentally feasible
spin transport studies in altermagnets where control over the
$\mathbf{L}$-orientation is critical. Second, our key result is that the
$\sigma_{AHE}$ in epitaxial Mn5Si3 films shows anisotropic behavior as a
function of the external field-induced orientation of $\mathbf{L}$ relative to
the underlying crystal structure, and this anisotropic behavior depends on the
crystal quality and is unlikely to relate to the magnetocrystalline
anisotropy. This observation of the anisotropic $\sigma_{AHE}$ adds to the
growing array of properties of epitaxial Mn5Si3 advocating the altermagnetic
phase in this material.

###### Acknowledgements.

This work was supported by the French national research agency (ANR) and the
Deutsche Forschungsgemeinschaft (DFG) (Project MATHEEIAS- Grant No.
ANR-20-CE92-0049-01 / DFG-445976410). H. R., E. S., S. T. B. G. and A. T. are
supported by the DFG-GACR grant no. 490730630. J. R. acknowledges MINECO for
the Margaritas Salas program. D. K. acknowledges the academy of sciences of
the Czech Republic for the Lumina quaeruntur program.

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# A need to revise stellar opacities from asteroseismology of $\delta$ Scuti
stars

Jadwiga Daszyńska-Daszkiewicz University of Wrocław, Faculty of Physics and
Astronomy
Astronomical Institute, ul. Kopernika 11
PL-51-622 Wrocław, Poland  Przemysław Walczak University of Wrocław, Faculty
of Physics and Astronomy
Astronomical Institute, ul. Kopernika 11
PL-51-622 Wrocław, Poland  Alexey Pamyatnykh Nicolaus Copernicus Astronomical
Center, Polish Academy of Sciences
ul. Bartycka 18, PL-00-716 Warsaw, Poland  Wojciech Szewczuk University of
Wrocław, Faculty of Physics and Astronomy
Astronomical Institute, ul. Kopernika 11
PL-51-622 Wrocław, Poland  Wojciech Niewiadomski  University of Wrocław,
Faculty of Physics and Astronomy
Astronomical Institute, ul. Kopernika 11
PL-51-622 Wrocław, Poland

###### Abstract

We construct seismic models of the four double-mode radial $\delta$ Scuti
stars adopting opacities from three databases: OPAL, OP and OPLIB. The aim is
to find the models that fit the observed frequencies of the two radial modes
and have the effective temperature and luminosity consistent with the observed
values. Using the Bayesian analysis based on Monte Carlo simulations, we
obtain that only the OPAL seismic models are caught within the observed error
box in the HR diagram. Seismic models computed with the OP and OPLIB data are
much cooler and less luminous. By including the relative amplitude of the
bolometric flux variations (the so-called parameter $f$) into these
simulations, we constrain the efficiency of convection in the envelopes,
described by the mixing length parameter $\alpha_{\rm MLT}$. We get
$\alpha_{\rm MLT}\approx 0.5$ for BP Peg, AE UMa and RV Ari (Population I
stars) and $\alpha_{\rm MLT}\approx 1.0$ for SX Phe (Population II star). For
all the stars, overshooting from the convective core seems inefficient. A
similar effect of opacity should occur also for classical Cepheids or RR Lyr
stars that are used as standard candles to measure the universe.

Stellar evolution(1599) — Stellar pulsations(1625) — $\delta$ Scuti variable
stars(370) — Atomic data(2216) — Asteroseismology(73)

##  1 Introduction

Asteroseismology provides the most stringent constraints on the theory of
stellar structure and evolution. It also offers a test of microphysics data,
in particular stellar opacities, which are among the major and still uncertain
components of modern astrophysics.

The opacity calculations are almost 100 years old and it seems that their
revision is an unfinished story. Extensive opacity calculations that included
for the first time bound-bound absorption started in Los Alamos (Cox &
Stewart, 1962; Cox, 1965) and were known as the Los Alamos Opacity Library
(LAOL) (Hübner et al., 1977). For many years, these data were widely used but
some disagreements were waiting to be explained, e.g., problems with the
standard solar model, an unknown mechanism of pulsations in B-type stars, too
large period ratios in classical Cepheids models. In the early nineties
stellar opacities were recalculated by two independent teams: OPAL (Iglesias
et al., 1992; Rogers & Iglesias, 1992) and OP (Seaton, 1993; Seaton et al.,
1994). The most spectacular result was the finding of a local maximum caused
by a huge number of transition lines of iron group elements. This maximum
occurs at temperature of about 200 000 K and is called the $Z$-bump.

The discovery of the Z-bump was a big step forward in stellar physics, however
there are still some uncertainties and many indications that something is
still missing and/or has not been correctly included (Blancard et al., 2016).
The first example is a disagreement between the standard solar model and the
helioseismic and neutrino-flux predictions (e.g., Turck-Chieze et al., 2004;
Guzik, 2008; Christensen-Dalsgaard et al., 2009) that arose after the revision
of solar chemical abundances (Asplund et al., 2005, 2009). The laboratory
measurements at physical conditions similar to the boundary of the solar
convection zone have indicated that the Rosseland mean opacities of iron
predicted by all codes are underestimated by 30 to 400 % (Bailey et al., 2015;
Pradhan & Nahar, 2018; Zhao et al., 2018). The $30–45$% underestimate of iron
opacity at stellar interior temperatures was also measured by Nagayama et al.
(2019). Another example is a presence of high-order gravity modes in $\beta$
Cephei and $\delta$ Scuti stars, that are not excited in standard-opacity
models (e.g., Pamyatnykh et al., 2004; Balona, 2014). Increasing the mean
opacity at temperature of about 290 000 K, where nickel has its maximum
contribution to the Z-bump, helped to excite g modes in $\beta$ Cep models
(Salmon et al., 2012; Daszyńska-Daszkiewicz et al., 2017). In the case of
$\delta$ Scuti stars, an increase of opacity at $T=115\,000$ K allowed to make
g modes unstable (Balona et al., 2015). This new opacity bump at $T=115\,000$
K was indeed identified by Cugier (2012, 2014) in the Rosseland mean opacities
taken from the model atmospheres of Castelli & Kurucz (2003).

Seismic models, that is, models that reproduce the observed frequencies of the
identified pulsational modes, are also sensitive to the adopted opacity tables
(e.g., Daszyńska-Daszkiewicz et al., 2017, 2020, 2021, 2022). Here, we present
this effect for the double-mode radially pulsating $\delta$ Sct stars. These
pulsators are of special interest because the period ratio of two consecutive
radial modes (usually the fundamental and first overtone) takes the value in a
very small range. On the other hand, we get the period ratio from space
observations with an accuracy of up to six decimal places. We performed
extensive seismic studies, based on the Monte Carlo simulations, for the four
high-amplitude $\delta$ Sct (HADS) stars: BP Peg, AE UMa, RV Ari and SX Phe.
The first three stars belong to Population I and the last one, SX Phe, belongs
to Population II. One often talks about a separate group of pulsating
variables with SX Phe as their prototype.

The reminder of the paper is organized as follows. In Section 2, we explain
the motivation for our studies. Section 3 contains the results of seismic
modelling for the four HADS stars. The last Section is the summary. Details of
seismic modelling with the Bayesian analysis based on the Monte Carlo
simulations are given in Appendix.

##  2 Motivation

Three opacity databases are commonly used in stellar evolution computations:
OPAL (Iglesias & Rogers, 1996), OP (Seaton, 1996, 2005) and OPLIB (Colgan et
al., 2015, 2016). There are some subtle differences between these data that
result from the adopted physics and methods of computations. However, they
have, in general, a minor effect on evolutionary tracks. This is shown in the
left panel of Fig. 1, where we plotted the tracks in the Hertzsprung-Russell
(HR) diagram, computed for a mass $M=1.8$ M⊙ with the OPAL, OP and OPLIB
opacities. For these tracks, we adopted a typical initial hydrogen abundance
by mass $X_{0}=0.70$ and metallicity by mass $Z=0.014$.

Nevertheless, we found that opacity data has a very significant effect on a
frequency ratio of consecutive radial pulsational modes. In the right panel of
Fig. 1, we show the corresponding evolution of a frequency ratio of the
fundamental and first overtone radial modes. Grey dots represent the models
with the same values of the effective temperature $T_{\rm eff}=7030$ K and
luminosity $\log L/$L${}_{\odot}=1.2$. We selected these models to present the
pure effect of opacity data not blurred by the effect of $T_{\rm eff}$ or
$\log L/$L⊙.

Evolutionary computations were performed using the Warsaw-New Jersey code,
(e.g., Pamyatnykh, 1999). The code takes into account the mean effect of the
centrifugal force, assuming solid-body rotation and constant global angular
momentum during evolution. All the stars analysed in this paper are slow
rotators, which is one of the main properties of HADS stars. Therefore, the
effect of differential rotation will be negligible.

The treatment of convection in the stellar envelope relies on the standard
mixing-length theory (MLT) and the efficiency of convection is described by
the mixing length parameter $\alpha_{\rm MLT}$. At lower temperature range,
i.e., for $\log T<3.95$, opacity data from Ferguson et al. (2005) were used.
The solar chemical mixture was adopted from Asplund et al. (2009) and the
OPAL2005 equation of state was used (Rogers et al., 1996; Rogers & Nayfonov,
2002). The evolutionary tracks in Fig. 1 were computed at zero-rotation and
for $\alpha_{\rm MLT}=0.5$, to focus only on the opacity effect.

Stellar pulsations were computed using linear non-adiabatic code of
Dziembowski (1977). This code adopts the frozen convection approximation,
i.e., the convective flux does not change during the pulsations, which is a
reasonable approach if convection is not very efficient in the envelope. The
effects of rotation on pulsational frequencies are taken into account up to
the second order in the framework of perturbation theory. In the case of
radial modes, the second order effects of rotation reduces to the factor
$4\nu_{\rm rot}^{2}/3\nu_{\rm puls}$ independently of the radial order $n$
(e.g., Simon, 1969; Kjeldsen et al., 2018), where $\nu_{\rm rot}$ is the
frequency of rotation. At a rotation rate of 18 km$\cdot$s-1, for the dominant
frequencies of the studied stars, this factor amounts to about $0.004\,\rm
d^{-1}$ in the case of SX Phe and about $0.002\,\rm d^{-1}$ in the case of
other three HADS stars. For higher frequencies these values will be smaller.

Figure 1: Left panel: evolutionary tracks of a 1.8 M⊙ star computed with the
OPAL, OP and OPLIB opacity tables. Right panel: the evolution of a frequency
ratio of the fundamental and first overtone radial modes. Grey dots represent
models with the effective temperature, $T_{\rm eff}=7020$ K, and luminosity,
$\log L/$L${}_{\odot}=1.2$, marked with a dot on the left panel.

Figure 2: Left panel: the run of the Rosseland mean opacity from OPAL, OP and
OPLIB data inside the model with a mass 1.8 M⊙, effective temperatures $T_{\rm
eff}=7020$ K and luminosity $\log L/$L${}_{\odot}=1.2$. Right panel: the
corresponding differences in $\log\kappa$.

For the models marked with grey dots in Fig. 1, we plotted in Fig. 2 the run
of the Rosseland mean opacity. As mentioned, all three models have the same
effective temperature $T_{\rm eff}=7030$ K and luminosity $\log
L/$L${}_{\odot}=1.2$. As one can see the overall run of $\log\kappa(T)$ from
the three databases is very similar. However, by drawing the differences, it
is possible to locate certain depths where three opacity profiles deviate from
each other. This is shown in the right panel of Fig. 2. For example, there are
significant differences between OPAL, OP and OPLIB around temperature
corresponding to the hydrogen and first helium ionization ($\log T\approx
4.0-4.45$). On the other hand, all three $\kappa$’s are very close in the
second helium ionization zone ($\log T\approx 4.67$), where main driving of
pulsations for $\delta$ Sct stars occurs. One can also see that OPAL and OP
are very close to each other near the $Z$-bump, i.e., near $\log T=5.2-5.3$.
Instead, the data from OP and OPLIB are very similar near $\log T\approx
5.46$, where nickel has its maximum contribution to the opacity around
$Z$-bump.

All these small differences in opacity tables (of a few precent) cause huge
differences in the frequency ratio, already in the third decimal place, while
modern observations give accuracy down to six decimal places. In the next
section, we will study how these differences affect seismic models of the
realistic stars.

##  3 Seismic models of the four high-amplitude $\delta$ Scuti pulsators

Table 1: The main parameters from observations of the four studied $\delta$ Sct stars, i.e., the range of effective temperature, luminosity, metallicity and the projected rotational velocity. star | Pop. | $T_{\rm eff}$ |  $\log L/$L⊙ | [m/H] | $V_{\rm rot}\sin i$
---|---|---|---|---|---
|  | [K ] |  | [dex] | [km$\cdot$s-1]
BP Peg | I | 6800$-$8100 | 1.247(120) | $0.2$ |  $\sim$18
AE UMa | I | 7100$-$8200 | 1.091(90) | $-0.3$ |  $<$10
RV Ari | I | 7000$-$8200 | 1.103(26) | $0.1$ |  $\sim$18
SX Phe | II | 7000$-$8600 | 0.899(17) | $-1.0$ | 18(2)

To study the effect of opacity, we selected four high-amplitude $\delta$ Sct
stars: BP Pegasi, AE Ursa Majoris, RV Arietis and SX Phoenicis. These are
relatively simple objects because: 1) they pulsate in the two radial modes:
fundamental and first overtone, 2) the linear theory of pulsations is still
applicable, because even in the case of classical Cepheids, which have the
light amplitudes about five times larger, nonlinear period ratios differ from
linear values only by several tenths of a per cent (e.g., Kollath & Buchler,
2001). 3) they are very slow rotators, 4) their effective temperatures are not
low enough for convection in their envelopes to be efficient, 5) mass loss can
be neglected, and 6) there are no observational evidences for anomalous
surface abundances, so in the first approximation the effect of diffusive
settling or radiative levitation can be safely neglected.

Some results for SX Phe and BP Peg were already presented in Daszyńska-
Daszkiewicz et al. (2020) and Daszyńska-Daszkiewicz et al. (2022),
respectively. Here, we included the Monte Carlo simulations for SX Phe and
increased the number of simulations for BP Peg.

The basic parameters of the studied stars are given in Table 1. The range of
the effective temperature were gathered from the literature and luminosity
were derived from the Gaia DR3 data (Gaia Collaboration et al., 2022) in this
paper. This $T_{\rm eff}$ range also includes the changes due to pulsations.
The last two columns contain the metallicity [m/H] and projected rotational
velocities $V_{\rm rot}\sin i$. These parameters were adopted from Rodriguez
et al. (1992) for BP Peg, AE UMa and RV Ari, and from Antoci et al. (2019) for
SX Phe.

In Table 2, we list the observed frequencies of the two radial modes and the
corresponding amplitudes. In the case of BP Peg, we performed the frequency
analysis using the ASAS photometry (Pojmanski, 1997). For the other three
stars TESS observations were available and we used them to determine the
frequencies. We know that in each case these two frequencies are radial modes,
based on the period ratio and on the independent mode identification from
multi-colour photometry (Daszyńska-Daszkiewicz et al., 2020, 2022, 2023).

Table 2: Frequencies of the fundamental ($\nu_{1}$) and first overtone ($\nu_{2}$) radial modes of the four studied HADS stars. The frequencies of BP Peg were derived from the ASAS data and their amplitudes are given in [mmag]. For the other three stars the frequencies were obtained from the TESS light curves and the amplitudes are expressed in [ppt].  star |  $\nu_{1}$ | Ampl. |  $\nu_{2}$ | Ampl. | $\nu_{1}/\nu_{2}$
---|---|---|---|---|---
|  [d-1] |  |  [d-1] |  |
BP Peg | 9.128797(4) | 208(4) | 11.83315(3) | 34(4) | 0.77146
AE UMa | 11.625598(2) | 131.65(2) | 15.031250(5) | 30.70(2) | 0.77343
RV Ari | 10.737888(55) | 128.84(3) | 13.899137(121) | 39.62(2) | 0.77256
SX Phe | 18.193566(3) | 133.64(1) | 23.379306(7) | 32.92(10) | 0.77819

Having the two radial modes, fundamental and first overtone, we constructed
seismic models that fit the observed frequencies within the errors. Besides,
we fitted also the non-adiabatic parameter $f$ for the dominant frequency.
This parameter is a relative amplitude of radiative flux variations at the
level of the photosphere. The theoretical value of $f$ for a given pulsational
mode is derived in the framework of non-adiabatic theory of stellar pulsations
and it is complex because there is a phase shift between the flux and radius
variations. In the case of $\delta$ Sct models, the parameter $f$ is very
sensitive to the adopted value of $\alpha_{\rm MLT}$ (e.g., Daszyńska-
Daszkiewicz et al., 2003). Therefore, from matching the theoretical and
empirical values of $f$ it is possible to get reliable constraints on the
efficiency of convection in sub-photospheric layers. The empirical values of
$f$ are derived from the photometric amplitudes and phases in at least three
passbands, see Daszyńska-Daszkiewicz et al. (2020, 2021, 2022) for recent
results.

Our extensive, complex seismic modelling was made using the Bayesian analysis
based on the Monte Carlo simulations. As a result, we obtained constraints on
the mass $M$, metallicity $Z$, initial hydrogen abundance $X_{0}$, rotational
velocity $V_{\rm rot}$ as well as on the mixing length parameter ($\alpha_{\rm
MLT}$). In the case of BP Peg, AE UMa and RV Ari (Population I stars), we got
$\alpha_{\rm MLT}\approx 0.6,~{}0.4,~{}0.5$, respectively, and in the case of
SX Phe (the Population II star), we obtained $\alpha_{\rm MLT}\approx 1.0$.
Details of our computations and detailed results are given in the Appendix.

Results for $X_{0}=0.70$ and some range of $Z$ are presented in Fig. 3 in the
HR diagram. As one can see, in the case of each star, only the OPAL seismic
models have effective temperatures and luminosities consistent with the
observational determinations. The OP and OPLIB seismic models have far too low
values of $\log T_{\rm eff}$ and $\log L/$L⊙. In the case of the Population I
stars, the OP models have parameters even lower than the OPLIB models. In the
case of SX Phe, the OP and OPLIB seismic models are in the same position in
the HR diagram. All depicted models, for each set of opacity data, are in the
post-main sequence phase of evolution. The vast majority of them burn hydrogen
in the shell surrounding the core and just one or two models are in the phase
of an overall contraction. All seismic models in the main sequence phase of
evolution had much too low $T_{\rm eff}$ and $\log L/$L⊙ in the case of each
opacity tables. Both radial modes, fundamental and first overtone, are
unstable.

Figure 3: Seismic models of BP Peg, AE UMa, RV Ari and SX Phe constructed with
the three opacity data: OPAL, OP and OPLIB. The initial hydrogen abundance by
mass was $X_{0}=0.70$ and the range of metallicity $Z$ is given in each case.

##  4 Summary

We presented the results of extensive seismic modelling for the four high-
amplitude $\delta$ Sct stars that pulsate in two radial modes, i.e, BP Peg, AE
UMa, RV Ari and SX Phe. To this aim we used the Bayesian analysis based on the
Monte Carlo simulations. Three opacity tables were adopted: OPAL, OP and
OPLIB. Besides frequencies, we fitted also the parameter $f$ for the dominant
frequency in order to constrain, in particular, the mixing length parameter
$\alpha_{\rm MLT}$ in the envelope. We obtained $\alpha_{\rm MLT}\approx
0.6,0.4,0.5$ for BP Peg AE UMa and RV Ari, respectively, and $\alpha_{\rm
MLT}\approx 1.0$ for SX Phe.

In the case of each studied HADS, only the OPAL seismic models are close to or
within the error box on the HR diagram whereas the OP and OPLIB seismic models
are far beyond. This ”seismic opacity” discrepancy is independent of the
metallicity as it was obtained for stars with different values of [m/H] and
even for a Population II HADS SX Phe. Thus, we have shown that there are
systematic differences in seismic models of double-mode radial $\delta$ Sct
pulsators, computed with different opacity data. However, the solution to this
puzzle is rather beyond our ability. Rather, this is one more message to
atomic physicists that something is still missing in stellar opacities. This
is also a warning to those who model double-mode classical Cepheids or RR Lyr
stars. Such huge effect of opacity can occur also for these variables which
are used as standard candles to measure distances.

The work was financially supported by the Polish National Science Centre grant
2018/29/B/ST9/02803. Calculations have been partly carried out using resources
provided by Wroclaw Centre for Networking and Supercomputing
(http://www.wcss.pl), grant No. 265. This work has made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Data collected by the
TESS mission and ASAS project were used. Funding for the TESS mission is
provided by the NASA’s Science Mission Directorate.

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##  Appendix A Asteroseismic modelling with Monte Carlo-based Bayesian
analysis

We computed extensive grids of seismic models for each star and employed Monte
Carlo–based Bayesian analysis to obtain constraints on various parameters. The
analysis was based on the Gaussian likelihood function defined as (e.g.,
Jørgensen & Lindegren, 2005; da Silva et al., 2006; Rodrigues et al., 2017;
Jiang & Gizon, 2021)

${\cal
L}(E|{\mathbf{H}})=\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\cdot{\rm
exp}\left(-\frac{({\cal O}_{i}-{\cal M}_{i})^{2}}{2\sigma_{i}^{2}}\right),$
(A1)

where ${\mathbf{H}}$ is the hypothesis that represents adjustable model and
theory parameters that in case of studied stars were: mass $M$, initial
hydrogen abundance $X_{0}$, metallicity $Z$, initial rotational velocity
$V_{\rm rot,0}$, convective overshooting parameter $\alpha_{\rm ov}$ and the
mixing length parameter $\alpha_{\rm MLT}$. The evidence $E$ represents the
calculated observables ${\cal M}_{i}$, that can be directly compared with the
observed parameters ${\cal O}_{i}$ determined with the errors $\sigma_{i}$.

Here, we used the following observations: effective temperature $T_{\rm eff}$,
luminosity $L/L_{\odot}$, the frequencies of the two radial modes $\nu_{1}$
and $\nu_{2}$, and the non-adiabatic parameter $f$ for the dominant mode. The
second mode $\nu_{2}$ had too low amplitudes to make use of it. To derive the
empirical values of $f$ we applied the method of Daszyńska-Daszkiewicz et al.
(2003). To this aim, we used the amplitude and phases in the four Strömgren
passbands from Rodriguez et al. (1992) for BP Peg, AE UMa and RV Ari, and from
Rolland et al. (1991) for SX Phe. The method requires also the model
atmospheres and here we relied on Vienna (NEMO) models Heiter et al. (2002)
that include turbulent convection treatment from Canuto et al. (1996).

Then, we made a huge number of simulations (from about 90 000 to 160 000
depending on the star) to maximize the likelihood function given in Eq. (A1)
in order to constrain the parameters for each star. For each randomly selected
set of parameters ($M,~{}X_{0},~{}Z,~{}V_{\rm rot,0},~{}\alpha_{\rm ov}$ and
$\alpha_{\rm MLT}$), we calculated evolutionary and pulsational models.

In the case of the initial hydrogen abundance $X_{0}$, we assumed a beta
function $B(2,2)$ as a prior probability, since we wanted to limit its value
to the reasonable range, i.e., from 0.65 to 0.75 with $X_{0}=0.7$ as the most
probable. For other parameters we used uninformative priors, i.e., a uniform
distribution. Moreover, already first simulations showed that overshooting for
the convective core is ineffective because in all runs the parameter
$\alpha_{\rm ov}$ tended to zero very fast and did not change. This is because
all models with $\alpha_{\rm ov}>0$ had much too low values of $\log L/$L⊙.
Therefore, we set $\alpha_{\rm ov}=0.0$ in further computations.

##  Appendix B Constraints on parameters

The most important result of our seismic modeling is that only with the OPAL
opacity tables, we were able to match all observables for each star. In the
case of OP and OPLIB data, it was often difficult to find any model that
reproduced both frequencies and the non-adiabatic $f$ for the dominant mode.
Even if such models were found, they were far beyond the error box in the HR
diagram.

The results of our simulations with the OPAL data for all studied stars are
presented as histograms. The histograms were normalised to 1.0 by the number
of all models, thus the numbers on the Y-axis times 100 are the percentage of
models with a given parameter range. In Fig. 4, there are shown histograms for
masses of BP Peg, AE UMa, RV Ari and SX Phe. The next Figures 5$-$8, present
histograms for $Z,~{}X_{0}$, the current value of rotation $V_{\rm rot}$ and
for $\alpha_{\rm MLT}$. The expected values of the parameters from these
distributions as well as of the initial rotation, effective temperature and
luminosity are given in Table 3. The errors in parentheses are standard
deviations. The current value of rotation $V_{\rm rot}$ corresponds to the
given seismic model. In Table 4, we give the median values for the determined
parameters. This statistic is more informative for skewed distributions or
distributions with outliers, which is the case for some parameters. For
example, in the case of BP Peg, AE UMa and RV Ari, rather an upper limit may
be given for the rotational velocity. One can see also the asymmetry for the
metallicity $Z$ in the case of BP Peg, AE UMa and RV Ari, while the histogram
of $Z$ for SX Phe shows quite a symmetric distribution. On the other hand, the
histograms for the mixing parameter $\alpha_{\rm MLT}$ are almost symmetric
for BP Peg , AE UMa and RV Ari, while in the case of SX Phe we have some
skewness. The errors in Table 4 were calculated as the 0.84-quantile minus the
median and the median minus the 0.16-quantile. These quantiles correspond to
estimates of values separated by one standard deviation from the mean value in
the case of a normal distribution.

In the case of each star, all presented models are in the post-main sequence
phase of evolution, mostly during the hydrogen-shell burning and very rare in
the overall contraction. Main sequence models had always too low effective
temperatures and luminosities. Efficiency of convection in the envelopes of
the Population I stars, parameterized by $\alpha_{\rm MLT}$, is rather low and
amounts to about 0.5. In the case of SX Phe, which is a lower-mass star and
belongs to Population II, convection is more efficient and $\alpha_{\rm
MLT}\approx 1.0$.

Figure 4: The normalized histograms for the mass (in M⊙) of BP Peg, AE UMa, RV
Ari and SX Phe. All seismic models were computed with the OPAL opacities.

Figure 5: The normalized histograms for the metallicity $Z$ for seismic models
computed with the OPAL opacities.

Figure 6: The normalized histograms for the initial hydrogen abundance $X_{0}$
for seismic models computed with the OPAL opacities.

Figure 7: The normalized histograms for the current rotational velocity
$V_{\rm rot}$ for seismic models computed with the OPAL opacities.

Figure 8: The normalized histograms for the mixing length parameter $\alpha_{\rm MLT}$ for seismic models computed with the OPAL opacities. Table 3: The expected values of the parameters of the four studied $\delta$ Sct stars from the Monte-Carlo simulations. The uncertainties were calculated as the square roots of the variance. Seismic models were computed with the OPAL opacities. star | $M$ | $Z$ | $X_{0}$ | $\alpha_{\rm MLT}$ | $V_{\rm rot,0}$ | $V_{\rm rot}$ | $\log(T_{\rm eff}/{\rm K})$ |  $\log L/$L⊙
---|---|---|---|---|---|---|---|---
| [M⊙] |  |  |  | [km$\cdot$s-1] | [km$\cdot$s-1] |  |
BP Peg | 1.81(4) | 0.0281(25) | 0.682(19) | 0.63(7) | 16.0(9.3) | 15.1(8.8) | 3.8353(16) | 1.161(10)
AE UMa | 1.55(3) | 0.0136(7) | 0.687(11) | 0.43(14) | 19.2(11.9) | 19.2(11.9) | 3.8608(36) | 1.085(18)
RV Ari | 1.62(3) | 0.0168(9) | 0.690(9) | 0.52(5) | 18.(9.3) | 17.6(9.2) | 3.8494(22) | 1.096(12)
SX Phe | 1.088(21) | 0.00199(5) | 0.677(14) | 1.04(58) | 12.1(7.5) | 14.1(8.7) | 3.8986(37) | 0.889(17)
Table 4: The median values of the parameters of the four studied $\delta$ Sct stars from the Monte-Carlo simulations. The uncertainties were calculated from quantiles 0.84 and 0.16. Seismic models were computed with the OPAL opacities. star | $M$ | $Z$ | $X_{0}$ | $\alpha_{\rm MLT}$ | $V_{\rm rot,0}$ | $V_{\rm rot}$ | $\log(T_{\rm eff}/{\rm K})$ |  $\log L/$L⊙
---|---|---|---|---|---|---|---|---
| [M⊙] |  |  |  | [km$\cdot$s-1] | [km$\cdot$s-1] |  |
BP Peg | $1.81^{+0.03}_{-0.04}$ | $0.0271^{+0.0028}_{-0.0018}$ | $0.682^{+0.015}_{-0.023}$ | $0.60^{+0.07}_{-0.06}$ | $14.9^{+9.8}_{-11.5}$ | $14.1^{+9.1}_{-10.9}$ | $3.8351^{+0.0013}_{-0.0018}$ | $1.158^{+0.010}_{-0.009}$
AE UMa | $1.54^{+0.03}_{-0.02}$ | $0.0135^{+0.0006}_{-0.0008}$ | $0.685^{+0.012}_{-0.011}$ | $0.40^{+0.13}_{-0.14}$ | $17.0^{+13.4}_{-12.9}$ | $17.0^{+13.5}_{-12.9}$ | $3.8606^{+0.0028}_{-0.0043}$ | $1.082^{+0.017}_{-0.019}$
RV Ari | $1.62^{+0.03}_{-0.03}$ | $0.0164^{+0.0013}_{-0.0007}$ | $0.689^{+0.008}_{-0.010}$ | $0.51^{+0.05}_{-0.05}$ | $16.8^{+9.1}_{-10.9}$ | $16.4^{+9.0}_{-10.6}$ | $3.8489^{+0.0020}_{-0.0023}$ | $1.092^{+0.013}_{-0.010}$
SX Phe | $1.083^{+0.024}_{-0.020}$ | $0.00197^{+0.00010}_{-0.00011}$ | $0.672^{+0.016}_{-0.012}$ | $0.92^{+0.64}_{-0.63}$ | $11.0^{+7.8}_{-8.5}$ | $12.8^{+9.1}_{-10.0}$ | $3.8973^{+0.0040}_{-0.0032}$ | $0.884^{+0.019}_{-0.014}$