# On the injection scale of the turbulence in the partially ionized very local
interstellar medium

Siyao Xu (徐思遥) Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ
08540, USA<EMAIL_ADDRESS>111NASA Hubble Fellow Hui Li (李晖) Los Alamos National
Laboratory, NM 87545, USA<EMAIL_ADDRESS>
###### Abstract

The cascade of magnetohydrodynamic (MHD) turbulence is subject to ion-neutral
collisional damping and neutral viscous damping in the partially ionized
interstellar medium. By examining the damping effects in the warm and
partially ionized local interstellar medium, we find that the interstellar
turbulence is damped by neutral viscosity at $\sim 261$ au and cannot account
for the turbulent magnetic fluctuations detected by Voyager 1 and 2. The MHD
turbulence measured by Voyager in the very local interstellar medium (VLISM)
should be locally injected in the regime where ions are decoupled from
neutrals for its cascade to survive the damping effects. With the imposed ion-
neutral decoupling condition and the strong turbulence condition for the
observed Kolmogorov magnetic energy spectrum, we find that the turbulence in
the VLISM is sub-Alfvénic, and its largest possible injection scale is $\sim
194$ au.

††software: MATLAB (MATLAB, 2021)

##  1 Introduction

Turbulent magnetic fluctuations following a Kolmogorov spectrum were observed
by Voyager 1 and 2 in the outer heliosheath (Burlaga et al., 2018; Zhao et
al., 2020; Lee & Lee, 2020; Fraternale & Pogorelov, 2021; Burlaga et al.,
2022). While the source(s) of the turbulence in the very local interstellar
medium (VLISM) are under debate (Holzer, 1989; Zank, 2015; Zank et al., 2019),
it is believed to play a crucial role in affecting the transport of energetic
particles and cosmic rays (Lazarian & Opher, 2009; Stone et al., 2013;
Krimigis et al., 2013; López-Barquero et al., 2017; Fraternale et al., 2022),
and the structure of the Interstellar Boundary Explorer (IBEX) ribbon
(Giacalone & Jokipii, 2015; Zirnstein et al., 2020). In particular, modeling
by Zirnstein et al. (2020) suggests that the turbulent magnetic fields with
scales $<100$ au are important for producing the ribbon structure similar to
IBEX observations, and such turbulence is likely not of pristine interstellar
origin, whereas turbulent fluctuations at scales $\geq 100$ au produce
features inconsistent with IBEX observations.

Turbulence and turbulent magnetic fields are ubiquitous in the ISM (e.g.,
Armstrong et al. 1995; Chepurnov & Lazarian 2010; Gaensler et al. 2011; Xu &
Zhang 2017; Lazarian et al. 2018), with the turbulent energy injected by
supernova explosions on large length scales ($\sim 100$ pc; Breitschwerdt et
al. 2017) and cascading down toward smaller and smaller scales. In the warm
and partially ionized local interstellar medium (LISM) (Frisch et al., 2011;
Slavin & Frisch, 2008), the magnetohydrodynamic (MHD) turbulence is subject to
the damping effects due to the frictional collisions between ions and neutrals
(i.e., ion-neutral collisional damping) and among neutrals (i.e. neutral
viscosity) (Xu & Lazarian, 2017). The damping effects cause the cutoff of MHD
turbulence cascade when the damping rate exceeds the cascading rate of MHD
turbulence.

The linear analysis of MHD waves in a partially ionized medium was performed
by Kulsrud & Pearce (1969) and more recently by e.g., Pudritz (1990); Balsara
(1996); Mouschovias et al. (2011); Zaqarashvili et al. (2011); Soler et al.
(2013). Different from linear MHD waves, MHD turbulence is characterized by
the nonlinear cascade of turbulent energy, with scale-dependent turbulence
anisotropy and limited timescale of turbulent motions (Goldreich & Sridhar,
1995; Lazarian & Vishniac, 1999). The ion-neutral collisional (IN) damping of
MHD turbulence in a highly ionized medium at a high plasma $\beta$ was studied
by Lithwick & Goldreich (2001). The neutral viscous (NV) damping of MHD
turbulence was analyzed by Lazarian et al. (2004), which tends to dominate
over the IN damping toward a higher ionization fraction and a higher
temperature (Xu & Lazarian, 2017). Xu et al. (2015, 2016) performed a general
analysis including both damping effects of MHD turbulence in different
interstellar phases with varying ionization fractions.

The range of length scales for the existence of MHD turbulence depends on the
coupling state between ions and neutrals and the corresponding damping
effects. Irrespective of the origin, the MHD turbulence measured in the VLISM
should survive the damping in the partially ionized medium. In this work, we
will examine the damping of the interstellar MHD turbulence driven at large
scales. More importantly, we will explore the constraint imposed by the
damping effects on the locally driven MHD (LMHD) turbulence in the VLISM, as
observed by Voyager. In Section 2, we analyze the damping of the interstellar
turbulence driven in the strongly coupled regime and the injection of the LMHD
turbulence in the decoupled regime with weak damping. In Section 3, we
determine the turbulence regime and the largest injection scale of the LMHD
turbulence constrained by the ion-neutral decoupling condition. Discussion and
conclusions are presented in Sections 4 and 5.

##  2  Source for turbulence in the VLISM

There are three possible sources for the turbulent magnetic fluctuations
measured by Voyager, including the interstellar turbulence driven by supernova
explosions at $\sim 100$ pc, the turbulence driven in the Local Interstellar
Cloud at $\sim 2$ pc (Zank et al., 2019), and the LMHD turbulence in the
VLISM. We will first focus on the former two possible sources, for which the
turbulence is driven on large scales in the regime where neutrals and ions are
strongly coupled together (Section 2.1). For the local source, the turbulence
with a small energy injection scale is expected to arise in the regime where
ions are decoupled from neutrals (Section 2.2).

###  2.1 Damping of interstellar turbulence driven in strongly coupled regime

The interstellar turbulence is driven by supernova explosions at $L\sim 100$
pc (Breitschwerdt et al., 2017) and cascades down to smaller and smaller
scales (Chepurnov et al., 2010; Yuen et al., 2022). Its damping scale varies
in different interstellar phases and regions, depending on the local physical
conditions. Given the approximate energy equipartition between turbulence and
magnetic fields in the ISM expected from small-scale nonlinear turbulent
dynamo (Xu & Lazarian, 2016) and indicated by observational measurements
(Pattle et al., 2022), we consider that the interstellar turbulence is trans-
Alfvénic, i.e., $M_{A}=V_{L}/V_{A}\approx 1$. Here $M_{A}$ is the turbulent
Alfvén Mach number, $V_{L}$ is the turbulent velocity at the injection scale
$L$ of turbulence, $V_{A}=B/\sqrt{4\pi\rho}$ is the Alfvén speed, $B$ is the
magnetic field strength, $\rho=\rho_{i}+\rho_{n}$ is the total mass density,
$\rho_{i}=m_{H}n_{i}=m_{H}n_{e}$ is the mass density of ions, $m_{H}$ is the
hydrogen atomic mass, $n_{i}$ and $n_{e}$ are the number densities of ions and
electrons, $\rho_{n}=m_{H}n_{H}$ is the mass density of neutrals, and $n_{H}$
is the number density of neutrals.

In a partially ionized medium, MHD turbulence is subject to both IN damping
and NV damping (Lazarian et al., 2004; Xu & Lazarian, 2017). They both depend
on the coupling state between ions and neutrals. When the driving rate of
turbulence is lower than the neutral-ion collisional frequency
$\nu_{ni}=\gamma_{d}\rho_{i}$, where $\gamma_{d}=5.5\times 10^{14}$ cm3 g-1
s-1 is the drag coefficient (Draine et al., 1983; Shu, 1992), the MHD
turbulence is driven in strongly coupled ions and neutrals, i.e., the strongly
coupled regime. Neutrals decouple from ions when the cascading rate of MHD
turbulence $\tau_{\text{cas}}^{-1}$ becomes larger than $\nu_{ni}$. The
neutral-ion decoupling entails significant collisional friction, causing the
damping of MHD turbulence and the cutoff of its energy cascade. The
corresponding damping scale for Alfvénic turbulence is (Xu et al. 2015, 2016;
see Appendix B for the derivation)

$l_{\text{dam,IN},\perp}=\Big{(}\frac{2\nu_{ni}}{\xi_{n}}\Big{)}^{-\frac{3}{2}}L^{-\frac{1}{2}}V_{L}^{\frac{3}{2}},$
(1)

where $\xi_{n}=\rho_{n}/\rho$. The subscript $\perp$ means that the length
scale is measured perpendicular to the local magnetic field. As the energy
cascade of MHD turbulence is mostly in the perpendicular direction,
$l_{\text{dam,IN},\perp}$ corresponds to the cutoff scale of MHD turbulence.

In the case when the NV damping dominates over the IN damping, the damping of
MHD turbulence occurs in the strongly coupled regime, with the damping scale
(Xu & Lazarian 2017; see Appendix B)

$l_{\text{dam,NV},\perp}=\Big{(}\frac{\xi_{n}}{2}\Big{)}^{\frac{3}{4}}\nu_{n}^{\frac{3}{4}}L^{\frac{1}{4}}V_{L}^{-\frac{3}{4}},$
(2)

where $\nu_{n}=v_{\text{th}}/(n_{H}\sigma_{nn})$ is the kinematic viscosity in
neutrals, $v_{\text{th}}$ is the neutral thermal speed, and $\sigma_{nn}$ is
the collisional cross-section of neutrals. We note that
$l_{\text{dam,NV},\perp}$ is in fact the damping scale of the turbulent
kinetic energy spectrum. The magnetic fluctuations in the sub-viscous range
below $l_{\text{dam,NV},\perp}$ is termed new regime of MHD turbulence
(Lazarian et al., 2004) (see Section 4.1).

We now discuss whether the magnetic turbulence in the VLISM can come from the
interstellar turbulence that is injected by supernova explosions. The LISM
near the sun is warm, low-density, and partially ionized, with the temperature
$T\approx 6300$ K, $n_{H}\approx 0.2$ cm-3, and $n_{e}\approx 0.07$ cm-3
(Slavin & Frisch, 2008; Swaczyna et al., 2020). In addition, we adopt
$L_{\text{ISM}}\approx 100$ pc, $V_{L,\text{ISM}}\approx V_{A}$, $B\approx
5~{}\mu$G as the typical driving conditions of interstellar turbulence and
interstellar magnetic field strength (Crutcher et al., 2010). With these high
temperature and moderate ionization fraction, we find that

$\displaystyle l_{\text{dam,IN},\perp}\approx$ $\displaystyle 7.6\times
10^{13}~{}\text{cm}\Big{(}\frac{n_{e}}{0.07~{}\text{cm}^{-3}}\Big{)}^{-\frac{3}{2}}\Big{(}\frac{n_{H}}{0.2~{}\text{cm}^{-3}}\Big{)}^{-\frac{3}{4}}$
(3)
$\displaystyle\Big{(}\frac{L_{\text{ISM}}}{100~{}\text{pc}}\Big{)}^{-\frac{1}{2}}\Big{(}\frac{B}{5~{}\mu\text{G}}\Big{)}^{\frac{3}{2}}$

is much smaller than

$\displaystyle l_{\text{dam,NV},\perp}$ $\displaystyle\approx 3.9\times
10^{15}~{}\text{cm}\Big{(}\frac{T}{6300~{}\text{K}}\Big{)}^{\frac{3}{8}}\Big{(}\frac{n_{H}}{0.2~{}\text{cm}^{-3}}\Big{)}^{-\frac{3}{8}}$
(4)
$\displaystyle~{}~{}~{}~{}\Big{(}\frac{L_{\text{ISM}}}{100~{}\text{pc}}\Big{)}^{\frac{1}{4}}\Big{(}\frac{B}{5~{}\mu\text{G}}\Big{)}^{-\frac{3}{4}},$

where we assume $n_{H}+n_{e}\sim n_{H}$, and $\sigma_{nn}\approx 10^{-14}$ cm2
(Krstic & Schultz, 1998; Vranjes & Krstic, 2013) is adopted. We note that the
gradients of quantities in the outer heliosheath (Zank et al., 2013) cause
uncertainties in the above estimates. By assuming the uncertainties
$\sigma(n_{e})\sim 0.01$ cm-3, $\sigma(n_{H})\sim 0.01$ cm-3, $\sigma(B)\sim
1~{}\mu$G, $\sigma(T)\sim 1000$ K, and $\sigma(L_{\text{ISM}})\sim 10~{}$pc,
we find $\sigma(l_{\text{dam,IN},\perp})\sim 2.8\times 10^{13}~{}$cm and
$\sigma(l_{\text{dam,NV},\perp})\sim 6.4\times 10^{14}~{}$ cm.

For the interstellar turbulence driven in the Local Interstellar Cloud, the
turbulent velocity of a few km s-1 is similar to that from the cascade of
supernova-driven turbulence at $\sim 2$ pc (Spangler et al., 2011). It might
be a part of the global cascade of the interstellar turbulence. Therefore, it
is also subject to the damping effects in the partially ionized LISM and
damped at the damping scale similar to that of the supernova-driven
interstellar turbulence.

The above calculations show that the damping of the MHD turbulence in the LISM
from the interstellar origin is dominated by NV in the strongly coupled
regime. The corresponding damping scale is $\sim 3.9\times 10^{15}$ cm, i.e.,
$261$ au. This means that the turbulence observed by Vogayer 1 and 2 is
unlikely due to the pristine interstellar origin.

###  2.2 LMHD turbulence in decoupled regime

As discussed above, for the MHD turbulence injected in the strongly coupled
regime, its cascade is cut off either by the IN or NV damping. When ions are
decoupled from neutrals, however, the driven MHD turbulence is no longer
subject to the NV damping, and the IN damping becomes constantly weak (Xu et
al., 2016). The MHD turbulence injected in ions that are decoupled from
neutrals, i.e., LMHD turbulence in the decoupled regime, is not cut off due to
the damping effects arising in a partially ionized medium.

Based on the in-situ measurements of turbulent magnetic energy spectrum by
Voyager 1 in the VLISM (Lee & Lee, 2020), we have the ratio of the turbulent
component to the average magnetic field strength $\delta
B_{\text{obs}}/B\approx 0.06$ measured at $l_{\text{obs}}\approx 3\times
10^{14}$ cm (corresponding to the smallest wavenumber of the measured spectrum
and assumed to be in the inertial range of turbulence). By assuming that the
magnetic fluctuations are mainly induced by Alfvénic turbulence (Cho &
Lazarian, 2002; Hu et al., 2022; Lee & Lee, 2020), the local turbulent
velocity can be estimated as

$\displaystyle v_{\text{obs}}$ $\displaystyle=\frac{\delta
B_{\text{obs}}}{B}V_{Ai}$ (5) $\displaystyle=2.5~{}\text{km
s}^{-1}\Big{(}\frac{\delta
B_{\text{obs}}/B}{0.06}\Big{)}\Big{(}\frac{n_{e}}{0.07~{}\text{cm}^{-3}}\Big{)}^{-\frac{1}{2}}\Big{(}\frac{B}{5~{}\mu\text{G}}\Big{)}.$

By assuming $\sigma(\delta B_{\text{obs}})\sim 0.1~{}\mu$G and the
uncertainties of other parameters (see Section 2.1), we find
$\sigma(v_{\text{obs}})\sim 0.85~{}$km s-1. The large uncertainties in our
calculations are mainly caused by the large uncertainties in magnetic field
strength measurements (Burlaga et al., 2018). Such a large turbulence level at
the measured length scale cannot be accounted for by the interstellar
turbulence, which is cut off at a larger length scale
$l_{\text{dam,NV},\perp}$ (Eq. (4)). So the measured turbulence is likely to
be driven in the VLISM. Given $v_{\text{obs}}/l_{\text{obs}}(\approx 8.3\times
10^{-10}~{}\text{s}^{-1})>\nu_{in}(\approx 1.8\times
10^{-10}~{}\text{s}^{-1})$, the LMHD turbulence measured by Voyager 1 should
be injected in the decoupled regime.

##  3  Constraint on the injection scale of the LMHD turbulence in the VLISM

We now discuss the constraint on the injection scale $L$ of the LMHD
turbulence. The three cases we consider are (a) super-Alfvénic turbulence with
isotropic injection scale, (b) sub-Alfvénic turbulence with isotropic
injection scale, and (c) sub-Alfvénic turbulence with anisotropic injection
scale. In Fig. 1, we illustrate the scalings of super- and sub-Alfvénic
turbulence (see also Appendix A) for isotropic and anisotropic injection.

(a) $M_{A}>1$, isotropic injection

(b) $M_{A}<1$, isotropic injection

(c) $M_{A}<1$, anisotropic injection

Figure 1:  Illustration for turbulent eddies in super-Alfvénic (a) and sub-
Alfvénic (b,c) turbulence. Strong MHD turbulence is indicated by shaded
regions. With anisotropic injection scale of turbulence in (c), the entire
turbulent cascade can be in the strong turbulence regime. The blue line
indicates the magnetic field.

The condition for the LMHD turbulence to arise in ions alone is

$\Gamma_{L}>\nu_{in},$ (6)

where $\Gamma_{L}$ is the driving rate of the turbulence, i.e., the cascading
rate of turbulence at $L$, and $\nu_{in}=\gamma_{d}\rho_{n}$ is the ion-
neutral collisional frequency.

Case (a). If the driven turbulence is super-Alfvénic, i.e.,
$M_{A}=V_{L}/V_{Ai}>1$, where $V_{Ai}=B/\sqrt{4\pi\rho_{i}}$ is the Alfvén
speed in ions, $V_{L}$ is related to $v_{l}$ by (Lazarian & Vishniac, 1999)

$v_{l}=V_{L}\Big{(}\frac{l}{L}\Big{)}^{\frac{1}{3}},$ (7)

where $v_{l}$ is the local turbulent velocity measured at length scale
$l(<L)$. With the cascade of turbulent energy, the turbulent velocity becomes
equal to $V_{Ai}$ at the Alfvénic scale $l_{A}=LM_{A}^{-3}$ (Lazarian, 2006).
Below $l_{A}$, the effect of magnetic fields on turbulence becomes important,
resulting in turbulence anisotropy (see Fig. 1). Therefore, $l$ in the above
equation should be replaced by $l_{\perp}$ when $l$ is smaller than $l_{A}$,
where $l_{\perp}$ is the length scale measured perpendicular to the local
magnetic field (Cho & Vishniac, 2000). By using $\Gamma_{L}=V_{L}/L$ and the
scaling relation in Eq. (7), the condition Eq. (6) in this case becomes

$\displaystyle
L<\Big{(}\nu_{in}v_{l}^{-1}l^{\frac{1}{3}}\Big{)}^{-\frac{3}{2}},~{}l_{A}<l<L,$
(8) $\displaystyle
L<\Big{(}\nu_{in}v_{l}^{-1}l_{\perp}^{\frac{1}{3}}\Big{)}^{-\frac{3}{2}},~{}l<l_{A}.$

The above expressions provide constraints on the maximum $L$ of the LMHD
turbulence in the decoupled regime if it is super-Alfvénic.

The in-situ measurements show that the LMHD turbulence has $v_{l}\approx
v_{\text{obs}}$ at $l_{\perp}\approx l_{\text{obs}}$ (Section 2.2). We
consider the largest observed scale $l_{\text{obs}}$ as the perpendicular
scale because the trajectory of Voyager 1 is nearly perpendicular to the
background magnetic field (Izmodenov & Alexashov, 2020; Fraternale &
Pogorelov, 2021; Dialynas et al., 2022). $v_{l}$ is smaller than $V_{Ai}$ (see
Eq. (5)) and related to $V_{Ai}$ by

$v_{l}=V_{Ai}\Big{(}\frac{l_{\perp}}{l_{A}}\Big{)}^{\frac{1}{3}},$ (9)

yielding

$\displaystyle l_{A}$
$\displaystyle=l_{\perp}\Big{(}\frac{v_{l}}{V_{Ai}}\Big{)}^{-3}$ (10)
$\displaystyle=1.4\times
10^{18}~{}\text{cm}\Big{(}\frac{n_{e}}{0.07~{}\text{cm}^{-3}}\Big{)}^{-\frac{3}{2}}\Big{(}\frac{v_{l}}{2.5~{}\text{km
s}^{-1}}\Big{)}^{-3}$
$\displaystyle~{}~{}~{}~{}\Big{(}\frac{l_{\perp}}{3.0\times
10^{14}~{}\text{cm}}\Big{)}\Big{(}\frac{B}{5~{}\mu\text{G}}\Big{)}^{3},$

with the estimated uncertainty $\sigma(l_{A})\sim 1.7\times 10^{18}~{}$cm. The
condition in Eq. (8) for super-Alfvénic turbulence can be written explicitly
as,

$\displaystyle L<$ $\displaystyle 2.9\times
10^{15}~{}\text{cm}\Big{(}\frac{n_{H}}{0.2~{}\text{cm}^{-3}}\Big{)}^{-\frac{3}{2}}\Big{(}\frac{v_{l}}{2.5~{}\text{km
s}^{-1}}\Big{)}^{\frac{3}{2}}$ (11)
$\displaystyle\Big{(}\frac{l_{\perp}}{3.0\times
10^{14}~{}\text{cm}}\Big{)}^{-\frac{1}{2}},$

with the estimated uncertainty $\sigma(L)\sim 1.5\times 10^{15}~{}$cm. The
super-Alfvénic condition $M_{A}>1$ requires $L>l_{A}$, which is not satisfied
by the above values. We conclude that the LMHD turbulence cannot be super-
Alfvénic.

Case (b). If the driven turbulence is sub-Alfvénic, i.e.,
$M_{A}=V_{L}/V_{Ai}<1$, the MHD turbulence is weak with weak interactions
between counterpropagating Alfvén wave packets (Galtier et al., 2000) over
scales $[L,l_{\text{tran}}]$, where $l_{\text{tran}}=LM_{A}^{2}$ is the
perpendicular transition scale from weak to strong MHD turbulence (Lazarian &
Vishniac, 1999) (see Fig. 1). For weak turbulence, there is no parallel
cascade. Its cascade to smaller perpendicular scales strengthens until the
cascade becomes strong (Goldreich & Sridhar, 1997). The scaling of weak
turbulence follows $v_{l}=V_{L}(l_{\perp}/L)^{1/2}$ (Lazarian & Vishniac,
1999), while in the strong turbulence regime, the local turbulent velocity
follows the scaling (Lazarian & Vishniac, 1999)

$v_{l}=V_{L}\Big{(}\frac{l_{\perp}}{L}\Big{)}^{\frac{1}{3}}M_{A}^{\frac{1}{3}}.$
(12)

For the weak turbulence at $L$, there is

$\Gamma_{L}=\frac{V_{L}}{L}M_{A}.$ (13)

By combining Eqs. (12) and (13), the condition in Eq. (6) becomes

$L<\nu_{in}^{-2}v_{l}^{3}l_{\perp}^{-1}V_{Ai}^{-1}.$ (14)

The spectral indices of magnetic fluctuations in weak and strong MHD
turbulence are $-2$ and $-5/3$, respectively. With the Kolmogorov slope
($-5/3$) found for the measured magnetic energy spectrum (Burlaga et al.,
2018; Lee & Lee, 2020), we consider that the locally measured turbulence is in
the strong MHD turbulence regime. For the driven sub-Alfvénic turbulence, the
condition in Eq. (14) gives

$\displaystyle L<$ $\displaystyle 3.6\times
10^{14}~{}\text{cm}\Big{(}\frac{n_{e}}{0.07~{}\text{cm}^{-3}}\Big{)}^{\frac{1}{2}}\Big{(}\frac{n_{H}}{0.2~{}\text{cm}^{-3}}\Big{)}^{-2}$
(15) $\displaystyle\Big{(}\frac{v_{l}}{2.5~{}\text{km
s}^{-1}}\Big{)}^{3}\Big{(}\frac{l_{\perp}}{3.0\times
10^{14}~{}\text{cm}}\Big{)}^{-1}\Big{(}\frac{B}{5~{}\mu\text{G}}\Big{)}^{-1},$

with $\sigma(L)\sim 3.8\times 10^{14}~{}$cm. By using Eqs. (12) and (13), the
condition in Eq. (6) can also be written as

$\displaystyle V_{L}$
$\displaystyle<(\nu_{in}^{-1}v_{l}^{3}l_{\perp}^{-1})^{\frac{1}{2}}$ (16)
$\displaystyle<5.2~{}\text{km
s}^{-1}\Big{(}\frac{n_{H}}{0.2~{}\text{cm}^{-3}}\Big{)}^{-\frac{1}{2}}\Big{(}\frac{v_{l}}{2.5~{}\text{km
s}^{-1}}\Big{)}^{\frac{3}{2}}$
$\displaystyle~{}~{}~{}~{}\Big{(}\frac{l_{\perp}}{3.0\times
10^{14}~{}\text{cm}}\Big{)}^{-\frac{1}{2}},$

with $\sigma(V_{L})\sim 2.7~{}$km s-1. The corresponding $M_{A}$ is

$\displaystyle M_{A}$
$\displaystyle=\frac{V_{L}}{V_{Ai}}<0.13\Big{(}\frac{n_{e}}{0.07~{}\text{cm}^{-3}}\Big{)}^{\frac{1}{2}}\Big{(}\frac{n_{H}}{0.2~{}\text{cm}^{-3}}\Big{)}^{-\frac{1}{2}}$
(17) $\displaystyle\Big{(}\frac{v_{l}}{2.5~{}\text{km
s}^{-1}}\Big{)}^{\frac{3}{2}}\Big{(}\frac{l_{\perp}}{3.0\times
10^{14}~{}\text{cm}}\Big{)}^{-\frac{1}{2}}\Big{(}\frac{B}{5~{}\mu\text{G}}\Big{)}^{-1},$

with $\sigma(M_{A})\sim 0.072$. Given such a small $M_{A}$ value, the implied
$l_{\text{tran}}=LM_{A}^{2}$ is $\lesssim 6.1\times 10^{12}~{}$cm. This means
that the observed turbulence would be in the weak MHD turbulence regime with
$l_{\text{obs}}>l_{\text{tran}}$. This is inconsistent with the observed
Kolmogorov spectrum for strong MHD turbulence.

Case (c). In the above calculations we assume that the injection scale of
turbulence is isotropic. In the presence of strong background magnetic fields,
the driven sub-Alfvénic turbulence is likely to have anisotropic $L$ (see
e.g., Pogorelov et al. 2017). With a sufficiently small perpendicular
component of injection scale $L_{\perp}$, the shear in the direction
perpendicular to the magnetic field can cause significant distortions of
magnetic field lines within the Alfvén wave period. If the anisotropy is
sufficiently large so that the nonlinear interaction between
counterpropagating Alfvén wave packets is strong and thus the critical balance
relation (Goldreich & Sridhar, 1995) is satisfied at $L$, the entire turbulent
cascade would be in the strong MHD turbulence regime (see Fig. 1). In this
case, we have

$v_{l}=V_{L}\Big{(}\frac{l_{\perp}}{L_{\perp}}\Big{)}^{\frac{1}{3}},$ (18)

and $\Gamma_{L}=V_{L}/L_{\perp}$. The condition in Eq. (6) leads to the
constraint on the perpendicular injection scale,

$\displaystyle L_{\perp}$
$\displaystyle<\Big{(}\nu_{in}v_{l}^{-1}l_{\perp}^{\frac{1}{3}}\Big{)}^{-\frac{3}{2}}$
(19) $\displaystyle\approx 2.9\times
10^{15}~{}\text{cm}\Big{(}\frac{n_{H}}{0.2~{}\text{cm}^{-3}}\Big{)}^{-\frac{3}{2}}\Big{(}\frac{v_{l}}{2.5~{}\text{km
s}^{-1}}\Big{)}^{\frac{3}{2}}$
$\displaystyle~{}~{}~{}~{}\Big{(}\frac{l_{\perp}}{3.0\times
10^{14}~{}\text{cm}}\Big{)}^{-\frac{1}{2}},$

with the uncertainty $\sigma(L_{\perp})\sim 1.5\times 10^{15}~{}$cm, and the
same constraint on $V_{L}$ as in Eq. (16).

Among the three cases, only Case (c), i.e., sub-Alfvénic turbulence with
anisotropic injection scale, provides the self-consistent result. In Fig. 2,
we present $2\pi/L_{\perp,\text{max}}$ together with the observationally
measured magnetic energy spectrum taken from Lee & Lee (2020) for the period
from 2012 August to 2019 December and Burlaga et al. (2018) for intervals
2013.3593-2014.6373 and 2015.3987-2016.6759. Here $L_{\perp,\text{max}}$ (Eq.
(19)) is the largest possible perpendicular injection scale of the LMHD
turbulence in the VLISM. We find that $L_{\perp,\text{max}}$ is close to
$l_{\text{dam,NV},\perp}$ of the interstellar turbulence given by Eq. (4).

Figure 2:  The data for the magnetic energy spectrum are taken from Lee & Lee
(2020) (black) and Burlaga et al. (2018) (red and blue). The dashed line
indicates the Kolmogorov slope. The vertical dashed and dash-dotted lines
correspond to the damping scales of interstellar turbulence
($l_{\text{dam,NV},\perp}$, Eq. (4)) and interstellar magnetic fluctuations
($l_{\text{dam,IN},\perp,\text{sub-v}}$, Eq. (22)). The vertical solid line
corresponds to the largest possible perpendicular injection scale
($L_{\perp,\text{max}}$, Eq. (19)) of the locally driven sub-Alfvénic
turbulence in the VLISM.

##  4 Discussion

###  4.1 New regime of MHD turbulence

In the case when the NV damping dominates over the IN damping (see Section
2.1), the turbulent motions are damped at the neutral viscous scale, but the
magnetic fluctuations can exist on smaller scales. There is a new regime of
MHD turbulence in the sub-viscous range (Cho et al., 2002; Lazarian et al.,
2004; Xu & Lazarian, 2016), where the kinetic energy spectrum is steep with
the spectral index $-4$ and the magnetic energy spectrum is flat with the
spectral index $-1$. We note that such a flat magnetic energy spectrum
corresponds to scale-independent magnetic fluctuations. The sub-viscous
magnetic fluctuations are caused by the shear of viscous-scale turbulent
eddies in the direction perpendicular to the magnetic field. As the IN damping
suppresses magnetic fluctuations, the damping scale of the sub-viscous
magnetic fluctuations is determined by the balance between the eddy-turnover
rate at $l_{\text{dam,NV},\perp}$ and the IN damping rate,

$\frac{v_{\nu}}{l_{\text{dam,NV},\perp}}=\omega_{\text{d,IN}},$ (20)

where $v_{\nu}=V_{L,\text{ISM}}(l_{\text{dam,NV},\perp}/L_{\text{ISM}})^{1/3}$
is the turbulent velocity at $l_{\text{dam,NV},\perp}$. The IN damping rate in
this case is

$\omega_{\text{d,IN}}=\frac{\xi_{n}k_{\perp}^{2}(\delta
V_{A})^{2}}{2\nu_{ni}},$ (21)

where $\delta V_{A}=\delta B/\sqrt{4\pi\rho}$, and $k_{\perp}^{2}(\delta
V_{A})^{2}$ is the magnetic force per unit mass per unit displacement
corresponding to the sub-viscous magnetic fluctuation $\delta B$ perpendicular
to the magnetic field. Eq. (21) is different from the expression for IN
damping rate of Alfvén waves in Kulsrud & Anderson (1992), as over sub-viscous
scales, there are no Alfvén wave motions of magnetic fields. By assuming that
the Alfvénic component dominates the turbulent motion, we approximately have
$\delta V_{A}\approx v_{\nu}$. Then we obtain from Eq. (20)

$\displaystyle l_{\text{dam,IN},\perp,\text{sub-v}}$
$\displaystyle=\frac{\xi_{n}}{2}\nu_{ni}^{-\frac{1}{2}}\nu_{n}^{\frac{1}{2}}$
(22) $\displaystyle\approx 1.0\times
10^{15}~{}\text{cm}\Big{(}\frac{T}{6300~{}\text{K}}\Big{)}^{\frac{1}{4}}\Big{(}\frac{n_{e}}{0.07~{}\text{cm}^{-3}}\Big{)}^{-\frac{1}{2}}$
$\displaystyle~{}~{}~{}~{}\Big{(}\frac{n_{H}}{0.2~{}\text{cm}^{-3}}\Big{)}^{-\frac{1}{2}},$

with $\sigma(l_{\text{dam,IN},\perp,\text{sub-v}})\sim 8.5\times
10^{13}~{}$cm, as the IN damping scale of sub-viscous magnetic fluctuations
(see Fig. 2). We see that $l_{\text{dam,IN},\perp,\text{sub-v}}$ is slightly
smaller than $l_{\text{dam,NV},\perp}$ in Eq. (4). So the cutoff scale of
magnetic fluctuations is close to the cutoff scale of the kinetic energy
spectrum of the interstellar turbulence.

For the LMHD turbulence driven in the VLISM, the new regime of MHD turbulence
is also expected below the (effective) ion viscous scale. As pointed out in
Fraternale & Pogorelov (2021), the small-scale spectral flattening with the
spectral index $-1$ (see also Fig. 2) may be accounted for by the sub-viscous
magnetic fluctuations in the new regime of LMHD turbulence (Cho et al., 2002;
Xu & Lazarian, 2016).

###  4.2 Compressibility of the LMHD turbulence

In our calculations, we assume that the Alfvénic component carries most of the
turbulent energy in compressible MHD turbulence. This is supported by
compressible MHD turbulence simulations (Cho & Lazarian, 2002; Hu et al.,
2022), as well as the higher power of perpendicular magnetic fluctuations
compared to that of parallel magnetic fluctuations found in Lee & Lee (2020).
We consider strong MHD turbulence with strong nonlinear interactions between
oppositely directed Alfvén wave packets and balanced cascade (Goldreich &
Sridhar, 1995) based on the Kolmogorov magnetic energy spectrum reported in
e.g., Burlaga et al. (2018); Lee & Lee (2020). But we also note the existence
of large-scale compressive component of magnetic fluctuations with the
spectral index close to $-2$ (Fraternale & Pogorelov, 2021), which probably
reflects the discontinuities in magnetic field distribution associated with
shock/compression waves. As the conversion from Alfvén modes to compressive
modes of MHD turbulence is inefficient (Cho & Lazarian, 2002), the compressive
component is likely to originate from the turbulence injection. By combining
the in-situ data of magnetic and electron density fluctuations, Lee & Lee
(2020) found that the observations cannot be explained by the linear
magnetohydrodynamic modes alone. In addition, within the limited range of
measured frequencies/scales, it is difficult to distinguish between the
spectral indices $-5/3$ and $-2$ in observations, and thus the possibility of
the existence of weak turbulence cannot be completely excluded.

Due to the damping effects in the presence of neutrals, we find that the
measured turbulence in the VLISM is likely to be entirely of heliospheric
origin. Moreover, the turbulent energy transfer rate of the LMHD turbulence is
$\epsilon_{\text{LMHD}}\sim\rho_{i}V_{L}^{3}/L_{\perp}\approx 5.7\times
10^{-24}~{}$erg cm-3 s-1, which is much larger than
$\epsilon_{\text{ISM}}\sim\rho V_{L,\text{ISM}}^{3}/L_{\text{ISM}}\approx
1.4\times 10^{-26}~{}$erg cm-3 s-1 of the interstellar turbulence. This is
consistent with the finding in Lee & Lee (2020); Ocker et al. (2021) that the
Kolmogorov electron density spectrum measured by Voyager 1 has a significantly
higher intensity than that measured in the interstellar warm ionized medium
(Armstrong et al., 1995; Chepurnov & Lazarian, 2010). The quasiperiodic
structures seen in the magnetic field fluctuations also indicate the
heliospheric forcing of the measured turbulence in the VLISM (Fraternale &
Pogorelov, 2021). Both solar rotation and solar cycle may play an important
role in driving the heliospheric turbulence (Zank et al., 2019).

With large observational uncertainties, the transition from the magnetic
fluctuations parallel to the mean magnetic field in an earlier interval to
those transverse to the mean magnetic field in later intervals was found from
Voyager 1 and 2 measurements (Burlaga et al., 2018; Zhao et al., 2020; Burlaga
et al., 2022). Magnetic fluctuations parallel to the local magnetic field
arising from slow and fast modes in compressible MHD turbulence and pseudo-
Alfvénic modes in incompressible MHD turbulence are important for the mirror
diffusion of particles (Xu & Lazarian, 2020; Lazarian & Xu, 2021). Zank et al.
(2019) has shown that the fluctuations measured by Voyager 1 and 2 can be
explained by a transmission of fast modes from inner heliosheath with a
conversion to incompressible Alfvén waves. They further assumed that the the
outer scale of this heliosphere-originated turbulence is around 120 au, and
the interstellar turbulence will continue down to scales smaller than those
measured by Voyager. Our work, by considering the ion-neutral interactions in
the LISM region, suggests that the interstellar turbulence should have a
cutoff around $261$ au, and a new heliosphere-related turbulence (such as the
scenario described by Zank et al. (2019)) with an injection scale smaller than
$\sim 2.9\times 10^{15}$ cm ($\approx 194$ au) is needed to explain the
Voyager measurements.

##  5 Conclusions

The damping effects in the partially ionized LISM determine the range of
length scales for the existence of interstellar MHD turbulence and the LMHD
turbulence. Due to the high temperature and moderate ionization fraction in
the LISM, we find that the dominant damping mechanism of the interstellar MHD
turbulence is the NV damping. The NV damping scale of the interstellar
turbulence is about $261$ au. Below the NV damping scale of turbulent cascade,
the new regime of MHD turbulence with constant magnetic fluctuations (Lazarian
et al., 2004) is expected to rise. We find that the sub-viscous magnetic
fluctuations are cut off due to the IN damping at a scale slightly smaller
than the NV damping scale.

For the LMHD turbulence, when the injection occurs in the regime with ions
decoupled from neutrals, the turbulent cascade can not be cut off by the
damping related to partial ionization. Given the turbulent velocity at the
largest observed length scale inferred from in-situ measurements, after
applying the ion-neutral decoupling condition, we find that the LMHD
turbulence in the VLISM is sub-Alfvénic with the injected turbulent energy
smaller than the magnetic energy. With the trajectory of Voyager 1
approximately perpendicular to the background magnetic field, by assuming
anisotropic injection scale of the LMHD turbulence, we further find the upper
limit of the perpendicular injection scale $L_{\perp,\text{max}}\approx
2.9\times 10^{15}$ cm $\approx 194$ au, which is close to the NV damping scale
of the interstellar turbulence. Our estimated largest outer scale of LMHD
turbulence is comparable to the extent of the heliosphere in the upstream
direction (Pogorelov et al., 2017) and one order of magnitude smaller than the
estimate given in Burlaga et al. (2018); Lee & Lee (2020) by extrapolating the
power-law slope to the equipartition between the turbulent fluctuation and the
average magnetic field strength. Note that other considerations such as mode
conversion (Zank et al., 2019) and IBEX modeling (Zirnstein et al., 2020)
could further limit the injection scale down to tens of au.

We thank the referees for very detailed and helpful comments, which improved
the manuscript perceptibly. S.X. acknowledges inspiring discussions with Alex
Lazarian, Ethan Vishniac, and the support for this work provided by NASA
through the NASA Hubble Fellowship grant # HST-HF2-51473.001-A awarded by the
Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Incorporated, under NASA contract
NAS5-26555. H.L. acknowledges useful discussions with Fan Guo, Eric Zirnstein,
and the support by LANL LDRD program.

##  Appendix A Anisotropy of strong MHD turbulence

The theoretically established scale-dependent anisotropy of trans-Alfvénic
turbulence (Goldreich & Sridhar, 1995) and sub- and super-Alfvénic turbulence
(Lazarian & Vishniac, 1999) has been tested by MHD turbulence simulations (Cho
& Lazarian, 2002, 2003) and supported by spacecraft measurements in the solar
wind (Horbury et al., 2008; Luo & Wu, 2010; Forman et al., 2011). Here we
briefly review the anisotropy of strong Alfvénic turbulence.

The cascading rate of Alfvénic turbulence in strong MHD turbulence regime is

$\tau_{\text{cas}}^{-1}=v_{l}l_{\perp}^{-1}=V_{\text{st}}L_{\text{st}}^{-\frac{1}{3}}l_{\perp}^{-\frac{2}{3}},$
(A1)

where $V_{\text{st}}$ is the turbulent velocity at the outer scale
$L_{\text{st}}$ of strong MHD turbulence. More specifically, there is

$V_{\text{st}}=V_{A},L_{\text{st}}=l_{A}=LM_{A}^{-3}$ (A2)

for super-Alfvénic turbulence with $M_{A}>1$, and

$V_{\text{st}}=V_{L}M_{A},L_{\text{st}}=l_{\text{tran}}=LM_{A}^{2}$ (A3)

for sub-Alfvénic turbulence with $M_{A}<1$,. We note that $V_{A}$ should be
replaced by $V_{Ai}$ when the turbulence is injected in the decoupled regime.

By combining the expression of $\tau_{\text{cas}}^{-1}$ with the critical
balance relation

$\tau_{\text{cas}}^{-1}=\frac{V_{A}}{l_{\|}},$ (A4)

we can obtain the anisotropic scaling relation of strong Alfvénic turbulence,

$l_{\|}=\frac{V_{A}}{V_{\text{st}}}L_{\text{st}}^{\frac{1}{3}}l_{\perp}^{\frac{2}{3}}.$
(A5)

It shows that smaller-scale turbulent eddies are more elongated along the
local magnetic field.

##  Appendix B Damping scales of MHD turbulence cascade in a partially ionized
medium

Under the consideration of both IN and NV damping effects in a partially
ionized medium, Xu et al. (2015, 2016); Xu & Lazarian (2017) derived the
general expression of the damping rate of Alfvénic turbulence. Here we briefly
review its approximate form in different coupling regimes.

In the weakly coupled regime, there is only the IN damping, with the damping
rate

$\omega_{d}=\frac{\nu_{in}}{2}.$ (B1)

For the MHD turbulence injected in the decoupled regime, as the cascading rate
is always larger than $\nu_{in}$ and thus larger than $\omega_{d}$, the
cascade is not cut off by the damping in a partially ionized medium.

In the strongly coupled regime, the damping rate can be approximately written
as

$\omega_{d}=\frac{\xi_{n}}{2}\Big{(}k^{2}\nu_{n}+\frac{k_{\perp}^{2}(\delta
V_{A})^{2}}{\nu_{ni}}\Big{)}=\frac{\xi_{n}}{2}\Big{(}k^{2}\nu_{n}+\frac{k_{\|}^{2}V_{A}^{2}}{\nu_{ni}}\Big{)},$
(B2)

where we assume that the magnetic fluctuations are mainly induced by Alfvénic
turbulence and apply the critical balance relation for strong MHD turbulence,
$k_{\perp}\delta V_{A}=k_{\perp}v_{k}=k_{\|}V_{A}$. Here $v_{k}$ is the
turbulent velocity at wavenumber $k$. The first and second terms of
$\omega_{d}$ correspond to NV and IN damping, respectively.

For the MHD turbulence injected in the strongly coupled regime, when the
damping rate exceeds the cascading rate, MHD turbulence cascade is damped. We
consider that the damping scale is in the strong MHD turbulence regime. By
comparing Eq. (B2) with Eq. (A1), we find the damping scale

$l_{\text{dam,NV},\perp}=\Big{(}\frac{\xi_{n}}{2}\Big{)}^{\frac{3}{4}}\nu_{n}^{\frac{3}{4}}L_{\text{st}}^{\frac{1}{4}}V_{\text{st}}^{-\frac{3}{4}}$
(B3)

when the NV damping dominates over the IN damping, where we assume $k\sim
k_{\perp}$, and

$l_{\text{dam,IN},\perp}=\Big{(}\frac{2\nu_{ni}}{\xi_{n}}\Big{)}^{-\frac{3}{2}}L_{\text{st}}^{-\frac{1}{2}}V_{\text{st}}^{\frac{3}{2}}$
(B4)

in the opposite case. If the MHD turbulence is damped due to neutral viscosity
in the strongly coupled regime, there is the so-called new regime of MHD
turbulence on scales below the NV damping scale (Lazarian et al., 2004). If
the damping of MHD turbulence is caused by neutral-ion decoupling,
hydrodynamic turbulent cascade can happen in neutrals that are decoupled from
ions on scales smaller than the IN damping scale (Xu et al., 2015; Burkhart et
al., 2015).

## References

  * Armstrong et al. (1995) Armstrong, J. W., Rickett, B. J., & Spangler, S. R. 1995, ApJ, 443, 209, doi: 10.1086/175515
  * Balsara (1996) Balsara, D. S. 1996, ApJ, 465, 775, doi: 10.1086/177462
  * Breitschwerdt et al. (2017) Breitschwerdt, D., Tautz, R. C., & de Avillez, M. A. 2017, Impact of Supernovae on the Interstellar Medium and the Heliosphere, ed. A. W. Alsabti & P. Murdin (Springer International Publishing AG), 2359, doi: 10.1007/978-3-319-21846-5_18
  * Burkhart et al. (2015) Burkhart, B., Lazarian, A., Balsara, D., Meyer, C., & Cho, J. 2015, ApJ, 805, 118, doi: 10.1088/0004-637X/805/2/118
  * Burlaga et al. (2018) Burlaga, L. F., Florinski, V., & Ness, N. F. 2018, ApJ, 854, 20, doi: 10.3847/1538-4357/aaa45a
  * Burlaga et al. (2022) Burlaga, L. F., Ness, N. F., Berdichevsky, D. B., et al. 2022, ApJ, 932, 59, doi: 10.3847/1538-4357/ac658e
  * Chepurnov & Lazarian (2010) Chepurnov, A., & Lazarian, A. 2010, ApJ, 710, 853, doi: 10.1088/0004-637X/710/1/853
  * Chepurnov et al. (2010) Chepurnov, A., Lazarian, A., Stanimirović, S., Heiles, C., & Peek, J. E. G. 2010, ApJ, 714, 1398, doi: 10.1088/0004-637X/714/2/1398
  * Cho & Lazarian (2002) Cho, J., & Lazarian, A. 2002, Physical Review Letters, 88, 245001, doi: 10.1103/PhysRevLett.88.245001
  * Cho & Lazarian (2003) —. 2003, MNRAS, 345, 325, doi: 10.1046/j.1365-8711.2003.06941.x
  * Cho et al. (2002) Cho, J., Lazarian, A., & Vishniac, E. T. 2002, ApJ, 566, L49, doi: 10.1086/339453
  * Cho & Vishniac (2000) Cho, J., & Vishniac, E. T. 2000, ApJ, 539, 273, doi: 10.1086/309213
  * Crutcher et al. (2010) Crutcher, R. M., Wandelt, B., Heiles, C., Falgarone, E., & Troland, T. H. 2010, ApJ, 725, 466, doi: 10.1088/0004-637X/725/1/466
  * Dialynas et al. (2022) Dialynas, K., Krimigis, S. M., Decker, R. B., et al. 2022, Space Sci. Rev., 218, 21, doi: 10.1007/s11214-022-00889-0
  * Draine et al. (1983) Draine, B. T., Roberge, W. G., & Dalgarno, A. 1983, ApJ, 264, 485, doi: 10.1086/160617
  * Forman et al. (2011) Forman, M. A., Wicks, R. T., & Horbury, T. S. 2011, ApJ, 733, 76, doi: 10.1088/0004-637X/733/2/76
  * Fraternale & Pogorelov (2021) Fraternale, F., & Pogorelov, N. V. 2021, ApJ, 906, 75, doi: 10.3847/1538-4357/abc88a
  * Fraternale et al. (2022) Fraternale, F., Adhikari, L., Fichtner, H., et al. 2022, arXiv e-prints, arXiv:2207.14115.  https://arxiv.org/abs/2207.14115
  * Frisch et al. (2011) Frisch, P. C., Redfield, S., & Slavin, J. D. 2011, ARA&A, 49, 237, doi: 10.1146/annurev-astro-081710-102613
  * Gaensler et al. (2011) Gaensler, B. M., Haverkorn, M., Burkhart, B., et al. 2011, Nature, 478, 214, doi: 10.1038/nature10446
  * Galtier et al. (2000) Galtier, S., Nazarenko, S. V., Newell, A. C., & Pouquet, A. 2000, Journal of Plasma Physics, 63, 447, doi: 10.1017/S0022377899008284
  * Giacalone & Jokipii (2015) Giacalone, J., & Jokipii, J. R. 2015, ApJ, 812, L9, doi: 10.1088/2041-8205/812/1/L9
  * Goldreich & Sridhar (1995) Goldreich, P., & Sridhar, S. 1995, ApJ, 438, 763, doi: 10.1086/175121
  * Goldreich & Sridhar (1997) —. 1997, ApJ, 485, 680, doi: 10.1086/304442
  * Holzer (1989) Holzer, T. E. 1989, ARA&A, 27, 199, doi: 10.1146/annurev.aa.27.090189.001215
  * Horbury et al. (2008) Horbury, T. S., Forman, M., & Oughton, S. 2008, Phys. Rev. Lett., 101, 175005, doi: 10.1103/PhysRevLett.101.175005
  * Hu et al. (2022) Hu, Y., Lazarian, A., & Xu, S. 2022, MNRAS, 512, 2111, doi: 10.1093/mnras/stac319
  * Izmodenov & Alexashov (2020) Izmodenov, V. V., & Alexashov, D. B. 2020, A&A, 633, L12, doi: 10.1051/0004-6361/201937058
  * Krimigis et al. (2013) Krimigis, S. M., Decker, R. B., Roelof, E. C., et al. 2013, Science, 341, 144, doi: 10.1126/science.1235721
  * Krstic & Schultz (1998) Krstic, P., & Schultz, D. 1998, Atomic and Plasma-Material Interaction Data for Fusion, 8
  * Kulsrud & Pearce (1969) Kulsrud, R., & Pearce, W. P. 1969, ApJ, 156, 445, doi: 10.1086/149981
  * Kulsrud & Anderson (1992) Kulsrud, R. M., & Anderson, S. W. 1992, ApJ, 396, 606, doi: 10.1086/171743
  * Lazarian (2006) Lazarian, A. 2006, ApJ, 645, L25, doi: 10.1086/505796
  * Lazarian & Opher (2009) Lazarian, A., & Opher, M. 2009, ApJ, 703, 8, doi: 10.1088/0004-637X/703/1/8
  * Lazarian & Vishniac (1999) Lazarian, A., & Vishniac, E. T. 1999, ApJ, 517, 700, doi: 10.1086/307233
  * Lazarian et al. (2004) Lazarian, A., Vishniac, E. T., & Cho, J. 2004, ApJ, 603, 180, doi: 10.1086/381383
  * Lazarian & Xu (2021) Lazarian, A., & Xu, S. 2021, ApJ, 923, 53, doi: 10.3847/1538-4357/ac2de9
  * Lazarian et al. (2018) Lazarian, A., Yuen, K. H., Ho, K. W., et al. 2018, ApJ, 865, 46, doi: 10.3847/1538-4357/aad7ff
  * Lee & Lee (2020) Lee, K. H., & Lee, L. C. 2020, ApJ, 904, 66, doi: 10.3847/1538-4357/abba20
  * Lithwick & Goldreich (2001) Lithwick, Y., & Goldreich, P. 2001, ApJ, 562, 279, doi: 10.1086/323470
  * López-Barquero et al. (2017) López-Barquero, V., Xu, S., Desiati, P., et al. 2017, ApJ, 842, 54, doi: 10.3847/1538-4357/aa74d1
  * Luo & Wu (2010) Luo, Q. Y., & Wu, D. J. 2010, ApJ, 714, L138, doi: 10.1088/2041-8205/714/1/L138
  * MATLAB (2021) MATLAB. 2021, MATLAB and Statistics Toolbox Release 2021b (Natick, Massachusetts: The MathWorks Inc.)
  * Mouschovias et al. (2011) Mouschovias, T. C., Ciolek, G. E., & Morton, S. A. 2011, MNRAS, 415, 1751, doi: 10.1111/j.1365-2966.2011.18817.x
  * Ocker et al. (2021) Ocker, S. K., Cordes, J. M., Chatterjee, S., et al. 2021, Nature Astronomy, 5, 761, doi: 10.1038/s41550-021-01363-7
  * Pattle et al. (2022) Pattle, K., Fissel, L., Tahani, M., Liu, T., & Ntormousi, E. 2022, arXiv:2203.11179, arXiv:2203.11179.  https://arxiv.org/abs/2203.11179
  * Pogorelov et al. (2017) Pogorelov, N. V., Heerikhuisen, J., Roytershteyn, V., et al. 2017, ApJ, 845, 9, doi: 10.3847/1538-4357/aa7d4f
  * Pudritz (1990) Pudritz, R. E. 1990, ApJ, 350, 195, doi: 10.1086/168373
  * Shu (1992) Shu, F. H. 1992, The physics of astrophysics. Volume II: Gas dynamics.
  * Slavin & Frisch (2008) Slavin, J. D., & Frisch, P. C. 2008, A&A, 491, 53, doi: 10.1051/0004-6361:20078101
  * Soler et al. (2013) Soler, R., Carbonell, M., Ballester, J. L., & Terradas, J. 2013, ApJ, 767, 171, doi: 10.1088/0004-637X/767/2/171
  * Spangler et al. (2011) Spangler, S. R., Savage, A. H., & Redfield, S. 2011, ApJ, 742, 30, doi: 10.1088/0004-637X/742/1/30
  * Stone et al. (2013) Stone, E. C., Cummings, A. C., McDonald, F. B., et al. 2013, Science, 341, 150, doi: 10.1126/science.1236408
  * Swaczyna et al. (2020) Swaczyna, P., McComas, D. J., Zirnstein, E. J., et al. 2020, ApJ, 903, 48, doi: 10.3847/1538-4357/abb80a
  * Vranjes & Krstic (2013) Vranjes, J., & Krstic, P. S. 2013, A&A, 554, A22, doi: 10.1051/0004-6361/201220738
  * Xu & Lazarian (2016) Xu, S., & Lazarian, A. 2016, ApJ, 833, 215, doi: 10.3847/1538-4357/833/2/215
  * Xu & Lazarian (2017) —. 2017, New Journal of Physics, 19, 065005, doi: 10.1088/1367-2630/aa6ec9
  * Xu & Lazarian (2020) —. 2020, ApJ, 894, 63, doi: 10.3847/1538-4357/ab8465
  * Xu et al. (2015) Xu, S., Lazarian, A., & Yan, H. 2015, ApJ, 810, 44, doi: 10.1088/0004-637X/810/1/44
  * Xu et al. (2016) Xu, S., Yan, H., & Lazarian, A. 2016, ApJ, 826, 166, doi: 10.3847/0004-637X/826/2/166
  * Xu & Zhang (2017) Xu, S., & Zhang, B. 2017, ApJ, 835, 2, doi: 10.3847/1538-4357/835/1/2
  * Yuen et al. (2022) Yuen, K. H., Ho, K. W., Law, C. Y., Chen, A., & Lazarian, A. 2022, arXiv:2204.13760, arXiv:2204.13760.  https://arxiv.org/abs/2204.13760
  * Zank (2015) Zank, G. P. 2015, ARA&A, 53, 449, doi: 10.1146/annurev-astro-082214-122254
  * Zank et al. (2013) Zank, G. P., Heerikhuisen, J., Wood, B. E., et al. 2013, ApJ, 763, 20, doi: 10.1088/0004-637X/763/1/20
  * Zank et al. (2019) Zank, G. P., Nakanotani, M., & Webb, G. M. 2019, ApJ, 887, 116, doi: 10.3847/1538-4357/ab528c
  * Zaqarashvili et al. (2011) Zaqarashvili, T. V., Khodachenko, M. L., & Rucker, H. O. 2011, A&A, 529, A82, doi: 10.1051/0004-6361/201016326
  * Zhao et al. (2020) Zhao, L. L., Zank, G. P., & Burlaga, L. F. 2020, ApJ, 900, 166, doi: 10.3847/1538-4357/ababa2
  * Zirnstein et al. (2020) Zirnstein, E. J., Giacalone, J., Kumar, R., et al. 2020, ApJ, 888, 29, doi: 10.3847/1538-4357/ab594d


# Macroscopic quantum tunneling in PdO nanoparticles

Francisco Ascencio  C. Reyes-Damián  Roberto Escudero
Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de
México
Mexico City, 04510 Mexico
<EMAIL_ADDRESS>
###### Abstract

We studied the physical behavior of PdO nanoparticles at low temperatures,
which presents an unusual behavior clearly related to macroscopic quantum
tunneling. The samples show a tetragonal single phase with P42/mmc space
group. Most importantly, the particle size was estimated at about 5.07 $\pm$
1.97 nm. Appropriate techniques were used to determine the characteristic of
these nanoparticles. The most important aspect of this study is the magnetic
characterization performed at low temperatures. It shows a peak at 50 K in
zero field cooling mode (ZFC) that corresponds to the Blocking temperature
($T_{B}$). These measurements in ZFC and field cooling (FC) indicate that the
peak behavior is due to different relaxation times of the asymmetrical
barriers when the electron changes from a metastable state to another. Below
$T_{B}$ in FC mode, the magnetization decreases with temperature until 36 K;
this temperature is the crossover temperature $T_{Cr}$ related to the
anisotropy of the barriers, indicative of macroscopic quantum tunneling.

##  1 Introduction

The study of the magnetic properties of materials has been a topic of great
interest in nanoparticles because of the unusual new properties. For instance,
molecular magnets offer interesting characteristics useful to different
systems. In general, compounds formed by nanostructures could be one of the
most simple manners to observe the effects of macroscopic quantum tunneling ,
MQT.

One relevant characteristic seen in ferromagnetic materials is that their
magnetic characteristics change when the size is reduced to a limit; this is
the critical value. In this limit, the magnetic behavior changes to
superparamagnetism, SP. At that point, the thermal assisted SP can be blocked
by random thermal fluctuations due to spin orientations and could be frozen as
the temperature decreases. At low temperatures, the thermal fluctuations are
frozen and consequently become trapped and can not surpass the magnetic
barriers of metastable states. At this point, quantum tunneling magnetic
effects, MQT takes place because nanoparticles show a dramatic increase in the
surfaced to volume ratio; the effect occurs because of anisotropy of the
barriers. At this point, MQT clearly is observed, and also other
characteristics as spin glass, behavior, paramagnetic, and or
superparamagnetism characteristics.

Experimentally some of those features related to the MQT can be seen at low
temperatures: the blocking and crossover temperatures. Both features are quite
dependent and related to the anisotropy of the arising energy barriers in the
metastable states that allow the spin transference between two states.

The characteristics and properties observed in nanomaterials are intrinsic
properties of electronic interaction between them. In 3d and 4d series atoms,
band structure changes dramatically, changing the surface because of the
narrowing of the bands and distances. One aspect that is important in the
study of the cumulous of nanoparticles. Accordingly, to carry out and
understand this behavior is quite important to know the coordination of the
structures formed by the cumulous.

Pd samples with macroscopic dimensions behave quite differently as nanosize
scale; this behavior was observed by many authors [1, 2, 3, 4].

When nonmagnetic material is in the critical size [5, 6] studies performed in
Metal-Oxygen compounds show that the magnetic behavior is strongly related to
oxygen vacancies, substitutional and interstitial defects, as well as surface
defects of the nanomaterials.[7, 8, 9] However, one of the most unusual
magnetic characteristics in nanomaterials is the observation of properties
that can be related to macroscopic quantum tunneling of the magnetization
(QTMs or MQT) [10, 11, 12, 13, 14, 15, 16, 17, 18]. The MQT has been observed
mainly in nanoparticles compounds of metal oxides; the two important
characteristics or features can be observed at low temperatures, they are the
blocking $T_{B}$ and the cross-over temperatures, $T_{Cr}$.[19, 20, 21, 22,
23]

It is important to mention that the study of metallic oxides is attractive
because of numerous applications that can be used in different fields; i.e.
catalysis, as precursors to the synthesis of superconductors, electronic
devices, biomedical applications, and gas sensors [24, 25, 26, 27, 28].
Additionally these studies become relevants when the compounds are low-
dimensional materials as in thin films, wires, or nanoparticles. Nowadays, it
is well known that size, shape, and surface structure, and physicochemical
properties as in optical, electronics, and magnetism behaves differently than
in bulk. Related to this study, Palladium Oxide, PdO, nanoparticles have
attracted attention because of the uniques physicochemical properties. PdO in
bulk is a nonmagnetic compound [29]. It presents a tetragonal symmetry,
Cooperite, in which Pd atoms occupy D2h and O atoms occupy D2d positions [30].
In the structure, the Pd atoms are coordinated by 4 oxygen atoms in a planar
manner, while O atoms are coordinated with 4 Pd atoms, forming a tetrahedron.
PdO is a P-type semiconductor, with a direct gap. In previous studies by many
authors has been observed that the bandgap is close to 2 eV [31, 32] which has
industrial interest mainly because its photocatalytic characteristics [33]
used in the oxidation of CH4 [34], and in sensors [35]. Actually , some
methods have been used to synthesize PdO nanostructures; as sol-
gel[36],thermo-hydrolyzation assisted by microwave [37]. Related to this work
the synthesize was performed using a simple method that a continuation will be
described.

##  2 Experimental Details

PdO nanoparticles were synthesized by an alkali salt method, synthesized with
concentrations of 0.125 g of Pd(NO3)2 * 2 H2O (Sigma Aldrich, 99 %), 0.125 g
of Li2CO3 (Ripley Scientific) and 2.5 g of NaCl (Sigma Aldrich) were mixed in
an agate mortar and milled for 30 minutes until obtain a fine powder. Samples
were heated at 400 ∘C in a furnace at atmospheric pressure. Finally, the
obtained samples were washed five times with distilled water in order to
eliminate residues of the reaction.

###  2.1 Characterization

Characterized of the samples were carried out by X$-$ray diffraction (XRD)
performed with a Bruker (D8 Advance) and Cu$-$Kα radiation, determined from
2$\Theta$ range of 20${}^{\circ}-120^{\circ}$. The phase was identified using
the Crystallographic Open Database (COD) and crystalline structure refined
with Rietveld method using the BGMN program [38] and the graphical interface
Profex. Peak profiles were modeled to determine the crystal size using the
Debye$–$Scherrer formula [39]. Transmission electron microscopy (TEM), JEOL
JEM-2010F FasTem microscope operated at 200 kV. The samples for TEM were
prepared, depositing a drop of the solution into the mesh Cu grid. The Raman
spectra were recorded in a Thermo Scientific confocal microscope equipped with
micro$-$Raman using three different excitation sources: 532 nm (2.33 eV), 633
nm (1.95 eV), and 780nm (1.58 eV); measurements were carried out in the
50$–$3000 cm-1 range. Optical properties were characterized by UV$–$visible
spectroscopy with a Thermo Evolution 220 spectrophotometer. Finally,
magnetization as a function of temperature (M(T)) and magnetization as a
function of the magnetic field (M(H)) were determined using a Quantum Design
magnetometer (MPMS).

##  3 Results and Discussion

###  3.1 X-Ray Diffraction

Figure 1: X-ray diffraction pattern measured with CuKα radiation and bragg
reflections of reference $96-100-9032$ taken from the Crystallographic Open
Database. The right inset shows the Rietveld refinement of PdO nanoparticles
diffraction pattern. The bottom line is the difference between experimental
and calculated pattern. A photograph of the powder sample is also shown in the
left inset.

The crystallographic structure of the PdO nanoparticles was determined by
X-ray diffraction (XRD). XRD pattern is shown in the figure 1, there we show
the XDR measurements of PdO (nano)powders in which the PdO tetragonal phase
was detected. The right inset of the figure 1 shows the Rietveld refinement of
the sample; the red line is the experimental data, in the inset, the black
line is the calculated pattern. At the bottom of the diagram, the differences
between the experimental and calculated points are shown in gray color.
Vertical blue marks at the bottom are displayed and correspond to the Bragg
positions for the phase. Finally, the background was modeled with a Lagrangian
polynomial. From Rietveld refinement, we see the phase identified and modeled
with tetragonal PdO phase according to the Crystallographic Open Database
(COD) number $96-100-9032$ described by the space group P42/mmc. The average
crystal size was 10.31 $\pm$ 0.83 nm. Cell parameters were calculated the
resulting lattice parameters were a= 0.3042 $\pm$ 0.0001 and c = 0.5356 $\pm$
0.0003 nm, according to the space group # 131. Finally, Rietveld residues
were: Rwp=6.91%, Rexp=6.51% and Rwp/Rexp = 1.06.

###  3.2 Transmission Electron Microscopy

Transmission electron microscopy was carried out to obtain structural and
morphological information of the samples. In the figure 2 (a), HRTEM
micrograph is presented. It is observed that the nanoparticles have an
irregular polyhedron shape. The first particle of around 5 nm presents the
interplanar distances d1 = 2.195 Å and d2 = 2.523 Å, with an angle between
planes of $\theta$= 52.07∘. These reflections correspond to the (1 1 0) and (0
1 1) planes oriented along the [-1 1 -1] axis zone from the FFT lattice
fringes are resolved. Figure 2 (b) shows another nanoparticle with 4 nm of
diameter with the FFT, is seen only the plane (0 1 1). In the figure 2 (c),
the nanoparticles average diameter were 5.07$\pm$1.97 nm. measured using a
total of 200 particles. The structure of these nanoparticles was confirmed
through electron diffraction, figure 2 (d), the pattern was matched with the
PdO crystal structure. The reflections were indexed as the (0 1 0), (0 1 1),
(0 1 2), (1 1 2), and (0 1 3) crystallographic planes, according to COD, #
$96-100-9032$. The optical and magnetic properties of palladium oxide
nanoparticles are related to grain boundaries or defects in the nanoparticle
structure; hence, the HRTEM images permit a clear observation of their fine
features. With TEM studies, it can be observed that the produced nanoparticles
are single crystals without the presence of twins or grain boundaries.

Figure 2: (a)HRTEM image of a very small PdO nanoparticles with the
corresponding FFT, (b) another HRTEM image for a representative nanoparticle
with the corresponding FFT,(c) the nanoparticles size distribution (d)
electron diffraction pattern.

###  3.3 Raman and UV-visible Spectroscopy

Raman spectra were collected at room temperature from the sample powders to
obtain detailed structural information related to normal vibrational modes of
the samples with three wavelengths $\lambda$= 532, 633, and 780 nm or
excitation energies of 2.33, 1.95, and 1.59 eV, in order to observe resonance
phenomena.

Assuming that the PdO compound has a tetragonal symmetry with space group
P42/mmc (#131) and point group D4h, and according to the factor group
analysis, this compound will display two vibrational Raman active modes (B1g
and Eg). The remaining normal modes of vibration are located in the IR region
(A2u, 2Eu) and a silence mode B2u [40, 32].

Figure 3: Full Raman spectra of PdO nanoparticles at room temperature for
three excitation energies 2.33, 1.95 and 1.59 eV

In the figure 3, the normalized Raman spectra of the samples measured at room
temperature with three different wavelengths (excitation energy) are shown. It
is observed that the spectra show some Raman peaks below 1500 cm-1, in Table 1
a list of the peaks are presented for the three excitation energies.

A careful analysis of the Raman spectra (Figure 4) for each energy shows that
the PdO nanoparticles have three main peaks related to B1g between the
frequencies: [620.62$-$630.83] cm-1, peaks between the frecuencies
[425.09$-$401.66] cm-1 which corresponds to Eg and finally the vibrational
mode between [273.01$-$278.24 ] cm-1 correspond to X8 mode [36]. Besides many
other peaks related to second order processes overtones, prohibited modes and
combinations of normal modes were observed [41]. It is clear in the figure 4
that when the sample was measured with the wavelength of 633 nm (1.95 eV) the
peaks were better defined and new peaks appeared at different frequencies
which could be the resonance effects. No peaks related to vibrations of
compounds containing carbon were detected. The absence of those peaks is
indicative of the purity of the samples.

Figure 4: Comparison of Raman spectra of PdO nanoparticles around the main peak for different excitation energy Table 1: Raman peaks obtained from Fig. 3. 2.33 eV |  1.95 eV |  1.59 eV
---|---|---
273.01 |  277.73 |  278.24
425.09 |  331.93 |  401.66
620.62 |  409.36 |  519.3
680.02 |  552.02 |  630.83
1062.13 |  628.80 |  661.04
1257.21 |  695.28 |  911.74
|  1057.43 |
|  1142.61 |
|  1254.29 |

The optical properties of the single crystal particles of PdO were measured.
Figure 5 (a) shows the optical absorption spectra of the sample perform at
room temperature. In this spectrum, we observe a strong absorption intensity
in the near UV and violet region. From this measurement, the optical band gap
was estimated from the absorbance spectrum using the Tauc$-$plot method [42,
43, 44]. Considering a direct transition, the bandgap value was determined to
be 1.90 eV as presented in the figure 5 (b). The estimated bandgap values are
in agreement with values proposed by [45] for PdO thin films, also in
theoretical studies by DFT, close values have been obtained [46]. In different
studies of PdO, some values have been found, which are in the range of
[0.8$-$2.67] eV. The discrepancy of the values could be related to the
structural defects at the surface, such as oxygen vacancies and confinement
effects related to particle size reduction.

Figure 5:  (a) Optical absorption of the PdO nanoparticles, (b) Tauc plot for
direct transition. The bandgap was estimated as 1.9 eV

###  3.4 Magnetism

The magnetization M(T) as function of temperature for PdO nanoparticles was
determined with an applied magnetic field of 1000 Oe in two modes of
magnetization, as usual; zero field cooling (ZFC) and field cooling (FC) modes
as can be see in the figure 6. The difference between ZFC and FC curves
indicates a large anisotopic field.

Figure 6:  Magnetization of PdO nanoparticles as function of temperature
measured at 1000 Oe in ZFC and FC modes. The arrows marks the blocking
temperature ($T_{B}$) and the crossover temperature ($T_{Cr}$)

In additionl we performed measurements of the inverse magnetization versus
temperature which gives additional information about the behavior shown figure
7. A clear paramgnetic behavior is seen from room temperature to about 100 K.
Below this temperature an incipient ferromagnetism araises. The para-
ferromagnetic transition has been observed in other nanoparticles and the
transition was associated with oxygen vacancies [47]. At low temperature the
Curie Weiss temperature ($\theta_{C-W}$) behavior is very small, fitting gives
values between -1.9 K or to about 4.5 K. However, additional structure at
around 50 K, is seen in figure 6. This characteristic we related to the
metastable states when the nanoparticles are transported from one metastable
state to the other, dependding of the applied magnetic field. We related this
characteristic or behavior with a macroscopic quantum tunneling [48, 49]. Two
features give information of the magnetization; the blocking temperature
$T_{B}$ [50] and the crossover temperature, $T_{Cr}$. The Fig. 6 shows those
characteristics temperatures at 50 K and 36 K respectively. Knowing $T_{B}=36$
K and the mean volume of the nanoparticles
($V=6.54$1\text{\times}{10}^{-20}\text{\,}\mathrm{c}$m^{3}$). The magnetic
anisotropy constant ($K$) can be calculated by the the formula 1 [51], were
$k_{B}$ is the Boltzman constant.

$K=25k_{B}T_{B}V^{-1}$ (1)

The value for $K$ is 1.89$1\text{\times}{10}^{6}\text{\,}\mathrm{e}$rg/cm3 is
not comparable with bulk PdO because this last is not magnetic.

Figure 7:  A paramagnetic behavior was determined in the temperature range 100
- 300 of temperature with the inverse of M as function of temperature. For
temperatures between 50-100 Curie-Weiss fits gives a small $\theta_{C-W}$.
Inset show the low temperature-depend of 1/M in order to appreciate the range
of values for $\theta_{C-W}$

Below $T_{B}$ the magnetization decrease with the temperature until 36 K, this
temperature is the crossover temperature and separate the thermally activated
regime $T_{Cr}$ from the quantum tunneling regime [13]. Below $T_{Cr}$ = 36 K,
MQT effects take place and one can re-observed superparamgnetic state even
when the thermal energy is smaller than the magnetic energy barrier [23, 48].

Figure 8: Magnetization of PdO nanoparticles as function of H at different
temperatures. Inset shows the low field-dependent magnetization of the
nanoparticles. Hysteresis loop shows small hysteresis at 40 K in the blocked
state

The magnetization of the nanoparticles was measure as a function of magnetic
field H in order to confirm the superparamagnetism state below $T_{Cr}$ and
above $T_{B}$. The field-dependent of magnetization was measured using three
different temperatures, see figure 8. At the lowest temperature, 5 K, there is
no hysteresis, and the large magnetization values are related to
superparamagnetic characteristics. This magnetic behavior corresponds with the
expected characteristic in a quantum tunneling regime ($T<T_{Crossover}$). The
value of the magnetization in the blocked state (40K) is less than the value
of the magnetization measured at 5 K. Measurements present small hysteresis
and a remanent magnetization Mr=$6.57\text{\times}{10}^{-4}$ emu/g with non
symmetric coercive field HC+=82 Oe and HC-=-128 Oe related with the possible
coexistence antiferromagnetic state. Above the blocking temperature, 55 K, the
behavior is superparamagnet, and the hysteresis is negligible. We assume that
superparamagnetism is due to the small size of the particle ($\sim 5$ nm).

##  4 Conclusions

In summary, this work presented a magnetic study of PdO nanoparticles in which
we observed macroscopic quantum tunneling phenomena due to very small
nanoparticles. The particle size was estimated at about 5.07 $\pm$ 1.97 nm
from the TEM measurements. The Samples were analyzed by transmission electron
microscopy in which the irregular shapes of the nanoparticles were observed.
Using UV-VIS and Raman spectroscopy, the sample was analyzed
spectroscopically. For the Raman studies, three excitation energies were used:
2.33, 1.95, and 1.59 eV ($\lambda$= 532, 633 and 780 nm), and observed the
main peaks of PdO, besides the resonance phenomena, were observed. The UV-VIS
measurements were performed, the behavior shows a strong absorption near to UV
region. The bandgap value was estimated as 1.90 eV using Tauc-plot method.
Finally, the most important aspect of this study is that we found a process
related to macroscopic quantum tunneling of magnetization at low temperatures.

##  5 acknowledgements

Thanks to M.Sc. A. Bobadilla for the He supply. To R. Hernandez for the
technical assistance in TEM. To A. Pompa and A. Lopez for technical support.
We acknowledge DGAPA-PAPIIT Grant No. IT101920, F. Ascencio thanks to the
DGAPA-UNAM, for the support through the Postdoctoral Scholarship

## References

  * [1] Y. Oba, T. Sato, T. Shinohara, Physical Review B 78(22), 224417 (2008)
  * [2] T. Shinohara, T. Sato, T. Taniyama, Physical Review Letters 91(19), 197201 (2003)
  * [3] C. Xiao, H. Ding, C. Shen, T. Yang, C. Hui, H.J. Gao, The Journal of Physical Chemistry C 113(31), 13466 (2009)
  * [4] Y.T. Jeon, G.H. Lee, Journal of Applied Physics 103(9), 094313 (2008)
  * [5] A. Cox, J. Louderback, S. Apsel, L. Bloomfield, Physical Review B 49(17), 12295 (1994)
  * [6] G.L. Nealon, B. Donnio, R. Greget, J.P. Kappler, E. Terazzi, J.L. Gallani, Nanoscale 4(17), 5244 (2012)
  * [7] D. Gao, J. Zhang, J. Zhu, J. Qi, Z. Zhang, W. Sui, H. Shi, D. Xue, Nanoscale Research Letters 5(4), 769 (2010)
  * [8] K. Punia, G. Lal, S.K. Barbar, S.N. Dolia, P.A. Alvi, S. Dalela, S. Kumar, Vacuum 184, 109921 (2021)
  * [9] B. Choudhury, A. Choudhury, Journal of Applied Physics 114(20), 203906 (2013)
  * [10] E.M. Chudnovsky, L. Gunther, Physical review letters 60(8), 661 (1988)
  * [11] E.M. Chudnovsky, Journal of applied physics 73(10), 6697 (1993)
  * [12] B. Barbara, W. Wernsdorfer, L. Sampaio, J. Park, C. Paulsen, M. Novak, R. Ferré, D. Mailly, R. Sessoli, A. Caneschi, et al., Journal of magnetism and magnetic materials 140, 1825 (1995)
  * [13] X. Zhang, J. Hernandez, J. Tejada, R. Ziolo, Physical Review B 54(6), 4101 (1996)
  * [14] R. Kodama, Journal of magnetism and magnetic materials 200(1-3), 359 (1999)
  * [15] L. Thomas, B. Barbara, Journal of low temperature physics 113(5), 1055 (1998)
  * [16] B. Barbara, L. Gunther, Physics World 12(3), 35 (1999)
  * [17] W. Wernsdorfer, E.B. Orozco, B. Barbara, A. Benoit, D. Mailly, N. Demoncy, H. Pascard, O. Kubo, H. Nakano, IEEE transactions on magnetics 34(4), 973 (1998)
  * [18] B. Barbara, W. Wernsdorfer, Current Opinion in Solid State and Materials Science 2(2), 220 (1997)
  * [19] X. Zhang, J. Tejada, Journal of Applied Physics 75(10), 5637 (1994)
  * [20] R. Kodama, C. Seaman, A. Berkowitz, M. Maple, Journal of Applied Physics 75(10), 5639 (1994)
  * [21] X. Zhang, R. Ziolo, E. Kroll, X. Bohigas, J. Tejada, Journal of magnetism and magnetic materials 140, 1853 (1995)
  * [22] M. Ibrahim, S. Darwish, M. Seehra, Physical Review B 51(5), 2955 (1995)
  * [23] C. Xiao, J. Zhang, J. Xu, W. Tong, B. Cao, K. Li, B. Pan, H. Su, Y. Xie, Scientific reports 2(1), 1 (2012)
  * [24] J.C. Védrine, ChemSusChem 12(3), 577 (2019)
  * [25] R.J. Cava, Journal of the American Ceramic Society 83(1), 5 (2000)
  * [26] M. Greiner, M. Helander, Z. Wang, W.M. Tang, J. Qiu, Z. Lu, Applied Physics Letters 96(21), 103 (2010)
  * [27] M.A. Shahbazi, L. Faghfouri, M.P. Ferreira, P. Figueiredo, H. Maleki, F. Sefat, J. Hirvonen, H.A. Santos, Chemical Society Reviews 49(4), 1253 (2020)
  * [28] D. Nunes, A. Pimentel, A. Gonçalves, S. Pereira, R. Branquinho, P. Barquinha, E. Fortunato, R. Martins, Semiconductor Science and Technology 34(4), 043001 (2019)
  * [29] R. Ahuja, S. Auluck, B. Johansson, M. Khan, Physical Review B 50(4), 2128 (1994)
  * [30] K. Hass, A. Carlsson, Physical Review B 46(7), 4246 (1992)
  * [31] T. Kumari, R. Gopal, A. Goyal, J. Joshi, Journal of Inorganic and Organometallic Polymers and Materials 29(2), 316 (2019)
  * [32] R. Bardhan, H.F. Zarick, A. Schwartzberg, C.L. Pint, The Journal of Physical Chemistry C 117(41), 21558 (2013)
  * [33] S. Veziroglu, J. Hwang, J. Drewes, I. Barg, J. Shondo, T. Strunskus, O. Polonskyi, F. Faupel, O. Aktas, Materials Today Chemistry 16, 100251 (2020)
  * [34] J.G. McCarty, Catalysis Today 26(3-4), 283 (1995)
  * [35] A. Das, D.H. Ko, C.H. Chen, L.B. Chang, C.S. Lai, F.C. Chu, L. Chow, R.M. Lin, Sensors and Actuators B: Chemical 205, 199 (2014)
  * [36] Q. Zhao, H. Li, L. Zhang, Y. Cao, Journal of Photochemistry and Photobiology A: Chemistry 384, 112032 (2019)
  * [37] Z. Li, J. Song, D.C. Lee, A. Abdelhafiz, Z. Xiao, Z. Hou, S. Liao, B. DeGlee, M. Liu, J. Zeng, et al., Electrochimica Acta 329, 135166 (2020)
  * [38] J. Bergmann, P. Friedel, R. Kleeberg, (1998)
  * [39] H.P. Klug, L.E. Alexander, _X-ray diffraction procedures: for polycrystalline and amorphous materials_ (1974)
  * [40] J. Remillard, W. Weber, J. McBride, R. Soltis, Journal of applied physics 71(9), 4515 (1992)
  * [41] J. McBride, K. Hass, W. Weber, Physical Review B 44(10), 5016 (1991)
  * [42] W. Li, Materials chemistry and physics 99(1), 174 (2006)
  * [43] N. Kumar, S.S. Parui, S. Limbu, D.K. Mahato, N. Tiwari, R.N. Chauhan, Materials Today: Proceedings (2020)
  * [44] F. Ascencio, J. Estevez, L. Rangel-Gamboa, L. Bazán-Díaz, R. Mendoza-Cruz, R. Herrera-Becerra, Materials Today Communications 25, 101614 (2020)
  * [45] C.J. Huang, F.M. Pan, H.Y. Chen, Li-Chang, Journal of Applied Physics 108(5), 053105 (2010)
  * [46] M.K. Bruska, I. Czekaj, B. Delley, J. Mantzaras, A. Wokaun, Physical Chemistry Chemical Physics 13(35), 15947 (2011)
  * [47] B. Sarkar, A. Bandyopadhyay, J. Mandal, A. Deb, P. Chakrabarti, Journal of Alloys and Compounds 656, 339 (2016)
  * [48] Y. Xu, B. Yao, Q. Cui, RSC advances 6(9), 7521 (2016)
  * [49] J. Tejada, R. Ziolo, X. Zhang, Chemistry of materials 8(8), 1784 (1996)
  * [50] I. Bruvera, P. Mendoza Zélis, M. Pilar Calatayud, G.F. Goya, F.H. Sánchez, Journal of Applied Physics 118(18), 184304 (2015)
  * [51] X.H. Li, C.L. Xu, X.H. Han, L. Qiao, T. Wang, F.S. Li, Nanoscale research letters 5(6), 1039 (2010)