# Episodic energy release during the main- and post-impulsive phase of a solar
flare

Yuqian Wei Institute for Space Weather Sciences, New Jersey Institute of
Technology, 323 Martin Luther King Blvd, Newark, NJ 07102-1982, USA  Bin Chen
(陈彬) Institute for Space Weather Sciences, New Jersey Institute of Technology,
323 Martin Luther King Blvd, Newark, NJ 07102-1982, USA  Sijie Yu (余思捷)
Institute for Space Weather Sciences, New Jersey Institute of Technology, 323
Martin Luther King Blvd, Newark, NJ 07102-1982, USA  Haimin Wang Institute for
Space Weather Sciences, New Jersey Institute of Technology, 323 Martin Luther
King Blvd, Newark, NJ 07102-1982, USA  Big Bear Solar Observatory, New Jersey
Institute of Technology, 40386 North Shore Lane, Big Bear City, CA 92314-9672,
USA  Yixian Zhang  School of Physics & Astronomy, University of Minnesota Twin
Cities, Minneapolis, MN 55455, USA  Lindsay Glesener  School of Physics &
Astronomy, University of Minnesota Twin Cities, Minneapolis, MN 55455, USA
Yuqian Wei<EMAIL_ADDRESS>
###### Abstract

When and where the magnetic field energy is released and converted in eruptive
solar flares remains an outstanding topic in solar physics. To shed light on
this question, here we report multi-wavelength observations of a C9.4-class
eruptive limb flare that occurred on 2017 August 20. The flare, accompanied by
a magnetic flux rope eruption and a white light coronal mass ejection,
features three post-impulsive X-ray and microwave bursts immediately following
its main impulsive phase. For each burst, both microwave and X-ray imaging
suggest that the non-thermal electrons are located in the above-the-loop-top
region. Interestingly, contrary to many other flares, the peak flux of the
three post-impulsive microwave and X-ray bursts shows an increase for later
bursts. Spectral analysis reveals that the sources have a hardening spectral
index, suggesting a more efficient electron acceleration into the later post-
impulsive bursts. We observe a positive correlation between the acceleration
of the magnetic flux rope and the non-thermal energy release during the post-
impulsive bursts in the same event. Intriguingly, different from some other
eruptive events, this correlation does not hold for the main impulse phase of
this event, which we interpret as energy release due to the tether-cutting
reconnection before the primary flux rope acceleration occurs. In addition,
using footpoint brightenings at conjugate flare ribbons, a weakening
reconnection guide field is inferred, which may also contribute to the
hardening of the non-thermal electrons during the post-impulsive phase.

Solar radio emission (1522), Solar magnetic reconnection (1504), Solar flares
(1496), Non-thermal radiation sources (1119), Solar coronal mass ejections,
(310), Solar filament eruptions (1981)

††software: SciPy (Virtanen et al., 2020), AstroPy (Robitaille et al., 2013),
CASA (McMullin et al., 2007), Lmfit (Newville et al., 2016), MGN (Morgan &
Druckmüller, 2014), NumPy (Harris et al., 2020), SunPy (Community et al.,
2015)

##  1 Introduction

The relationship between the kinematics of coronal mass ejections (CMEs) and
the corresponding flaring emissions is important in understanding how magnetic
energy is released and subsequently converted into different forms of energy
in solar eruptions (e.g. Lin & Forbes, 2000). The close temporal correlation
between flare X-ray emission and the early kinematics evolution of the
erupting magnetic flux rope/filament during the main impulsive phase (MIP) has
been widely observed (Zhang et al., 2001; Maričić et al., 2007; Temmer et al.,
2008). As a direct indicator of the flare reconnection, the rate of the
magnetic flux change inferred using the advancing flare ribbon at the
photosphere, is found to show a temporal correlation with the acceleration of
the associated filament eruption/CME (Qiu et al., 2004; Hu et al., 2014). In a
statistical study, close correlations are found between the acceleration of
the erupting filaments and the magnetic flux change rate (Jing et al., 2005).
More recently, by analyzing a large sample of CME-associated flare events, Zhu
et al. (2020) confirmed the correlation between the peak filament/CME
acceleration and the peak rate of magnetic flux change. In addition, they
revealed a positive correlation between the total reconnected magnetic flux
and the maximum CME velocity in events accompanied with fast CMEs ($>\\!600\
\mathrm{km}\ \mathrm{s}^{-1}$). Such a correlation suggests that the magnetic
energy release rate is closely related to flux rope eruption. The correlation
is also revealed in resistive magnetohydrodynamic (MHD) simulations (Cheng et
al., 2003; Reeves, 2006; Reeves & Moats, 2010).

As discussed by Welsch (2018), the mechanisms of the flux rope/CME
acceleration can be grouped into two general categories: (1) Lorentz force in
and around the magnetic flux rope that directly drives its acceleration. These
models suggest the increasing Lorentz force during the flare impulsive phase
is attributed to the flare reconnection that adds poloidal magnetic flux to
the flux rope and, at the same time, reduces the tension force from the
overlaying constraining magnetic fields (Lin & Forbes, 2000; Lin et al.,
2005). (2) Acceleration of the flux rope that is due to momentum transferred
from the upward-directed reconnection outflows. The highly bent post-
reconnection field lines coming out of the diffusion region bear a large
magnetic tension force and are accelerated to near the Alfvén speed (Parker,
1957). After joining the flux rope and becoming “dipolarized,” the upward-
directed reconnection outflows transfer the momentum to the flux rope and
facilitate its acceleration (Wang et al., 2007; Xue et al., 2016; Jiang et
al., 2021). Although multiple models have been proposed to account for the
initiation and acceleration of the flux rope, such as the loss of equilibrium
model (Lin & Forbes, 2000), tether-cutting reconnection model (Moore et al.,
2001), breakout reconnection model (Antiochos et al., 1999; Karpen et al.,
2012), the net change of the upward Lorentz force and/or the added momentum
transfer from the upward reconnection outflows are both related to the
magnetic flux change due to the ongoing flare reconnection. Hence, a positive
correlation between the flux rope acceleration and flare energy release is
expected.

The electron acceleration rate, implicated by the intensity of nonthermal hard
X-ray (HXR) and/or microwave emission, is also found to be correlated to the
rate of the flare energy release. By measuring the photospheric magnetic field
and ribbon expansion, Qiu et al. (2004) inferred the rate of the magnetic flux
change and reconnection electric field evolution during the impulsive phase of
two two-ribbon flares, which are found to be temporally correlated with the
microwave emission and the derivative of the SXR emission (as a proxy for HXR
emission assuming the Neupert effect (Neupert, 1968)). Adopting a similar
method, Liu & Wang (2009) analyzed 13 two-ribbon flares and found an anti-
correlation between the average reconnection electric field and minimum
overall photon spectral index from HXR observation. Temmer et al. (2007) found
that the local reconnection electric field inferred from the ribbon expansion
in H$\alpha$/UV observations is uneven along the direction of the magnetic
polarity inversion line (PIL). The locations of the spatially resolved HXR
footpoint sources were found to be spatially correlated with the local
electric field. With the observation of the flare ribbon from the Interface
Region Imaging Spectrograph (IRIS; De Pontieu et al. 2014, Naus et al. (2022)
also revealed the strong correlation between the local magnetic flux change
rate and the production of the nonthermal electrons inferred from RHESSI HXR
data. These studies suggest that the electron acceleration in flares and its
temporal and spatial evolution are intimately related to the local electric
field in the reconnection region.

Recent modeling studies have suggested that, in addition to the absolute
magnetic energy release rate, the guide field, defined as the magnetic field
component in the same direction as the reconnection current, also plays a key
role in determining the efficiency of particle energization. In particular, a
strong guide field can suppress the acceleration of electrons to high
energies, resulting in an overall soft spectrum (Pritchett & Coroniti, 2004;
Dahlin et al., 2014, 2016; Li et al., 2017; Arnold et al., 2021). At present,
direct means of measuring the guide field in the reconnection region has not
been available, although the present microwave imaging spectroscopy
observations, made possible by the Expanded Owens Valley Solar Array (EOVSA;
Gary et al. 2018), have provided constraints of the guide field by comparing
the overall magnetic field profile along the current sheet to model
predictions (Chen et al., 2020b)111Prospects of achieving more direct
measurements of the guide field can be realized by using microwave imaging
spectropolarimetry.. Alternatively, the inclination angle of the post
reconnection flare arcade, usually constrained by comparing the orientation of
the conjugate ribbon brightenings with respect to the magnetic PIL, can be
used as a proxy to infer the guide field component in the coronal region (e.g.
Qiu et al., 2017, 2023). Recently, using three-dimensional MHD simulations,
Dahlin et al. (2022) revealed a prominent decrease of the guide field over the
impulsive phase of eruptive flares, conforming to the observational evidence
reported by Qiu et al. (2010).

However, the correlations discussed above have focused largely on the peak
rates derived during the main impulsive phases of the flare-CME events and are
based on statistical studies of multiple different events. In this work, we
report a new finding that such a correlation is also present during the post-
impulsive phases (PIP) of a single eruptive flare event. We also compare the
difference in the geometry of the magnetic reconnection in the main- and post-
impulsive flare phases and its possible impact on flux rope acceleration and
electron energization.

In Section 2.1, we provide an overview of the event observed in multiple
wavelengths and present the early evolution of the erupting flux rope. In
Section 2.2, we present microwave and X-ray imaging spectroscopy observations
and spectral analysis of the main- and post-impulsive phase bursts. In Section
2.3, we report measurements of the kinematics of the erupting magnetic flux
rope. In Section 3, we interpret the observational results and discuss the
implications based on the correlation between the flux rope acceleration,
electron acceleration, and flare emission during the main- and post-impulsive
phase.

##  2 Observations

###  2.1 Event Overview

Figure 1:  (a) The eruptive solar flare event under study as observed in EUV
by the SDO/AIA 131 Å filterband on 2017 August 20 at 19:41:00 UT. The white
box shows the field of view (FOV) that is used in Figure 4. (b) The event was
observed in STEREO/EUVI 304 Å at 19:26:00 UT, with an enlarged view shown in
(d) (whose FOV is indicated by the white box in (b)). (c) Relative location of
the STEREO-A spacecraft (red cube), the Earth/SDO spacecraft (green cube), and
the longitudinal direction of the event in the frame of the HEE coordinate
system (produced using the Solar-MACH software; Gieseler et al. 2023). Figure
2:  (a) RHESSI 20–35 keV X-ray (orange curve), and GOES 1–8 Å soft X-ray (SXR)
(blue solid curve) light curves and its derivative (blue dashed curve) from
19:20 UT to 19:45 UT on 2017 August 20. Other color curves show EOVSA
microwave flaring-region-integrated light curves at five selected frequencies
(3.4, 6.4, 9.4, 12.4, 15.9 GHz) from 19:31 UT to 19:45 UT. The blue arrows
indicate the peaks of the main-impulsive phase and the three post-impulsive
phase bursts, while the black arrows indicate the corresponding pre-burst time
of each burst used for background subtraction. (b) RHESSI 20–35 keV X-ray
(solid orange curve) decomposed into the main-impulsive phase burst and the
three post-impulsive phase bursts (solid black curves). The light curve is
fitted using four components described in the text (solid purple curve). The
background is the flare-region-integrated (same as that of the light curves in
(a)) microwave dynamic spectrum from 19:31 UT to 19:45 UT.

The C9.4-class event under study occurred on the east solar limb on 2017
August 20. The event was well observed in extreme ultraviolet (EUV) by the
Atmospheric Imaging Assembly on board the Solar Dynamics Observatory (SDO/AIA;
Pesnell et al. 2012; Lemen et al. 2012) and the Extreme UltraViolet Imager
(EUVI; Wuelser et al. 2004) onboard STEREO-A, one of the two Solar Terrestrial
Relations Observatory (STEREO; Kaiser et al. 2008) spacecraft. SDO/AIA
observed the flare event near the east limb from the Earth’s viewing
perspective, while STEREO-A was $\sim$230∘ west from the Earth and provided
observations from another vantage point (Figure 1). The Geostationary
Operational Environmental Satellite (GOES) and the Reuven Ramaty High-Energy
Solar Spectroscopic Imager (RHESSI; Lin et al. 2002) had full coverage of the
event in X-rays. Meanwhile, EOVSA observed the post-impulsive phase and the
decay phase of the event in the microwaves from 1–18 GHz (Figure 2). It missed
the main impulsive peak because the antennas went off the Sun for calibration
during that time.

The event was associated with a white-light CME observed by the K-coronagraph
of the Mauna Loa Solar Observatory (MLSO/K-Cor; Elmore et al. 2003), as well
as the Large Angle Spectroscopic COronagraph on board the Solar and
Heliospheric Observatory (SOHO/LASCO; Brueckner et al. 1995). Figure 3(a)
shows a faint, slow ($\sim$$250\ \mathrm{km}\ \mathrm{s}^{-1}$), and narrow
($\sim$32∘) CME in the LASCO C2 difference image. In MLSO/K-Cor white light
images, the CME, as shown in Figure 3(b), displays a typical three-part-
structure: a leading front, a relatively darker cavity, and a bright core,
which are typically explained as the plasma pileup at the leading edge of the
erupting magnetic flux rope, the flux rope body, and the embedded
filament/prominence, respectively (Illing & Hundhausen, 1985; Chen, 2011;
Vourlidas et al., 2013), although certain studies suggested that the bright
core may instead be attributed to the flux rope itself (Howard et al., 2017;
Veronig et al., 2018; Song et al., 2019). In addition, another recent study,
based on laboratory experiments, proposed an alternative model to interpret
the cavity as a result of induced reverse current evacuating the background
plasma away from the core (Haw et al., 2018).

Figure 3:  The associated CME observed in white light by MLSO/K-cor and
SOHO/LASCO C2. (a) The narrow CME as observed in LASCO/C2 running-difference
image at 20:48 UT (81 minutes after the flare peak), whose front is indicated
by the black arrow and the blue dashed curve. The white box shows the FOV of
the MLSO/K-cor image in (b). (b) The three-part-structure CME as observed in
MLSO/K-cor white light image at 19:46 UT (19 minutes after the flare peak).

Before the flare, a dark filament can be clearly identified in the EUV
passbands (Figure 4(a)). At 19:14 UT, two minutes before the HXR flux shows an
early rise, two new loops appear in the SDO/AIA 131 Å images (highlighted by
the yellow/orange dotted lines in Figure 4(b)). Four minutes later, at 19:18
UT, a flare arcade started to appear, which connected the two inner footpoints
of the two loops (highlighted by the pink dotted lines in Figure 4(c)).
Meanwhile, a new coronal structure also appears between the two loops
(indicated by the white arrow in Figure 4(c) and (d)). The event enters the
main impulsive phase at 19:24 UT (c.f., Figure 2(a)), when the flare arcade
further develops and brightens. The viewing perspective of the STEREO-A/EUVI
reveals two J-shaped flare ribbons in a typical two-ribbon configuration
(Figure 1(d)). The bright coronal structure seen during the pre-impulsive
phase disappeared when the event entered the post-impulsive phase (Figure
4(f)).

Figure 4:  Close-up view of the coronal evolution as observed by SDO/AIA 131 Å
from the pre-flare phase to the beginning of the post-impulsive phase. (a) The
dark filament which exists before the event, is marked by the white arrow. (b)
The appearance of two loops during the pre-flare phase is highlighted by the
yellow/orange dotted curves. (c) The newly formed coronal structure and the
flare arcade (pink dashed curve) appear as the result of a tether-cutting
reconnection during the pre-flare phase (indicated by the white arrows). (d)
The gradually brightening coronal structure and a flare arcade appear during
the main-impulsive phase, indicated by the white arrow and red dashed curve,
respectively. (e) and (f) An Enlarged view of the flare region at the SXR
flare peak and the beginning of the post-impulsive phase, respectively. Figure
5:  Evolution of the rising flux rope during the post-impulsive phase of the
event as observed by SDO/AIA in EUV and MLSO/K-Cor in white light. (a)–(c)
SDO/AIA 131 Å images showing the evolution of the erupting flux rope cavity,
indicated by the white arrow in (a). (d)–(f) SDO/AIA 94 Å images enhanced with
the MGN method. The overlying loop-like feature at the front of the cavity is
indicated by the white arrow in (d) and the pink dashed curve in (d)–(f),
while the center of the flux rope cavity is indicated by the pink open circle.
See Animation 14 for the visualization of the rising overlying loop-like
feature. (g)–(h) Contours of the MLSO/K-Cor white light images (the contour
levels are same as that of Figure 3(b) ) overlaid on SDO/AIA 94 Å images. The
same pink open circles in (d)–(f) are also shown. The lower boundary of
MLSO/K-Cor’s FOV at 1.07 solar radii is indicated by the red dashed curve. The
FOV that is used in (a)–(f) is shown as the white box in (i).

As the event entered the main-impulsive phase, a group of large-scale
overlying loops, which bridged the northern and southern ends of the active
region, started to rise towards the southwest direction in succession
(indicated by the white arrow in Figure 4(e)). The eruption started to appear
during the post-impulsive phase, when an oval-shaped cavity became visible in
the SDO/AIA 131 Å images ((Figure 5(a))), which we interpret as the cross-
section of the erupting magnetic flux rope.

To show the evolution of the erupting cavity more clearly, in Figures
5(d)–(f), we show SDO/AIA 94 Å images enhanced with the multi-scale Gaussian
normalization (MGN) method (Morgan & Druckmüller, 2014). A kernel size of
4′′.8 is selected to sharpen the edge and reveal the loop-like structure at
the front of the cavity. This loop-like structure will be used to measure the
kinematics of the eruption, which will be discussed in detail in Section 2.3.

When the cavity, whose center is indicated by the pink circle in Figure
5(D)–(I)), rises into the inner field of view (FOV) of the MLSO/K-Cor at 1.07
$R_{\odot}$ (red dashed line in Figure 5(G)–(I)), it moves synchronously with
the CME bright core seen in the MLSO/K-Cor white light images (the innermost
contour in Figure 5(h), (i)). The synchronized motion of the cavity center in
SDO/AIA 94 Å images and the bright core of the K-Cor CME corroborate our
interpretation of the structure as an erupting magnetic flux rope.

###  2.2 Microwave and X-ray Bursts During the Main- and Post-impulsive Phase

Shortly after the impulsive X-ray peak at 19:27 UT, the event entered its
post-impulsive phase. Almost at the same time, EOVSA went back to the Sun at
19:31 UT and fully covered the post-impulsive phase. In this phase, three
broadband bursts can be observed in the EOVSA 1–18 GHz dynamic spectrum
(Figure 2(b)), which peak at 19:33 UT, 19:35 UT, and 19:38 UT respectively.
The dynamic spectrum is produced by integrating the total flux of the flaring
region using images integrated from 19:32 UT to 19:45 UT with 134 frequencies
in 2.5–18 GHz over 31 evenly spaced spectral windows (referred to as SPW 0 to
SPW 30) and is conducted with a temporal resolution of 10-s. While post-
impulsive phase microwave bursts have been reported in the literature (e.g. Yu
et al., 2020; Kou et al., 2022), this event shows an increase in the peak
intensity of bursts that occur later. The peak flux density at, 9.4 GHz for
example, increases from 50 sfu for the first burst to 114 sfu for the last
one.

As shown in Figure 2, the bursts also have a response in GOES 1–8 Å SXR light
curve derivative and RHESSI 20–35 keV. To investigate the relationship between
the microwave and HXR bursts, we carried out a forward fitting on the RHESSI
20–35 keV light curve using a ‘heating-decay” function following the method
described in Gryciuk et al. (2017). For each burst, the time profile is
prescribed as:

$f(t)=\int_{0}^{t}g(t^{\prime})h(t-t^{\prime})\mathrm{d}t^{\prime},$ (1)

which is the result of a Gaussian-shaped pulse $g(t)$ convoluted with an
exponential decay term $h(t)$. They are, respectively,

$g(t)=g_{0}\exp\left(-\left(t-t_{0}\right)^{2}/2\sigma^{2}\right)$ (2)

and

$h(t)=\exp(-Dt),$ (3)

where $g_{0}$, $t_{0}$, $\sigma$, and D are the parameterized amplitude, peak
time, the standard deviation of the Gaussian function, and exponential decay
coefficient, respectively.

The observed RHESSI 20–35 keV light curve is fitted with four pulses, which
correspond to the main-impulsive peak and the three post-impulsive bursts,
respectively. Similar to post-impulsive microwave bursts, RHESSI 20–35 keV
X-ray bursts also show an increase in the peak intensity for later post-
impulsive bursts, as shown in Figure 2.

####  2.2.1 X-ray Imaging and Spectral Analysis

We reconstruct the RHESSI 6–12 keV images using the CLEAN algorithm (Hurford
et al., 2002) with a 40-s integration time, based on measurements from
detectors 3, and 8. The X-ray sources are plotted as green open contours in
Figure 7(a), (c)–(e). During the main-impulsive phase (Figure 7(a)), three
sources can be distinguished. The main source is located at the top of the
flare arcade. The lower (western) source coincides with the southern footpoint
of the arcade, while the upper (eastern) source coincides with the bright
coronal structure discussed in Section 2.1. During the post-impulsive phase,
the upper X-ray source quickly fades away following the rise of its EUV
counterpart, while the looptop and the footpoint sources remain. The detailed
spatial evolution of the HXR source during the post-impulsive phase will be
further presented and discussed in Section 2.4.

We utilize the OSPEX tool, which is part of the SolarSoft IDL package (sswidl;
Freeland & Handy 1998) distribution, to perform the X-ray spectral analysis.
Due to the increasingly more severe pulse pileup effect that affected RHESSI
X-ray measurements toward the end of its operations, we limited our spectral
analysis to data obtained from detector 3. This particular detector, thanks to
its low sensitivity at the time of observation, showed the least amount of
pileup effect among all active detectors. We note that the time profiles,
especially those at the $>\\!50\ \mathrm{keV}$ energy bands, have a time-
varying, but smooth, background associated with each orbit. A polynomial fit
to the time-varying background using the prior orbit is performed, which is
subtracted from the time profiles of interest at all energies. After the
background subtraction, no detectable emission remains at $>\\!50\
\mathrm{keV}$. We fit three components to each X-ray pulse, which include a
single-temperature thermal bremsstrahlung function vth, a broken power-law
function bpow, and a pseudo function that accounts for the pileup effect
pileup_mod 222A more detailed description of the pileup-correction module can
be found at https://sprg.ssl.berkeley.edu/~tohban/wiki/index.php/Pileup_mod_-
_Pseudo_function_for_correcting_pileup, to the observed photon count rate
spectrum. The fitting results for the X-ray pulses are presented and
summarized in Table 1 and Figure 6. The associated uncertainties are estimated
using the built-in Monte Carlo module in OSPEX.

Figure 6:  RHESSI photon flux spectra and spectral fitting results for the bursts during the main-impulsive phase (a) and the three post-impulsive phase burst (b)–(d). A single-temperature thermal bremsstrahlung model (vth, orange), a broken power-law model (bpow, green), and a pseudo-model to account for the pileup effect (pileup_mod, purple) are fitted to the background-subtracted data (black). The background is depicted by the grey curve. The key fit parameters are summarized in Table 1. |  2.5cmEmission Measure [$10^{49}$$cm^{-3}$] |  2.5cmPlasma Temperature [$keV$] |  2.5cmNormalization at Epivot |  2.5cmBreak Energy [$keV$] |  2.5cmNegative Power-law Index
---|---|---|---|---|---
main-impulsive Phase | 0.020$\pm$$5\mathrm{e}{-4}$ | 2.84$\pm$0.05 | 0.70$\pm$0.038 | 18.9$\pm$0.5 | 3.77$\pm$0.25
1st PIP Impulse | 0.019$\pm$$4\mathrm{e}{-4}$ | 2.17$\pm$0.05 | 0.423$\pm$0.018 | 20.7$\pm$2.4 | 2.49$\pm$0.24
2nd PIP Impulse | 0.019$\pm$$2\mathrm{e}{-4}$ | 2.10$\pm$0.06 | 0.328$\pm$0.010 | 20.0$\pm$2.0 | 2.92$\pm$0.30
3rd PIP Impulse | 0.023$\pm$$3\mathrm{e}{-4}$ | 2.08$\pm$0.06 | 0.366$\pm$0.008 | 20.3$\pm$2.0 | 2.87$\pm$0.31
Table 1:  RHESSI X-ray spectral fitting results at the main-impulsive phase
and the three post-impulsive phase bursts. The 1-$\sigma$ uncertainties are
estimated by running the built-in Monte Carlo analysis in OSPEX.

As shown in Figure 6, the bpow component (yellow curves) represents a
necessary contribution to the observed X-ray count spectra at energies above
$\sim 20\ \mathrm{keV}$ for each time period, signifying the presence of a
nonthermal electron population throughout the main- and post-impulsive phase
bursts. The nonthermal bpow component joins the thermal core at $\sim 20\
\mathrm{keV}$ for all the four analyzed time intervals. Meanwhile, the
background count rate dominates the spectra at $>\\!35\ \mathrm{keV}$.
Therefore, we integrate the 20–35 keV range to obtain the HXR light curve
shown in Figure 2, which is used as a proxy for the time variation of the
nonthermal component during the event. The power-law index of the observed
X-ray photon spectrum above $\sim$20 keV (taken from bpow) decreases from
$3.77$ during the main-impulsive phase to less than $3$ during the post-
impulsive phase, suggesting a notably harder HXR photon spectrum in the post-
impulsive phase.

####  2.2.2 Microwave Imaging Spectroscopy

EOVSA missed the main-impulsive phase of the event but had full coverage of
the post-impulse phase. Figure 7(b) shows EOVSA microwave images as open
contours (50% of the maximum brightness at each frequency) at 19:32 UT just
before the first post-impulsive phase burst. At high frequencies, the
microwave source is concentrated on the northern part of the flare arcade. The
sources extend from north to south (southeast in the viewing perspective of
STEREO-A/EUVI images) and align with the ridge of the post-flare arcades. The
microwave sources evolved rapidly during the post-impulsive phase. To better
reveal their evolution, we performed difference imaging against the pre-burst
background time in the visibility domain, with the selected background times
pointed by the black arrows in Figure 2. The difference imaging results for
the peaks of the three post-impulsive bursts (blue arrows in Figure 2) are
shown in Figures 7(c), (d), and (e), respectively. We verified that the
sources we used as the background are stable over time, which show very
similar morphology as that shown in Figure 7(b) as well as nearly uniform
spectral properties (black symbols in Figure 8.

Figure 7:  EOVSA multi-frequency microwave images and RHESSI 6–12 keV sources
during the main impulsive phase and the three post-impulsive bursts. (a)
RHESSI 6–12 keV source at the peak of the main-impulsive phase. The contour
levels are 30% and 70% of the maximum. (b) Multi-frequency microwave images at
the background time just prior to the first post-impulsive burst (the time is
indicated by the first black arrow in Figure 2). The open contours are at the
level of 50% of the maximum brightness at each frequency. The restoring beam
size of each frequency is shown in the bottom left corner. The microwave
source morphology at all the selected background times prior to the three
post-impulsive bursts is similar to each other. (c)–(e) Background-subtracted
multi-frequency microwave images at the peak of the three post-impulsive
bursts (indicated by the three blue arrows in Figure 2). The level of the
filled contours is as same as that of open contours in (b). The background
gray-scale images are from SDO/AIA 131 Å at the corresponding times.  Figure
8:  Microwave spectra derived from the above-the-loop-top (ALT) region during
the three post-impulsive bursts and corresponding spectral fitting results.
(a)–(c) The evolution of the best-fit values of magnetic field strength $B$,
power-law index of the electron energy distribution $\delta^{\prime}$, and
total nonthermal electron density $n_{\rm nth}$ above 20 keV, respectively.
(d)–(f) EOVSA microwave brightness temperature spectra (blue dots) at the
above-the-loop-top region during the post-impulsive phase bursts. The solid
orange curve shows the best-fit results while the solid purple curve is the
distribution of the MCMC runs within 1-$\sigma$ of the median MCMC values. The
corresponding residuals are shown at the bottom of each panel.

The background-subtracted microwave sources during three post-impulsive bursts
are mainly located above the top of the bright post-flare arcades seen in
SDO/AIA 131 Å. For bursts 2 and 3, the above-the-looptop microwave source
shows an obvious dispersion in height, with the high-frequency sources located
at lower heights than the low-frequency ones. Such a microwave source
morphology has already been reported by Gary et al. (2018); Chen et al.
(2020a) for the early impulsive phase of the X8.2 solar flare on September 10,
2017. The source dispersion above the looptop was interpreted as the signature
of nonthermal electrons distributing along the reconnection current sheet,
with the higher frequency source generally originating from sources regions
with a greater magnetic field strength. In this event, however, we do not see
a clear plasma sheet above the flare arcade, yet the physical origin of the
frequency-dependent distribution of the source could be similar. During the
peak of the three post-impulsive bursts, a secondary source, only seen at low
frequencies ($<$5 GHz), appears to the south of the above-the-looptop source.
By comparing to the EUV images, we attribute the secondary microwave source as
emission from the southern footpoint of the erupting flux rope, similar to the
interpretation in Chen et al. (2020a). We will return to the discussion of the
multi-wavelength flare context in Section 3. Figure 8(d)–(f) shows the
brightness temperature spectra derived from the above-loop-top (ALT) region
(black box in Figure 7(c)) at the peak time of each post-impulsive phase burst
(blue dots with error bars). The spectra show characteristics of the
nonthermal gyrosynchrontron radiation (Dulk, 1985). We fit the observed
spectra using a nonthermal gyrosynchrontron radiation model from a homogeneous
source with a power-law electron energy distribution based on the fast
gyrosynchrotron codes developed by (Fleishman & Kuznetsov, 2010). Three free
parameters are used in the spectral fitting: magnetic field strength $B$, the
total number density of nonthermal electrons $n_{\rm nth}$, and the power-law
index $\delta^{\prime}$ of the nonthermal electron energy distribution
($f(\varepsilon)=dn_{\rm
nth}(\varepsilon)/d\varepsilon\propto\varepsilon^{-\delta^{\prime}})$. The
energy range of the nonthermal electron distribution is fixed to 20 keV–10
MeV. The thermal electron density $n_{\rm th}$ and the plasma temperature $T$
is fixed to $1.7\times 10^{11}~{}\mathrm{cm}^{-3}$ and 13 MK, respectively,
which are based on the differential emission measure (DEM) analysis results
within the same region using data from six SDO/AIA EUV channels (94, 131, 171,
193, 211, and 335 Å). For the DEM analysis, we adopt the technique developed
by Hannah & Kontar (2012). The column depth is assumed to be
$13^{\prime\prime}$ based on the observed source size on the plane of the sky
at 6.4 GHz.

The best-fit spectra and the corresponding residual are shown as the thick
solid orange curves in Figures 8(d)–(f). Following Chen et al. (2020b), we
adopt the Markov Chain Monte Carlo (MCMC) method to estimate the uncertainties
of the parameters and verify that we have achieved global minimization in the
multi-dimensional parameter space. The spectra calculated from individual MCMC
samples and the corresponding residuals are shown as thin purple curves in
Figures 8(d)–(f). Figures 8(a)–(c) show the temporal evolution of the three
fit parameters at the peak time of each post-impulsive phase microwave burst.
The corresponding errors are estimated by the MCMC method. Of the three fit
parameters, the only one that undergoes a significant change (against the
uncertainties) is the power-law index of the nonthermal electron energy
distribution $\delta^{\prime}$. The power-law index shows a notable hardening
for later post-impulsive microwave bursts, which increases from $\sim 12$ for
burst #1 to $\sim 5.5$ for burst #3.

The brightness temperature spectra derived from the selected background times
prior to the bursts are plotted as the black crosses in Figure 8(a)–(c).
Spectral analysis indicates that the source is dominated by thermal emission.
This thermal loop-top source is very similar to that reported by Yu et al.
(2020) for the post-impulsive phase of the X8.2 solar flare on September 10,
2017.

###  2.3 Magnetic Flux Rope Kinematics

To accurately track the kinematics of the erupting magnetic flux rope seen in
SDO/AIA images in the low corona and the associated white-light CME observed
in MLSO/K-Cor images, we produce a time-distance stack plot by obtaining the
intensity along a slice shown as the dashed white lines in Figure 5(g). The
slice crosses the boundary of the FOV of SDO/AIA and extends to that of the
MLSO/K-Cor. The eruption changes its course slightly during its ascent. Hence
the slice is slightly bent toward the direction of the flux rope eruption. The
width of the slice, defined as the perpendicular distance across the slit
varying along the curved trajectory increases linearly from $3^{\prime\prime}$
at the bottom to $13.8^{\prime\prime}$ at the top. This adjustment compensates
for the general expansion of coronal structures. In order to clearly show the
dynamic features, we also apply the running-differential method on the SDO/AIA
94 Å and MLSO/K-Cor white light images and then enhance the edge of those
features with a high-pass filter technique.

The resulting time-distance stack plot is shown in Figure 9. A number of
rising tracks are visible at different heights, which correspond to not only
structures that correspond to the core of the erupting magnetic flux rope, but
also the overlying loops that enclose the flux rope cavity. We select one of
the most visible tracks at the immediate front of the erupting cavity-like
structure to measure the kinematics of the flux rope. The selected track is
denoted by the red symbols in Figure 9(a) and the corresponding feature in the
difference images is marked by pink dashed curves in Figure 5(d)–(f). At any
given time, the height of the tracked feature is determined by finding the
peak in the intensity–height profile. To estimate the uncertainty of the
tracked trajectory, we fit a skewed Gaussian function to the intensity–height
profile around the peak. The full width at half maximum of this Gaussian
function is used as an estimate of the uncertainty, shown as the vertical
extension of the red symbols in Figure 10(a).

A similar time-distance stack plot is derived using the MLSO/K-Cor images at
the same selected slice, shown in the upper portion of Figure 10(a). It can be
seen that the trajectories of the cavity front tracked in the SDO/AIA images
and the upper edge of the white light CME core seen by MLSO/K-Cor are well
aligned, which further demonstrates that we are tracking a coherent flux rope
structure erupting from the low to middle corona.

Figure 9:  (a) Composite time–distance stack plot of MLSO/K-Cor white light
(upper) and SDO/AIA 94 Å (lower; background-detrended) images series made
along a slice as indicated by the dashed white curve in Figure 5(g). Red
symbols indicate the tracked eruption feature that represents the flux rope
front in SDO/AIA 94 Å images. The vertical lengths of the symbols indicate the
corresponding uncertainties. (b) Same as (a) but shows the complete
time–distance stack plot of the SDO/AIA 94 Å image series. The trajectory of
the tracked feature is sandwiched between the red dashed-dotted lines. An
accompanying video (Animation 14) shows the tracked feature on the SDO/AIA 94
Å running-differential images and the corresponding time–distance plot. Figure
10:  (a) Evolution of the height (red symbols), velocity (green symbols), and
acceleration (orange symbols) of the erupting flux rope during the main- and
post-impulsive phase. The peak time of the main- and post-impulsive phase
bursts are indicated by the blue arrows. (b) Microwave and X-ray light curves
during the same period (similar to Figure 2(b) but with the EOVSA microwave
light curves shown instead). Also shown are the microwave-constrained power-
law index of the nonthermal electron distribution at the peaks of the three
post-impulsive phase bursts (purple symbols).

The speed and acceleration are derived from the height–time profile (red
symbols), which are denoted by the light green and light orange symbols in
Figure 10(a), respectively.

The speed and acceleration are derived from the height–time profile (red
symbols), which are denoted by the light green and light orange symbols in
Figure 10(a), respectively. In our methodology, the derivatives are smoothed
through rebinning after each derivation step, equivalently reducing the
temporal resolution by a factor of two. When estimating the error in the
first- and second-order derivative (velocity and acceleration), in addition to
the error propagated from the height measurement itself, we include the
smoothing error which is the root square (RMS) of the difference between the
original profile and its smoothed counterpart, indicating the average
smoothing error in both derivation steps.

Despite having a relatively large uncertainty, the acceleration of the flux
rope in this event extends to and peaks at the post-impulsive phase. In
particular, the last post-impulsive burst around 19:37 UT, which features the
strongest microwave/X-ray flux and hardest electron energy spectrum (purple
symbols in Figure 10(b) and red symbols in Figure 8(b)), corresponds to the
largest peak acceleration value. However, the relation in the main-impulsive
phase does not appear to follow the same trend as in the post-impulsive phase.
The maximum count rate of the RHESSI 6–20 keV time profile at the main-
impulsive phase is 16 times larger than that at the first post-impulsive
phase, while the corresponding maximum acceleration values are similar.

###  2.4 Source motion at flare looptops and footpoints

During the post-impulsive phase, we also observed a synchronized motion of
energy release signatures above the flare looptops and at the footpoints.
First, as shown in Figures 7(c)–(e), despite the dispersion in height (largely
in the east–west direction) as a function of frequency, the high-frequency
background-subtracted microwave sources display an evident systematic motion
from the southern to northern side of the flare arcade during the post-
impulsive bursts. To better demonstrate the overall motion of the microwave
source, in Figure 11(a), we show the centroid locations of the corresponding
microwave sources as filled color circles. The centroid locations at each
frequency are estimated by fitting the corresponding background-subtracted
microwave source to a two-dimensional Gaussian ellipse using CASA task IMFIT
(McMullin et al., 2007). The locations shown in the figure are obtained by
averaging the centroid locations derived from individual images in 6.4–14.4
GHz. The lower-frequency sources are excluded because of their lower angular
resolution and their sensitivity to lower magnetic field regions, which are
presumably located further away from the flare arcade. The uncertainties are
determined by the standard deviation of the image source centroids across the
frequency range.

Figure 11:  Synchronized northward motion of the microwave/X-ray/EUV looptop
sources and corresponding UV footpoint brightenings. (a) Centroid locations of
background-subtracted EOVSA microwave sources (filled circle), RHESSI 6–12 keV
X-ray looptop sources (dashed open circle), EUV looptop brightening observed
by SDO/AIA 131 Å images(stars). Also shown are the northern ($R_{n}$)/
southern ($R_{s}$) ribbon brightenings in SDO/AIA 1600 Å UV images (crosses)
at the peak time of three post-impulsive bursts. The symbols are color-coded
by time, as written in (a). The uncertainties of the microwave and X-ray
sources are indicated by the radius of the corresponding filled/open circle.
(b) Same as (a), but all the measured locations in (a) are re-projected into
STEREO-A/EUVI’s viewing perspective. (c)–(e) Detailed view of the northern
ribbon region ($R_{n}$; c), flare looptop (LT; d), and southern ribbon region
($R_{s}$; e) at the peak time of the main impulsive phase and the three post-
impulsive bursts. The corresponding FOV are indicated by the white boxes in
(a).

Similarly, both the centroid location of the RHESSI 6–12 keV X-ray looptop
source and the looptop EUV brightening show a northward trend during the three
post-impulsive bursts. In Figure 11, the open circles mark the centroids of
the RHESSI 6–12 keV source during the main-impulsive phase (red) and the post-
impulsive phase bursts (the color code is as same as the microwave circles),
with their sizes corresponding to the uncertainties. The centroid location of
the X-ray sources and their uncertainties are determined using Detector 3
images made with the VIS_FWDFIT algorithm. Meanwhile, the star symbols in
Figure 11(a) represent the locations (pixels) of the brightest SDO/AIA 131 Å
emission in the looptop region (within the box labeled “LT”) at the
corresponding times. The star symbols are also plotted in the SDO/AIA 131
Åimages in Figure 11(d).

The synchronized northward motion as observed in the looptop region is also
observed at the conjugate footpoints of the post-flare arcade. In Figure
11(a), the straight ends of the two ribbons are enclosed in the white boxes
with “$R_{n}$” and “$R_{s}$”. The crosses mark the brightest points on the
flare ribbons in SDO/AIA 1600 Åimages. An enlarged view of the southern and
northern ribbons are shown in Figures 11(c) and (e), respectively. Similar to
the coronal emissions, the conjugate footpoints have a unidirectional
northward motion.

During the main-impulsive phase, UV brightening is also observed at the far
end of the conjugate ribbons. The locations are shown as red diamonds in
Figure 11. The southern footpoint brightening is plotted in the inset in
Figure 11(a) with a larger FOV, in which the full FOV of Figure 11(a) is shown
as the white box.

The STEREO-A/EUVI images provide further confirmation of the locations of the
footpoint brightenings. In Figure 11(b), we show the same ribbon brightnenings
identified in SDO/AIA 1600 Å images reprojected to STEREO-A/EUVI’s viewing
perspective (color cross symbols). For the re-projection, we have assumed that
the footpoint brightnenings occur at a chromospheric height of 1000 km. Re-
projection of the flare arcades seen by SDO/AIA is less straightforward owing
to their unknown heights. For illustration purposes, in Figure 11(b), we show
re-projected flare arcades and the looptop EUV brightenings at the respective
times in STEREO-A/EUVI’s view, by assuming that all the flare arcades stand
vertically above the solar surface.

Figure 12:  (a) Reconnection geometry and the inclination angle during the
main impulsive phase. The images are the STEREO-A/EUVI 304 Å image at 19:06 UT
re-projected to the heliographic Carrington coordinates. The red crosses show
the represented locations of the footpoint brightenings (similar to Figure
11(b)). The dashed black line shows the estimated location of the polarity
inversion line (PIL). The solid green and red curves demonstrate the pre-
reconnection field lines, while the solid pink line shows the post-
reconnection arcade. (b) Same as (a), but demonstrating the reconnection
geometry (the pre-reconnection field line is demonstrated by the blue curve)
and the inclination angle during the three post-impulsive phase bursts. (c)
Evolution of the inclination angle $\theta$ during the three post-impulsive
bursts and the corresponding normalized guide field estimates
$B_{g}/B_{t}^{LT}$. The color code in (b) and (c) follows Figures 11.

The footpoint motion can be used to estimate the inclination angle of the
post-reconnection flare arcade with respect to the magnetic polarity inversion
line (PIL; Qiu et al. 2010; Qiu & Cheng 2022; Dahlin et al. 2022). However, it
is difficult to do so from SDO’s viewing perspective owing to the close-to-
limb location of the event. To make an estimate, in Figure 12(b), we use the
mid-point location between the parallel portion of the two ribbons seen by
STEREO-A/EUVI as a proxy for the PIL (shown as a black dashed line). Then, we
use straight lines that connect the three pairs of post-impulsive conjugate
footpoint brightenings, shown in Figure 12(b) as the orange, yellow, and green
dashed lines, respectively, to represent the orientations of the post-
reconnection flare arcade. It can be seen that the lines connecting the post-
impulsive footpoints become more and more perpendicular with regard to the PIL
as the flare progresses, indicating a smaller and smaller shear between the
reconnecting magnetic field lines. In order to quantify the evolution of the
normalized guide field, following Qiu et al. 2010, 2023, we define the
inclination angle $\theta$ as the acute angle between the post-reconnection
arcade and the PIL. The inclination angle $\theta$ can be used to estimate the
normalized guide field: $R_{g}\approx B_{g}/B_{t}^{LT}\approx\cot{\theta}$,
where $B_{g}$ and $B_{t}^{LT}$ are the guide field (parallel to the PIL) and
the transverse component (perpendicular to the PIL) of the magnetic field at
the looptop region, respectively. In Figure 12(c), we show the evolution of
the inclination angle $\theta$ and $R_{g}$ during the post-impulsive bursts.
The $\theta$ value increases from $64^{\circ}$ during the first burst to
$83^{\circ}$ during the last burst, demonstrating a decreasing shear of the
flare arcade333A larger $\theta$ value means a smaller shear. For example, the
limiting case of $\theta=90^{\circ}$ corresponds to the case in which the
reconnecting field lines are completely anti-parallel to each other, while
$\theta=0^{\circ}$ corresponds to purely parallel reconnecting field lines..

In the main-impulsive phase, quantifying the shear and the guide field is less
straightforward, as illustrated in Figure 4, the geometry is relatively
complex. However, we can use the observed ribbon brightenings and coronal
loops as constraints. The pre-reconnection loops during the main-impulsive
phase, illustrated as the red and green curves in Figure 12(a), correspond to
the two loops highlighted in Figure 4(b) observed by SDO/AIA 131 Å. The post-
reconnection arcade, displayed as the pink solid arcade in Figure 12(a),
corresponds to the arcade in SDO/AIA 131 Å outlined by a pink dashed curve in
Figure 4(c). The outer footpoints of the two pre-reconnection loops located in
the hook regions of the ribbons are indicated by a pair of red diamonds in
Figure 12(a), and the inner footpoints are shown by a pair of red crosses. In
contrast to the post-impulsive phase in which the reconnecting field lines are
nearly anti-parallel to each other, the reconnecting loops during the main
impulsive phase likely have a very small inclination angle ($\theta\ll
45^{\circ}$), suggestive of a much greater shear or guide field component.

###  2.5 Summary of the Observations

The main observational phenomena are summarized as follows:

  * •

The eruptive flare features three post-impulsive bursts in both microwaves and
X-rays. Imaging reveals that the source is located at and above the top of the
post-flare arcade.

  * •

The power-law index $\delta^{\prime}$ of the nonthermal electron energy
distribution diagnosed using the microwave data at the ALT region shows a
notable hardening for later post-impulsive bursts.

  * •

The time evolution of the erupting flux rope features multiple episodes of
acceleration during the post-impulsive bursts. The last one appears to have
the strongest acceleration, perhaps even greater than that during the main-
impulsive phase.

  * •

A synchronized northward motion of the microwave/X-ray/EUV looptop source and
UV footpoint brightening is observed during the post-impulsive phase.

  * •

The inclination angle of the flare arcades with respect to PIL, inferred by
the UV brightening on the flare ribbons, shows an increase for later post-
impulsive bursts. In other words, the shear of the post-reconnection flare
arcade decreases in time.

  * •

The RHESSI observations suggest a softer spectrum during the main-impulsive
phase than the post-impulsive phase.

##  3 Discussion and Conclusions

Our observations suggest a positive correlation between the microwave electron
spectral hardness diagnosed by EOVSA microwave imaging spectroscopy and the
acceleration of the associated flux rope in multiple microwave/X-ray bursts
during the post-impulsive phase of a single eruptive flare event. The
observation resembles the widely observed temporal correlation between the CME
acceleration and the hard X-ray flux (or SXR derivative) in previous studies
(e.g. Zhang et al., 2001; Qiu et al., 2004; Jing et al., 2005; Maričić et al.,
2007; Temmer et al., 2008). However, most, if not all, of the previous reports
were made during the main-impulsive phase of eruptive flares. In contrast, the
main-impulsive phase of our event does not correspond to the strongest flux
rope acceleration compared to its post-impulsive counterpart. Coincidentally,
despite featuring a brighter flare emission in X-rays, the main-impulsive
phase has a softer spectrum than the post-impulsive phase.

Figure 13:  Schematic cartoon of the flare geometry in the main-impulsive
phase (a)–(b) and post-impulsive phase (c)–(d). (a) The flare geometry from
the viewing perspective of the SDO/AIA during the main-impulsive phase. The
rising flux rope is marked by the twisted yellow curves. Red and green curves
represent magnetic field lines reconnecting in a tether-cutting scenario. The
reconnection forms a new field line adding to the flux rope (upper pink curve)
and a post-flare arcade (lower pink curve). The orange $X$ denotes the
reconnection point. (b) Same as (a), but is plotted in the viewing perspective
of STEREO-A/EUVI on 304 Å image. (c) The flare geometry in the viewing
perspective of the SDO/AIA during the post-impulsive phase. The overlying
field line around the erupting flux rope cavity is marked by the solid blue
curve. The EOVSA microwave source (green oval), RHESSI X-ray source (blue
oval), post-flare arcades (solid pink curves), and ribbon brightening (orange
ovals) are also shown. The EOVSA microwave source at the southern footpoint of
the erupting flux rope is shown as the pink oval. The background image is the
same as in Figure 5(a), showing the cross-section of the flux rope cavity. (d)
Same as (c), but is plotted in the viewing perspective of STEREO-A/EUVI on 304
Å image.

Figure 13 places the various observed features into the flare context for the
pre- and main-impulsive phase (panels (a) and (b)) and the post-impulsive
phase (panels (c) and (d)). During the pre- and main-impulsive phases, two
highly sheared magnetic field lines (red and green solid curves in Figure
13(a), (b)) reconnect with each other in a “tether-cutting” fashion, with
their inner footpoints coincide with the ribbon brightenings. After the
reconnection, one set of the overlying, highly sheared post-reconnection field
lines (upper solid pink line) join the flux rope, and the other set of lower-
lying field lines become the bright flare arcade (lower solid pink line). The
released energy from the reconnection results in the heating of both the flux
rope and the flare arcade, which are observed as the bright coronal structure
and the flare arcade seen in EUV (Figure 4(d)).

The heating and associated electron acceleration during this period may be
also responsible for the observation of the multiple RHESSI 6–12 keV sources
at the bright coronal structure, the looptop, and footpoint (Figure 7(a)). We
conclude that the observations during the pre- and main-impulsive phase are
broadly consistent with the tether-cutting reconnection scenario (see, e.g.,
Chen et al. 2014, for similar observations), which may also contribute to the
slow rise motion of the magnetic flux rope.

During the post-impulsive phase, the reconnection geometry represents that of
the standard scenario for eruptive flares, in which the overlying magnetic
fields reconnect below the flux rope. In line with suggestions made in other
studies (Tripathi et al., 2006; Aulanier et al., 2012, 2013; Janvier et al.,
2013; Qiu et al., 2017) that expand the essentially two-dimensional standard
flare model to three dimensions, our observation of a synchronized northward
motion of the microwave/X-ray/EUV looptop source and the UV footpoints favors
a scenario in which the flux rope erupts in a zipper-like fashion. In this
scenario, the primary reconnection site moves nearly parallel to the ribbon.
The motion can be attributed to asymmetric flux rope eruption, which may be
caused by an asymmetric external magnetic confinement (e.g. Tripathi et al.,
2006; Liu et al., 2009, 2010; Zimovets et al., 2021). Episodic energy release
events during the zipper reconnection may be also responsible for the observed
multiple episodes of microwave and X-ray bursts.

We suggest that the observed kinematics of the erupting flux rope is also
consistent with the two reconnection scenarios during the pre- to post-
impulsive phase of this event. The flux rope starts to accelerate after the
event enters the main-impulsive phase. Compared to that in the post-impulsive
phase (especially bursts # 2 and 3), the increase in acceleration during the
main-impulsive phase is relatively insignificant (Figure 10). In contrast, the
acceleration shows a clear increase when entering the post-impulsive phase
bursts. In an eruptive flare, the kinematic evolution of the flux rope usually
starts with a slow-rise phase followed by an impulsive acceleration phase
(Zhang et al., 2001). The slow-rise phase is often found to have an
approximately linear height-time profile, and the process is usually
attributed to the tether-cutting reconnection scenario (Sterling et al., 2007;
Schrijver et al., 2008; Cheng et al., 2020). By contrast, the fast-rise phase
is often attributed to either runaway reconnection (Sterling et al., 2011) or
positive feedback from the fast flare reconnection below the flux rope (Lin &
Forbes, 2000; Cheng et al., 2020; Liu & Su, 2021; Jiang et al., 2021). We
suggest that the differences we observe in the main- and post-impulsive phase
in the flux rope acceleration are generally similar to those that distinguish
the slow- and fast-rise phase in other eruptive flares.

Similar to the previous studies, we attribute the acceleration of the flux
rope to the positive feedback from the reconnection occurring in the current
sheet trailing the flux rope. However, rather than a dominant and impulsive
driver that gives rise to a prominent acceleration period, in our event, the
driver may be intermittent during the zipper-like reconnection. The rise of
the flux rope is likely asymmetric. It propagates from the active leg on the
southwestern side to the anchored leg on the northeastern side (Figure 13(c)
and (d)), giving rise to the systematic motion of the looptop X-ray,
microwave, and EUV sources and the footpoint brightenings. Meanwhile, the
intermittent reconnection also drives the multiple acceleration episodes
during the post-impulsive phase (see, e.g., Liu et al., 2009).

Now we turn our attention to the energization of nonthermal electrons during
the main- and post-impulsive phase. As shown in Figure 6 and Table 1, the
power-law index of the observed X-ray spectrum during the main-impulsive phase
seems even larger (softer) than that during the post-impulsive phase despite
having a much brighter flare emission at $<$ $\sim$20 keV. During the
impulsive phase, the reconnection between magnetic loops usually has a large
shear (Moore & Labonte, 1980; Moore et al., 2001) (Figure 12(a)).
Consequently, the guide field is large, which leads to the low productivity of
nonthermal electrons during the main impulsive phase (Dahlin et al., 2016; Li
et al., 2017; Arnold et al., 2021; Qiu & Cheng, 2022). For the post-impulsive
phase bursts, we find that the power-law index $\delta^{\prime}$ of the
nonthermal electron energy distribution is harder for later bursts. The
hardening of the electron energy spectrum coincides with an increasing
acceleration of the flux rope during these post-impulsive bursts. Previous
observational and modeling studies have suggested that, when the eruption is
well underway, the flux rope acceleration serves as an excellent proxy for the
rate of magnetic energy release via reconnection. Therefore, we attribute the
hardening of the nonthermal electron spectra to an increasing magnetic energy
release rate which, in turn, facilitates the acceleration of nonthermal
electrons to higher energies. We also note that the inclination angle $\theta$
of the post-flare arcade with regard to the PIL appears to increase throughout
the post-impulsive bursts (Figure 12(b)). Such a change implies a smaller
shear of the reconnecting magnetic field and a decreasing guide field. We
suggest that such a decreasing guide field component may also contributes to
the hardening of the nonthermal electron spectra. In summary, we have
presented an eruptive flare event that features three post-impulsive X-ray and
microwave bursts immediately following its main impulsive phase. We have
investigated the relationship between the flux rope acceleration and the
electron energization in the context of the flare geometry and its evolution.
We have found a positive correlation between the flux rope acceleration and
electron energization during the post-impulsive phase bursts, conforming to
the standard CME-flare scenario in which positive feedback between flare
reconnection and flux rope acceleration is expected. In contrast, such a
correlation does not seem to hold during its main impulsive phase. We
attribute the lack of flux rope acceleration during the main impulsive phase
to the tether-cutting reconnection scenario when the flux rope eruption has
not been fully underway. Our observations also suggest a weakening guide field
may contribute to the hardening of the nonthermal electron spectrum throughout
the main- and post-impulsive phases of the event.

The Expanded Owens Valley Solar Array (EOVSA) was designed, built, and is now
operated by the New Jersey Institute of Technology (NJIT) as a community
facility. EOVSA operations are supported by NSF grant AGS-2130832 and NASA
grant 80NSSC20K0026 to NJIT. This work is primarily supported by NASA grant
80NSSC20K1318 to NJIT. Additionally, it receives support by NSF under grants
AGS-1654382, AGS-1954737, AGS-1821294, AST-2108853, AST-2204384, and NASA
under grants 80NSSC19K0068, 80NSSC20K0627, 80NSSC20K1282, 80NSSC21K1671, and
80NSSC21K0003 to NJIT. Animated Figure 14 shows the time-evolution of the
loop-like feature at the front of the rising magnetic flux rope observed by
SDO/AIA 94 Å as well as its trajectory in the time-distance plot.

Figure 14:  This 7-second animation illustrates the ascent of the tracked
eruptive feature, identified as the flux rope front, as captured in SDO/AIA 94
Å images. The animation also depicts its trajectory in the corresponding time-
distance plot. This animation serves as a supplementary visual aid to Figure
10. (a) Time-distance plot of SDO/AIA 94 Å background-subtracted images with a
cut as shown as the white dashed line in Figure 5 (g). The orange curve shows
the EOVSA microwave flaring-region-integrated light curve at 7.9 GHz. The blue
solid line indicates the selected feature for tracking. (b) SDO/AIA 94 Å
background-subtracted images. The blue dot is the tracked point along the slit
used in (a).

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# Mixed spin S=1 and S=½ layered lattice in Cu2F5

Dmitry M. Korotin<EMAIL_ADDRESS>M.N. Mikheev Institute of Metal Physics
of Ural Branch of Russian Academy of Sciences, 18 S. Kovalevskaya St.,
Yekaterinburg, 620137, Russia.  Skolkovo Institute of Science and Technology,
30 Bolshoy Boulevard, bld.1, Moscow, 121205, Russia  Dmitry Y. Novoselov  M.N.
Mikheev Institute of Metal Physics of Ural Branch of Russian Academy of
Sciences, 18 S. Kovalevskaya St., Yekaterinburg, 620137, Russia.  Skolkovo
Institute of Science and Technology, 30 Bolshoy Boulevard, bld.1, Moscow,
121205, Russia  Department of Theoretical Physics and Applied Mathematics,
Ural Federal University, 19 Mira St., Yekaterinburg 620002, Russia  Vladimir
I. Anisimov  M.N. Mikheev Institute of Metal Physics of Ural Branch of Russian
Academy of Sciences, 18 S. Kovalevskaya St., Yekaterinburg, 620137, Russia.
Skolkovo Institute of Science and Technology, 30 Bolshoy Boulevard, bld.1,
Moscow, 121205, Russia  Department of Theoretical Physics and Applied
Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg 620002,
Russia  Artem R. Oganov  Skolkovo Institute of Science and Technology, 30
Bolshoy Boulevard, bld.1, Moscow, 121205, Russia

###### Abstract

The electronic and magnetic structure, including the Heisenberg model exchange
interaction parameters, was explored for the recently proposed novel cuprate
Cu2F5. Using the DFT+U calculation, it is shown that the compound is formed by
two types of copper ions with $d^{9}$ and $d^{8}$ electronic configurations.
We have found a very stable antiferromagnetic ordering with strong anisotropy
of exchange interaction that results in the appearance of an unusual
2D-magnetism: within the (100)-plane the exchange between the S=1 and S=1/2 Cu
ions has almost the same strength as between the two S=1 ions. The interplane
magnetic interaction is five times weaker than the in-plane one.

##  I Introduction

The low-dimensional spin-lattices exist in a plethora of forms such as spin-
ladders [1, 2, 3], plaquettes [4], dimers and zig-zags [5, 6]. The compounds
owning these magnetic structures demonstrate unusual magnetic excitation
spectra, including the appearance of the spin-gap. In the present work, we’ve
found a new type of 2D spin-lattice in Cu2F5– the novel stable copper fluoride
Cu2F5 predicted theoretically recently in our work [7]. Here we analyze its
electronic and magnetic structure in detail, relying on the known crystal
structure.

The novel copper fluorides Cu2F5 and CuF3, proposed earlier [7] have
structural and electronic similarity with high-$T_{c}$ cuprates, such as
La2CuO4 and with prototypical correlated perovskite compound KCuF3. Following
the analogy to the mentioned compounds, one can assume the existence in Cu2F5
of AFM ordered spin-lattice of the Cu ions moments. At the same time, the type
of the ordering and dimensionality of the spin-lattice is unclear and should
be obtained in calculations until experimental data appear.

The proposed analogy between Cu2F5 and La2CuO4 inspires us to analyze the spin
state of Cu ions in Cu2F5 and to determine the orbital content of the
electronic states right below the Fermi level. We have also explored the
magnetic exchange interactions within the Cu2F5 crystal to find some 2D
magnetic structures.

Figure 1: Crystal structure of Cu2F5. Blue spheres denote Cu ions inside the
ligand’s octahedron, magenta spheres denote Cu ions in the center of
plaquettes, grey spheres – F ions.

The theoretically predicted [7] (using DFT) crystal structure of Cu2F5 has
monoclinic C2/m symmetry and contains Cu ions with two types of coordination
(see Fig. 1). The first type of Cu ions (Cu1) is in the center of a slightly
distorted octahedron with three pairwise Cu-F distances (1.98, 1.96, 1.89
$\AA$) and only one F-Cu-F angle equal to 91.3$\degree$ (other F-Cu-F angles
are 90$\degree$). Another type of Cu ions (Cu2) is in the center of a square
formed by four F ions with two different Cu-F distances (1.83 and 1.86 $\AA$).
All F-Cu-F angles within such square plaquette are equal to 90$\degree$. Cu1
and Cu2 ions alternate along $a$ and $b$ crystal axes and form chains of the
same type ions along the $c$ axis.

If one neglects (not in calculations, but for simplicity of interpretation)
the deviation of the F-Cu1-F angle from the 90$\degree$ value then the Cu1 ion
type has the D4h point group symmetry. It means that the Cu $d$-level splits
into the $t_{2g}$ ($d_{zx}$, $d_{zy}$, $d_{xy}$) and $e_{g}$
($d_{x^{2}-y^{2}}$, $d_{3z^{2}-r^{2}}$) subshells. For the Cu2 type of ions
the square-planar surrounding splits the $d$-level into (from highest to
lowest): $b_{1g}(d_{x^{2}-y^{2}})$, $b_{2g}(d_{xy})$, $e_{g}(d_{zx},d_{zy})$
and $a_{1g}(d_{3z^{2}-r^{2}})$ subshells.

To interpret our results in terms of Cu $d$-orbitals, we have defined the
local coordinate system (LCS) for each Cu ion as shown in Fig. 6 with the
$z$-direction perpendicular to the plaquette plane for the Cu2 ion and the
$z$-direction along with the crystal $b$-vector for the Cu1 ion. Below,
analyzing densities of states, hopping integrals, exchange interaction, etc.
we will refer to the $d$-orbitals defined in this LCS.

##  II Methods

All calculations were performed using Quantum-ESPRESSO [8] package with
pseudopotentials from pslibrary set [9]. The exchange-correlation functional
was chosen to be in Perdew-Burke-Ernzerhof [10] form. The energy cutoff for
plane wave wavefunctions and charge-density expansion has been set to 50 Ry
and 400 Ry, respectively. Integration in the reciprocal space was done on a
regular $8\times 8\times 8$ $k$-points mesh in the irreducible part of the
Brillouin zone.

In Section III we first explore the electronic structure within DFT and then,
keeping in mind the partially filled $d$-shell of the Cu ions, continue with
the DFT+U approximation. The DFT+U method is a compromise between the ability
to describe correlated $d$-states and computational cost of the calculations.
At each value of Hubbard’s $U$ parameter we relaxed the crystal structure. At
the same time, Hund’s parameter $J$ was kept fixed at 0.9 eV in all DFT+U
calculations; the relation between crystal field splitting of the $e_{g}$
level of Cu1 ion and $J$ parameter could, in principle, affect the stability
of the $S=1$ magnetic configuration, but we do not expect such a case because
the splitting of $e_{g}$ levels is small. $J$ = 0.9 eV is typical for cuprates
[11, 12]. As soon as experimental crystal structure of Cu2F5 will become
available, it will be intriguing to calculate the $U$ and $J$ parameters with
the constrained DFT [13] or linear response [14] methods and investigate the
electronic structure of this compound with a more advanced approach such as
DFT+DMFT [15, 16].

The convergence criteria used for the crystal cell relaxation within DFT+U
are: total energy $<10^{-6}$ Ry, total force $<10^{-3}$ Ry/Bohr, pressure
$<0.5$ kbar.

##  III  Results

Partial densities of states (pDOSes) obtained within spin-unpolarized DFT
calculation are shown in Fig. 2. For the Cu1 ions, there is the filled
$t_{2g}$ energy band with the width $\approx$4.5 eV. The same three orbitals
($d_{xy}$, $d_{zx}$ and $d_{zy}$,) are filled for the Cu2 ions too, and the
corresponding energy bands are located in the same energy region.

The situation is different for the other two $d-$states. Both $e_{g}$-states
of the Cu1 ions are partially filled, their energy bands cross the Fermi
level, and the occupations (from DFT) are 0.8 $e$ and 0.74 $e$. Contrary, the
$d_{3z^{2}-r^{2}}$ orbital of the Cu2 ion (in the center of the flat
plaquette) is almost filled, and the corresponding peak of DOS is located
$\approx$1.2 eV below the Fermi level. And the $d_{x^{2}-y^{2}}$-orbital of
the Cu2 ions is partially filled with the occupation 0.62 $e$.

Keeping in mind the existence of partially filled $d$-states, one should
consider possible magnetic configurations. Also, the strong electronic
correlations between the Cu electrons better be taken into account. We have
compared the total energies of the Cu2F5 cell calculated for ferromagnetic
(FM) and two types of antiferromagnetic (AFM) orderings (shown in Fig. 3).
Since this compound has not yet been made in an experiment, and it is not
known is it metallic or insulating, we used various values for the Hubbard $U$
parameter from 2 to 8 eV. The AFM G-type of ordering is always favorable and
has the total energy $\approx$300 meV lower than the other two phases.

Figure 2: Partial densities of states for Cu2F5 obtained within spin-
unpolarized DFT calculation. Figure 3: Two types of considered AFM orderings
of Cu ions magnetic moments in Cu2F5. Dark blue octahedra are CuF6 octahedra,
light-magenta rhombuses are CuF4 plaquettes. Figure 4: Partial densities of
states for Cu2F5 obtained within DFT+U for the AFM-G ordered phase.
Positive/negative pDOSes correspond to spin-up/down states respectively.

The partial densities of states for AFM-G ordered magnetic phase of the Cu2F5
calculated within the DFT+U approach are shown in Fig. 4. For each value of
the $U$ parameter, the cell volume and atomic positions were relaxed to obtain
the ground-state crystal structure. Starting from $U$=4 eV the Cu2F5 is an
insulator. The band gap at $U$=4 eV is 0.48 eV and it broadens with increasing
of the $U$ parameter. For this $U$ value it is an antiferromagnetic insulator
with the top of the valence band formed by $x^{2}-y^{2}$ states of the Cu2
ion. The increase of $U$ results in the shifting of the occupied Cu-$d$ states
into the fluorine band. For $U$=6 and 8 eV, Cu2F5 is a charge-transfer
insulator with the band gap equals to 0.9 eV.

From the left panel of Fig. 4, it is seen that the Cu ion inside the fluorine
octahedron (Cu1 ion) has unusual Cu3+ valence with electronic configuration
$d^{8}$. The two peaks in spin-down DOSes for $3z^{2}-r^{2}$ and $x^{2}-y^{2}$
orbitals at +0.9 eV should be interpreted as an empty $e_{g}$-states of Cu and
the $t_{2g}$ states are filled. The second type of Cu ion (in the center of
fluorine plaquettes) has the $d^{9}$ electronic configuration with an empty
spin-down $x^{2}-y^{2}$ orbital located at 2-3 eV above the top of the valence
band. Consequently, there is S=1 spin-state for Cu1 ion and S=1/2 spin-state
for the plaquette-centered Cu ion. The position of the bottom of the
conduction band is only slightly affected by the electron-electron interaction
power due to strong hybridization between the F-$p$ and Cu-$d$ states in the
corresponding energy interval.

Using Green’s function method based on magnetic-force linear response theory
[17] we computed the Heisenberg exchange interaction between Cu ions up to the
9th nearest neighbor. The model Hamiltonian has the form:

$H=-\sum_{\langle ij\rangle}J_{ij}{\bf e}_{i}{\bf e}_{j},$ (1)

where ${\bf e}_{i}$ are the unit vectors pointing in the direction of the
$i$th site magnetization, and the summation runs once over each ion pair.

Figure 5: The pattern of the exchange interaction within the (100) layer of Cu
ions. Open circles denote Cu1 ions with S=1 in the center of CuF6 octahedron;
filled circles are Cu2 ions with S=1/2 (plaquettes centered). The strongest
exchanges are: $J_{2D}^{b}$ (green lines) and $J_{2D}^{c}$ (violet lines). The
interlayer exchange is $J_{\bot}$ is five times smaller than intralayer ones.
Other exchange interactions are negligible. Fluorine ions are not shown for
clarity.

The obtained $J_{i}$ values are presented in Table 1 (only values larger than
1.2 meV are included). The spatial illustration of the exchange interaction
directions is shown in Fig. 5.

$U$ |  $J_{2D}^{b}$ (meV) |  $J_{2D}^{c}$ (meV) |  $J_{\bot}$ (meV)
---|---|---|---
4 eV | -32.5 | -34.2 | -6.8
6 eV | -34.5 | -40.2 | -6.9
8 eV | -32.7 | -42.8 | -6.3
Table 1: Calculated values of exchange interaction parameters within the (100)
layer ($J_{2D}^{b}$ and $J_{2D}^{c}$) and between these layers ($J_{\bot}$)
Figure 6: Upper panel: The scheme of the half-filled Cu $d_{3z^{2}-r^{2}}$ →
F-p → the half-filled Cu $d_{x^{2}-y^{2}}$ superexchange mechanism. Lower
panel: Two half-filled $d$-orbitals providing the AFM superexchange
interaction along the $b$ crystal axis ($J_{2D}^{b}$). The $d$-orbitals are
defined in own local coordinate systems for the octahedron (x’y’z’) and
plaquette (x”y”z”). The fluoride $p$-orbital, that mediates the superexchange
interaction, located between the $d$-orbitals isn’t drawn for picture
simplicity.

All $J$ values have the same sign that corresponds to the AFM exchange. Both:
the Cu1-Cu2 interaction along the $b$-axis ($J_{2D}^{b}$) together with
Cu1-Cu1 interaction along the $c$-axis ($J_{2D}^{c}$), are five times larger
than the exchange interaction along the $a$-crystal axis ($J_{\bot}$). The
superexchange interaction along the $b$ crystal axis ($J_{2D}^{b}$) is
provided by the hopping of electrons between the half-filled Cu1
$3z^{2}-r^{2}$ orbital, the $p$-orbital of fluorine (located between the Cu
ions), and the half-filled Cu2 $x^{2}-y^{2}$ orbital as illustrated in Fig. 6.
The second half-filled $e_{g}$ orbital of the Cu1 ion – $x^{2}-y^{2}$
maintains Cu1-F-Cu1 superexchange interaction along the $c$ crystal axis
($J_{2D}^{c}$). In both interactions described above the Cu-F-Cu angle is
180o. Since this bond angle defines the $p$-$d$ orbitals overlap it,
consequently, determines the exchange strength. The Cu1-F-Cu2 along $a$ angle
is 129o only. That sets the much weaker exchange interaction in the direction
of the crystal $a$-axis ($J_{\bot}$). As for the Cu2-Cu2 exchange along the
$c$-axis, it is negligible because no fluorine ions are providing the
superexchange path.

Thereby a strong superexchange anisotropy in Cu2F5 exists in a way when the
in-plane magnetic interactions are significantly larger than the interplane
one. The absolute values of the exchange parameters are comparable with the
superexchange in 2D [18, 19], ladder cuprates [19] and 1D chain cuprates [19,
20]. The unusual thing is that the exchange energy between Cu ions with S=1 is
almost the same as between the S=1 and S=½ions.

One can suggest that doping of Cu2F5 with electrons would result in filling
the Cu1 $3z^{2}-r^{2}$ orbital since the bottom of the conduction band is
formed by these states according to Fig. 4. That will shift the Cu1 ion from
the $d^{8}$ configuration and from the half-filled $e_{g}$-subshell. According
to the Goodenough-Kanamori rule [21], it will suppress the $J_{2D}^{b}$ AFM
superexchange mechanism illustrated in the upper panel of Fig. 6. As a result,
1D magnetic chains could arise along the $c$-crystal axis.

In conclusion, we have presented a DFT+U study of the electronic and magnetic
structure of the novel copper fluoride. The Cu ions in this compound have two
different valences: 2+ for Cu ion in the square coordination and 3+ for Cu in
octahedral coordination. We have tested various Hubbard interaction parameters
$U$ for Cu2F5 and showed that the compound becomes an insulator starting from
$U$=4 eV. The value of the energy gap depends only slightly on the $U$ value.

Calculated values of superexchange interaction parameters indicate that the
significant magnetic interaction anisotropy exists in Cu2F5. The new 2D spin-
lattice is obtained: the exchange interactions between Cu ions in the (100)
planes are five times larger than along the $a$-crystal axis, and within the
layer, the exchange between ions with different spins (S=1 and S=½) has the
same magnitude.

## Acknowledgments

The presented results were obtained with support of Russian Science Foundation
(Project 19-12-00012).

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