# Crystal structures and superconducting properties of metallic double-chain
based cuprate Pr2Ba4Cu7O15-δ

Masahide Hagawa1, Michiaki Matsukawa1, Kota Niinuma1, Reiya Kudo1, Yuto
Mizusihima1, Naohisa Kawarada1, Hajime Yamamoto2, Kazuhiro Sano3, Yoshiaki
Ōno4 and Takahiko Sasaki5<EMAIL_ADDRESS>1Faculty of Science and
Engineering, Iwate University, Morioka 020-8551, Japan
2Institute of Multidisciplinary Research for Advanced Materials, Tohoku
University, Sendai 980-8577, Japan
3Department of Physics Engineering, Mie University, Tsu 514-8507, Japan
4Department of Physics, Niigata University, Niigata 950-2181, Japan
5Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

###### Abstract

We demonstrated the lattice structures and the superconducting phases of
metallic double-chain based cuprate Pr2Ba4Cu7O15-δ exhibiting higher
$T\mathrm{{}_{c}}$. After the oxygen heat treatment on citrate pyrolysis
precursors, their reduction treatment followed by a quench procedure caused
higher $T\mathrm{{}_{c}}$ samples with 26.5$\sim$30 K. The crystal structural
parameters for the superconducting sample ($\delta$ = 0.81) were analyzed from
the powder synchrotron X-ray diffraction data using RIETAN-FP program. The
effect of magnetic field on the superconducting phase of these samples with
different oxygen defects ($\delta$ =0.73, 0.81 and 0.87) was examined, for our
understanding of the superconducting magnetic field-temperature phase diagram.
For the $\delta$ = 0.87 sample with $T\mathrm{{}_{c}}$ $\sim$30 K, the
resistive critical field $H\mathrm{{}_{c}}^{*}$ was estimated to be 13 T at
4.2 K. The resistive transition broadening under the applied magnetic field
was not due to the motion of magnetic flux driven by Lorenz force. The oxygen
deficiency dependence on $T\mathrm{{}_{c,on}}$ for our samples was compared
with the data of several other groups.

##  I Introduction

Since the discovery of superconductivity in the metallic double-chain based
cuprate Pr2Ba4Cu7O15-δ (Pr247)[1], experimental and theoretical studies on
this system have been extensively reported [2, 3, 4, 5, 6, 7, 8]. The Pr247
system is considered to be an intermediate compound between the 2D high-
$T\mathrm{{}_{c}}$ layered cuprate superconductors and the quasi-1D organic
superconductors, as well as the spin-ladder compounds. It is well known that
the spin ladder system exhibits superconductivity with
$T\mathrm{{}_{c}}\sim$12 K only under an external pressure above $\sim$3.0 GPa
[9]. Pr2Ba4Cu7O15-δ cuprate is an electron-doped superconductor with
$T\mathrm{{}_{c}}\sim$15 K at ambient pressure through controlling the amount
of oxygen deficiency due to a reduction treatment.

For the crystal structure of Pr2Ba4Cu7O${}_{15-\delta}\ $ (Fig.1), the
PrBa2Cu3O7-δ (Pr123)-type blocks containing the CuO sinlgle chains are
alternatively stacked along the $c$-axis with the PrBa2Cu4O8 (Pr123)-type ones
containing the CuO double chains. The Pr123 and Pr124 compounds show no
superconductivity associated with the CuO2 planes because the strong
hybridization between Pr-4f and O-2p orbitals suppresses their metallic
conduction [10]. Furthermore, an angle-resolved photoemission study revealed
the CuO single chain in Pr123 and the CuO double chain in Pr124 exhibit
insulating and metallic characters, respectively, although both chains are
approximately quarter filling [11]. Recently, a Cu nuclear quadrupole
resonance study on Pr247 with a full volume fraction of superconductive phase
has revealed that the CuO2 plane is an antiferromagnetically insulating state
at 2 K despite its bulk superconductivity[12, 13]. This finding strongly
indicates that the superconductivity in Pr247 appears along the metallic CuO
double chains, considering the oxygen deficiency at the CuO single chain site.
For our further understanding of dimensionality in the electronic state of CuO
double chains, the lattice, transport, and magnetic properties of Pr247 have
been examined under an external pressure. Both the resistivity and
magnetization measurements revealed that the superconducting behaviors in
Pr247 are suppressed by the pressure effect[14, 15]. The negative pressure
dependences of the superconducting phase are qualitatively explained by the
normal to superconductive phase diagram of CuO double chains on the basis of
Tomonaga-Luttinger liquid theory[6]. The magneto resistance (MR) effect of the
normal phase of Pr247 is enhanced due to applied pressure, which is probably
related to a warped Fermi surface induced by the pressure. Taniguchi et al,
found out from in-situ XRD experiments under pressure that the $a$-axis
lattice parameter across the CuO double chain along the $b$-axis is elongated
against pressure above 2.0 GPa[16].

Figure 1: Projection along (100) of the crystal structure of Pr2Ba4Cu7O${}_{15-\delta}\ $. The CuO sinlgle-chain 123- and double-chain 124-blocks are alternatively stacked along the $c$-axis. The nomenclature of atomic positions is displayed on the left hand side. The polyhedral planar part represents CuO2 plane site. Figure 2: Powder synchrotron X-ray diffraction pattern on the superconducting sample Pr2Ba4Cu7O15-δ ($\delta$ = 0.81) at room temperature. The calculated and residual curves are also shown using RIETAN-FP program for two-phase Rietveld refinement. The upper and lower tick marks represent peak positions for the Pr2Ba4Cu7O15-δ and BaCuO2 phases, respectively. The mass fractions of Pr247 and BaCuO2 were estimated to be $96.0\%:4.0\%$. $\lambda$ = 0.42 Å.  Table 1: Lattice parameters and superconducting properties for the as-sintered and oxygen-reduction samples of Pr2Ba4Cu7O15-δ ($\delta$ =0, 0.73, 0.81, and 0.87). In details, see the corresponding text and references. $f_{\mathrm{SC}}$: the superconducting volume fraction estimated from the ZFC magnetization data at low temperatures. $\mathrm{T_{c,on}}$: defined as the onset temperature of resistive drop. The reliability factors for four samples are listed.  Sample | oxygen |  ${a}$ |  ${b}$ |  ${c}$ | $f\mathrm{{}_{SC}}$ | $\mathrm{T_{c,on}}$ |  $R\mathrm{{}_{wp}}$ | $R\mathrm{{}_{p}}$ | $R\mathrm{{}_{B}}$
---|---|---|---|---|---|---|---|---|---
No. | defect $\delta$ |  (Å) |  (Å) |  (Å) | (%) | ($K$) | (%) | (%) | (%)
#1 | - | 3.87958 (5) | 3.89975 (1) | 50.6281(2) |  - |  - | 7.95 | 5.61 | 1.54
#2-0 | - | 3.8791(5)* | 3.8997(5) * | 50.628(6)* |  - |  - | 11.9* | 9.14* | 4.24*
#2-1 | 0.81 | 3.88287(4) | 3.89003(4) | 50.8485(4) | 63 | 28.0 | 6.62 | 4.86 | 2.83
#3 | 0.73 | 3.8821(1) | 3.8899(1) | 50.8160(6) | 59 | 26.5 | 4.88 | 3.69 | 4.36
#4 | 0.87 | - | - | - | 53 | 30.0 | - | - | -
ref. [2] | - | 3.89127(9) | 3.90714(8) | 50.695(1) |  - |  - | 4.51 | 3.31 |  -
*Ultima IV | Cu-K$\alpha$ |  |  |  |  |  |  |  |

In this paper, we demonstrate crystal structures and superconducting
properties of metallic double-chain based cuprate Pr2Ba4Cu7O15-δ. In the next
section, the experimental outline is described. In Section 3, we present the
powder synchrotron X-ray diffraction data on the superconducting sample
Pr2Ba4Cu7O15-δ ($\delta$ = 0.81) using RIETAN-FP program for two-phase
Pr2Ba4Cu7O15-δ and BaCuO2. We examine the magnetic field effect on the
superconducting phase of these three samples with different oxygen defects
($\delta$ =0.73, 0.81 and 0.87). The superconducting magnetic field-
temperature phase diagram is determined from the low-temperature MR effect.
Concluding remarks are given in the final section.

##  II Experiment

Precursors for the superconducting cuprates Pr2Ba4Cu7O15-δ were prepared
through spontaneous combustion reactions using a citrate pyrolysis method[3,
17]. (refer to a video file recording the typical self-ignition process
attached in Appendix A in ref.[18]). The fine powders crushed from their
precursors were annealed under ambient oxygen atmosphere at 890-891 ∘C for
96-110 hours in an electric tube furnace. A narrow stability range of
annealing temperature between 890 and 891 ∘C allowed synthesis of pure phase
of Pr247 with the aid of a three-zone temperature controller. As for the as-
sintered samples, oxygen removed procedures are further needed in order to
realize their superconducting phases. The oxygen contents were reduced through
heat treatment in a vacuum atmosphere at 500-600 ∘C for 3-5 days. We conducted
gravimetric analysis on the present three samples and their oxygen defects
were estimated to be $\delta=$ 0.73, 0.81, and 0.87 (referred as samples #2-1,
#3, and #4). A quench process in air from 300 ∘C down to room temperature was
an effective approach to achieve better superconducting properties, in
comparison to furnace cooling procedures in our previous synthesis. We carried
out powder synchrotron X-ray diffraction measurements on the fine powders
crashed from the superconducting bulk samples at SPring-8 beamline BL02B2 at
room temperature. The wavelength of the incident X-ray, $\lambda$ = 0.42 Å,
was utilized. The collected diffraction patterns were fitted using RIETAN-FP
program two-phase Rietveld refinement[19]. The electric resistivity
measurement was measured by the $dc$ four-terminal method. The magneto-
transport up to 9 T was measured by the $ac$ four-probe method using a
physical property measuring system (PPMS, Quantum Design), from 4 K to 40 K at
several magnetic fields ($\mu_{0}H$=0, 1, 3, 6, and 9 T) after the zero-field-
cooling (ZFC) scan . The low-temperature magneto-resistance (MR) effect (up to
14 T) was measured in a superconducting magnet (15T-SM) at Institute for
Materials Research, Tohoku University. The electric current $I$ was applied
longitudinally to the sample ; consequently, the applied magnetic field $H$
was transverse to the sample (because $H\perp I$). The $dc$ magnetization was
performed under ZFC in a commercial superconducting quantum interference
device magnetometer (Quantum Design, MPMS).

Table 2: Selected bond lengths for Pr2Ba4Cu7O15-δ. determined by the powder synchrotron X-ray diffraction patterns. See Fig. 1 for positional notation of Pr247. The atomic positions for Pr 124 and Pr123 are referred to the published data[21, 22]. Cupl and Cuch denote the planar and chain copper sites, respectively. Oap and Och are the apical and double-chain oxygen sites.  bond length | Pr247/$\delta$=0.81 | #2-1 | Pr247 ref. [2] | as-sintered | Pr124 | Pr123
---|---|---|---|---|---|---
(Å) | 124-block | 123-block | 124-block | 123-block | ref. [21] | ref. [22]
Cupl-Cupl |  Cu2-Cu3 |  |  Cu2-Cu3 |  | Cu2-Cu2 | Cu2-Cu2
|  3.528(5) |  |  3.525 |  | 3.508 | 3.554
Cupl-Oap | Cu3-O6 | Cu2-O1 | Cu3-O6 | Cu2-O1 | Cu2-O1 | Cu2-O2
| 2.183(4) | 2.095(20) | 2.196 | 2.136 | 2.234 | 2.232
Cuch-Oap | Cu4-O6 | Cu1-O1 | Cu4-O6 | Cu1-O1 | Cu1-O1 | Cu1-O2
| 1.835(3) | 2.010(20) | 1.884 | 1.957 | 1.836 | 1.854
Cuch-Och($\parallel c$) | Cu4-O7 | - | Cu4-O7 | - | Cu1-O4 | -
| 1.735(16) | - | 1.807 | - | 1.876 | -
Cuch-Och($\parallel b$) | Cu4-O7 | - | Cu4-O7 | - | Cu1-O4 | -
| 1.975(3) | - | 1.960 | - | 1.955 | -
Table 3: Atomic position parameters of Pr2.2Ba3.8Cu7O15-δ ($\delta$=0.8). In our Rietveld refinement of the synchrotron and conventional X-ray diffraction patterns for all the samples investigated, similar atomic position parameters are adopted. $B$ is an isotropic thermal parameter. For oxygen atoms, $B$ is fixed to be 1.0 $\AA^{2}$.  #2-1 |  |  $\delta$=0.81 |  | (Ammm) |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---
Atom |  | Site |  | Occ. |  | $x$ |  |  $y$ |  |  $z$ |  |  $B$ ($\AA^{2}$)
Pr1 |  | 4$j$ |  | 1.0 |  | 0.5 |  | 0.5 |  | 0.11559(4) |  | 0.15(2)
Ba1 |  | 4$j$ |  | 0.95 |  | 0.5 |  | 0.04188(3) |  | 0.5 |  | 0.51(2)
Pr2 |  | 4$j$ |  | 0.05 |  | 0.5 |  | 0.04188(3) |  | 0.5 |  | 0.15(2)
Ba2 |  | 4$j$ |  | 0.95 |  | 0.5 |  | 0.18865(3) |  | 0.5 |  | 0.51(2)
Pr3 |  | 4$j$ |  | 0.05 |  | 0.5 |  | 0.18865(3) |  | 0.5 |  | 0.15(2)
Cu1 |  | 2$a$ |  | 1.0 |  | 0.0 |  | 0.0 |  | 0.0 |  | 0.33(2)
Cu2 |  | 4$i$ |  | 1.0 |  | 0.0 |  | 0.0 |  | 0.08073(7) |  | 0.33(2)
Cu3 |  | 4$i$ |  | 1.0 |  | 0.0 |  | 0.5 |  | 0.15027(7) |  | 0.33(2)
Cu4 |  | 4$i$ |  | 1.0 |  | 0.0 |  | 0.0 |  | 0.22965(6) |  | 0.33(2)
O1 |  | 4$i$ |  | 1.0 |  | 0.0 |  | 0.0 |  | 0.0386(4) |  | 1.0
O2 |  | 4$j$ |  | 1.0 |  | 0.5 |  | 0.0 |  | 0.0828(4) |  | 1.0
O3 |  | 4$i$ |  | 1.0 |  | 0.0 |  | 0.5 |  | 0.0861(3) |  | 1.0
O4 |  | 4$j$ |  | 1.0 |  | 0.5 |  | 0.0 |  | 0.1486(4) |  | 1.0
O5 |  | 4$i$ |  | 1.0 |  | 0.0 |  | 0.5 |  | 0.1421(3) |  | 1.0
O6 |  | 4$i$ |  | 1.0 |  | 0.0 |  | 0.0 |  | 0.1935(3) |  | 1.0
O7 |  | 4$i$ |  | 1.0 |  | 0.0 |  | 0.5 |  | 0.2355(3) |  | 1.0
O8 |  | 2$b$ |  | 0.1 |  | 0.0 |  | 0.5 |  | 0.0 |  | 1.0
O9 |  | 2$d$ |  | 0.1 |  | 0.5 |  | 0.0 |  | 0.0 |  | 1.0
Figure 3: Temperature dependences of electric resistivities for the
superconducting Pr2Ba4Cu7O15-δ. ($a$) sample #2-1 with $\delta$=0.81, ($b$)
sample #3 with $\delta$=0.74, and ($c$) sample #4 with $\delta$=0.87. In the
inset, the low-temperature magnetization curves measured under a zero field
cooling (ZFC) scan are plotted, to clarify a bulk superconductivity.  Figure
4: Low temperature dependences of electric resistivities for the
superconducting Pr2Ba4Cu7O15-δ measured at the several applied fields
($\mu_{0}H$=0, 1, 3, 6, and 9 T). ($a$) sample #2-1 with $\delta$=0.81, ($b$)
sample #3 with $\delta$=0.74, and ($c$) sample #4 with $\delta$=0.8. In ($d$),
the data for the sample #4 are taken at 0T and 9T for the applied current
$I$=1, 0.1 and 0.02 mA . Broken lines are guides for the eyes.  Figure 5: Low-
temperature magneto-resistance effect as a function of magnetic field up to 14
T for the superconducting Pr2Ba4Cu7O15-δ. ($a$) sample #2-1 and ($b$) sample
#3 at $T$ = 4.2 K, 5 K, and 7 K. ($c$) sample #4 at $T$= 4.2 K$\sim$11 K . The
arrow symbols denote the critical fields $H\mathrm{{}_{c}}^{*}$, where a zero-
resistance state is violated at selected temperatures upon increasing applied
field. Lines are guides for the eyes.

##  III Results and Discussion

Figure 1 shows the schematic projection along (100) of the crystal structure
of Pr2Ba4Cu7O15-δ. The CuO sinlgle-chain and double-chain blocks are
alternatively stacked along the c-axis. The two-phase Rietveld refinement
results for the as-sintered and oxygen-reduction samples of Pr2Ba4Cu7O15-δ
($\delta$ =0, 0.73, and 0.81) are listed in Table LABEL:TA1. The typical
powder synchrotron X-ray diffraction data on the superconducting sample
Pr2Ba4Cu7O15-δ ($\delta$ = 0.81) at room temperature are given in Fig. 2 . The
calculated and residual curves are also plotted using RIETAN-FP program for
two-phase Pr2Ba4Cu7O15-δ and BaCuO2 . The mass fractions of Pr247 and BaCuO2
were estimated to be $96.0\%:4.0\%$.

Here, we give some comments on the atomic position parameters of
Pr2.2Ba3.8Cu7O15-δ ($\delta$=0.8) listed in Table 3. In the present Rietveld
refinement, we assume that the Ba1 and Ba2 sites are partially replaced by Pr
ions with same occupancy rates of 0.05 in the 123 and 124 blocks,
respectively. For the Pr123 compound, it has been reported that the Pr rich
and Ba poor compositions Pr1+xBa2-xCu3O7-δ are stable against the
stoichiometric one from the single phase diagram studies [23]. These findings
strongly suggest that to form the Pr123 single phase, the Ba sites must have
partial Pr ions. This tendency is naturally extended to the Pr124 and Pr247
systems. The occupancy rates at the oxygen sites O8 and O9 in the 123 block
are set to be 0.1, to match the oxygen defect.

A vacuum reduction treatment of the as-sintered samples introduces oxygen
defects, resulting in carrier doping effect associated with the appearance of
superconducting phases. It is well known that higher $T_{c}$ samples are
closely related to larger oxygen defects accompanied by the $c$-axis
elongation of Pr247. The variations of the lattice parameters due to oxygen
removal are almost similar to the previous ones[2]. In Table 2, the selected
bond lengths for Pr2Ba4Cu7O15-δ determined by the powder synchrotron X-ray
diffraction patterns are listed. Comparing the interatomic distances for the
superconducting and non-superconducting samples, we detected no clear
differences in all the bond lengths between both the samples. As for the Cuch-
Och(Cu4-O7 $\parallel c$) bond length corresponding to the inter-chain
distance along the c-axis in the 124 block, it seems to be slightly shorter in
the $\delta$ = 0.81 sample with $T_{c,on}$=28 K than that in the non-
superconducting as-sintered sample. The normal to superconducting phase
diagram of a CuO double- chain has been investigated on the basis of Tomonaga-
Luttinger liquid theory[6]. In the $n-t_{pp}$ binary phase diagram, $n$ is
electron density in the $d$ band of the CuO double chain and $t_{pp}$ is
hopping energy between nearest neighbor oxygen sites occupied by 2$p_{\sigma}$
orbitals. While the system initially remains in the normal phase, the electron
carrier doping effect causes a phase transition of it from the normal to
superconducting state. Furthermore, a shrinkage of the distance between two
chains of a CuO double chain gives rise to an increase in the hopping energy
$t_{pp}$. As a result of the heavy oxygen defects, it is expected that the
double-chain system in the normal phase is varied diagonally towards the
superconducting phase on the $n-t_{pp}$ plane, realizing double-chain driven
superconductivity.

Next, in Fig. 3, we show the temperature dependences of electric resistivities
for the superconducting Pr2Ba4Cu7O15-δ . In the inset, the low-temperature
magnetization curves measured under a zero field cooling (ZFC) scan are
plotted, to clarify a bulk superconductivity. In addition, we examined the
magnetic field effect on the superconducting phase of these three samples with
different oxygen defects, as shown in Figures 4, 5 and 6($a$).

First of all, the reduction heat treatment increased the onset temperature of
superconductive transition $T\mathrm{{}_{c,on}}$ from 26.5 K at $\delta$ =0.7,
through 28.0 K at $\delta$ =0.81 one, up to 30.0 K at $\delta$ =0.87. From our
magnetization data for the three samples, their superconducting volume
fractions $f_{\mathrm{SC}}$ reached over 50 %. Furthermore, a zero-resistance
state of these samples was observed below $T\mathrm{{}_{c,zero}}$=20$\sim$22 K
at 0 T. On the other hand, the room-temperature resistivity was enhanced from
$\sim 12$ m$\Omega$ cm to $\sim 26$ m$\Omega$ with increasing the oxygen
defect. This behavior is probably related to the highly insulating state of
the CuO2 planes due to the removal of single-chain site oxygen in 123 block,
as suggested in the oxygen content study on the Pr123 compound [22] . In the
PrBa2Cu3O7-δ system, it was reported that the room-temperature resistivity is
increased by about three orders of magnitude from $\delta=$0 to
$\delta\sim$0.5. The present Pr247 reduced samples showed the higher residual
resistance ratios (RRR), $\rho$ (RT)/ $\rho$ (30K), ranging from 10 to 12, as
pointed out in our previous study[24]. Assuming that a model of parallel
resistors along the $b$ axis consists of CuO chain and CuO2 plane resistances,
it is reasonable to consider the contribution of a doped double chain alone to
$\rho$(T), because of ignoring electric conducting path in a highly insulating
plane.

The application of external magnetic field up to 9 T caused widely broadening
of the resistive transition curves accompanied by disappearance of a zero
resistance state. $T\mathrm{{}_{c,on}}$ seems to be stable with respect to the
applied field, in spite of a rapid degradation of $T\mathrm{{}_{c,zero}}$. To
check the motion of magnetic flux due to Lorenz force, we examined the
resistive transition profiles under 9 T as a function of the electric current
ranging from 0.02 mA to 1 mA. In our present experiment (Fig. 4 ($d$)) [25],
there were no significant differences in the resistive drops among 0.02, 0.1
and 1mA. These findings strongly indicated no influence of the motion of
magnetic flux on these broadening behaviors. Our results are reminded of the
magnetic field effect on the superconducting phases in the hole doped two-leg
ladder compound Sr2Ca12Cu24O4 and high-$T\mathrm{{}_{c}}$ cuprate (La1-xSrx)2
CuO4[26, 27].

From the low-temperature electric resistivity data for the superconducting
sample #4 under several applied fields ($\mu_{0}H$=0, 1, 3, 6, and 9 T) , we
obtained the onset critical temperature as a function of magnetic field
$T\mathrm{{}_{c,on}}$($H$). Using the the Werthamer-Helfand-Hohenberg (WHH)
formula in the dirty limit,
$\mu_{0}H_{\mathrm{c2}}(0)=-0.69T_{\mathrm{c}}(dH_{\mathrm{c2}}(T)/dT)|_{T=T_{c}}$[28],
the upper critical field at $T$= 0 K, $H_{\mathrm{c2}}$(0) was evaluated to be
$\sim 45$ T. According to $H_{\mathrm{c2}}$=$\phi_{\mathrm{0}}/2\pi\xi^{2}$,
we obtained the Ginsburg-Landau coherence length $\xi\sim 27$ Å, which is
comparable to the in-plane coherence length for (La1-xSrx)2 CuO4 single-
crystal thin films[29]. To further establish the superconducting high-field
phase diagram, we determined from the low-temperature MR effect the resistive
critical field as a function of temperature $H\mathrm{{}_{c}}^{*}$($T$) ,
where a zero-resistance state is violated at selected temperatures upon
increasing applied field. For the present samples, $H\mathrm{{}_{c}}^{*}$ is
estimated to be 12-13 T at 4.2 K. In the resistive crossover region from
$T\mathrm{{}_{c,on}}$($H$) to $H\mathrm{{}_{c}}^{*}$($T$) line, it is hard to
distinguish between dissipation due to a melting of vortex lattice and an
intrinsic effect associated with the double-chain induced superconductivity.
The higher critical fields are probably attributed to the improved sample
preparation process using the three-zone electric furnace, compared with the
lower $H\mathrm{{}_{c}}^{*}$ value ($\sim$4 T at 4 K) for the previous samples
with higher $T\mathrm{{}_{c,on}}=$ 26.5 K [15]. Additionally, the quenched
treatment just after vacuum reduction annealing contributes to the enhanced
superconducting properties for the present samples. These procedures are
related to further improvements of weak links between the superconducting
grains.

Finally, in Fig. 6 ($c$), we summarize $T\mathrm{{}_{c,on}}$ vs oxygen
deficiency for our samples, compared with the data of several other groups
[30, 14, 12]. It is noted that the higher $T\mathrm{{}_{c}}$ samples around
$\sim 26$ K are synthesized by using the citrate pyrolysis based precursors
under the different heat treatment conditions. On the other hand, the lower
$T\mathrm{{}_{c}}$ samples are annealed under ambient and high pressure oxygen
atmospheres by using a solid-state reaction method for mixed fine powders. It
is true that the value of $T\mathrm{{}_{c}}$ is closely dependent on the
oxygen defects. However, the differences between two-types of the lower ad
higher $T\mathrm{{}_{c}}$ samples are difficult to explain on the basis of the
oxygen occupancy at the CuO single chain sites only. According to high-
resolution electron microscopy studies [30, 25] on Pr247, there exit stacking
faults in the -123-124-123-124- (-S-D-S-D-) regular sequence along the $c$
axis as shown in Fig.1. Here, S and D correspond to the single and double CuO
chain blocks. For instances, the stacking faults such as the -123-123-123-124-
(-S-S-S-D-) irregular sequence are reported for the citrate pyrolysis samples.
For the Y247 system[31, 32], it is also concluded that the difference in
$T\mathrm{{}_{c}}$ values is not attributed to the oxygen occupancy but micro
structures. Through TEM measurements on different samples with two-types of
preparation methods, stacking faults along the $c$-axis are observed more
frequently in a polymerized-complex sample with $T\mathrm{{}_{c}}$ = 93 K than
in a solid-state reaction sample with $T\mathrm{{}_{c}}$ = 65 K. Accordingly,
we expect that higher $T\mathrm{{}_{c}}$ samples driven by CuO double chain
superconductivity are related to stacking faults, although CuO2 based layered
superconductivity of Y247 is quite different from that of the present Pr247
system.

Figure 6: ($a$) Low-temperature magneto-resistance effect as a function of
magnetic field up to 14 T for the superconducting Pr2Ba4Cu7O15-δ (sample #4)
at $T$=12$\sim$ 15 K. ($b$) Superconducting high-field phase diagram.
$T\mathrm{{}_{c,on}}$($H$) and $H\mathrm{{}_{c}}^{*}$($T$) were taken from the
temperature scan and magnetic field scan data. ($c$) $T\mathrm{{}_{c,on}}$ vs
oxygen deficiency for our samples ($\bigcirc$). For comparison,
$\blacklozenge$, $\blacksquare$, and $\blacktriangle$ correspond to the data
of Hagiwara et al.[30], Ishikawa et al.[14], and Nishioka et al.[12]
,respectively. Lines are guides for the eyes.

##  IV Summary

We demonstrated the lattice structures and the superconducting phases of
metallic double-chain based cuprate Pr2Ba4Cu7O15-δ exhibiting higher
$T\mathrm{{}_{c}}$. After the oxygen heat treatment on the citrate pyrolysis
precursors in the thee-zone temperature controlled furnace, their reduction
treatment followed by a rapidly cooling procedure caused higher
$T\mathrm{{}_{c}}$ samples with 28$\sim$30 K. We carried out the powder
synchrotron X-ray diffraction measurements on the as-sintered and vacuum
reduction samples at Spring-8 beamline. In particular, the crystal structural
parameters for the superconducting sample ($\delta$ = 0.81) were investigated
using RIETAN-FP program for two-phase Pr2Ba4Cu7O15-δ and BaCuO2.

We examined the magnetic field effect on the superconducting phase of these
three samples with different oxygen defects ($\delta$ =0.73, 0.81 and 0.87)
and established the superconducting magnetic field-temperature phase diagram
from the low-temperature MR effect. For the $\delta$ = 0.87 sample with
$T\mathrm{{}_{c}}$ $\sim$30 K, $H\mathrm{{}_{c}}^{*}$ was estimated to be 13 T
at 4.2 K. $H_{\mathrm{c2}}$(0) was evaluated to be $\sim 45$ T, resulting in
the Ginsburg-Landau coherence length $\xi\sim 27$ Å. The resistive transition
broadening under the applied magnetic field was not due to the motion of
magnetic flux driven by Lorenz force. When oxygen deficiency dependence on
$T\mathrm{{}_{c,on}}$ for our samples was compared with the data of several
other groups [30, 14, 12], better superconductive samples driven by CuO double
chains have a close relationship with stacking faults along the $c$-axis.

The authors are grateful for M. Nakamura for his assistance in PPMS
experiments at Center for Regional Collaboration in Research and Education,
Iwate University. M. M thanks Prof. M. Hagiwara for his valuable comments. The
synchrotron radiation experiments were performed at the BL02B2 of SPring-8.

## References

  * [1] M. Matsukawa, Y. Yamada, M. Chiba, H. Ogasawara, T. Shibata, A. Matsushita, and Y. Takano, Physica C 411 (2004) 101.
  * [2] Y. Yamada and A. Matsushita, Physica C 426-431 (2005) 213.
  * [3] M. Hagiwara, T. Shima, T. Sugano, K. Koyama, and M. Matsuura, Physica C 445-448 (2006) 111.
  * [4] Y. Wakisaka, K. Takubo, T. Sudayama, J. Y. Son, T. Mizokawa, M. Arita, H. Namatame, M. Taniguchi, S. Sekiya, K. Fukuda, F. Ishikawa, and Y. Yamada, J. Phys. Soc. Jpn. 77 (2008) 074710 .
  * [5] J. Tada, M. Matsukawa, T. Konno, S. Kobayashi, M. Hagiwara, T. Miyazaki, K. Sano, Y. Ono, and A. Matsushita, J. Phys. Soc. Jpn. 82 (2013) 105003.
  * [6] K. Sano, Y. Ono, and Y. Yamada, J. Phys. Soc. Jpn. 74 (2005) 2885.
  * [7] T. Nakano, K. Kuroki, and S. Onari, Phys. Rev. B 76 (2007) 014515.
  * [8] K. Okunishi, Phys. Rev. B 75 (2007) 174514.
  * [9] M. Uehara, T. Nagata, J. Akimitsu, H. Takahashi, N. Mori, and K. Kinoshita, J. Phys. Soc. Jpn. 65 (1996) 2764.
  * [10] R. Fehrenbacher and T. M. Rice, Phys. Rev. Lett. 70 (1993) 3471.
  * [11] T. Mizokawa, C. Kim, Z.-X. Shen, A. Ino, T. Yoshida, A. Fujimori, M. Goto, H. Eisaki, S. Uchida, M. Tagami, K. Yoshida, A. I. Rykov, Y. Siohara, K. Tomimoto, S. Tajima, Y. Yamada, S. Horii, N. Yamada, Y. Yamada, and I. Hirabayashi, Phys. Rev. Lett. 85 (2000 )4779.
  * [12] S. Nishioka, S. Sasaki, S. Nakagawa, M. Yashima, H. Mukuda, M. Yogi and J. Shimoyama, Appl. Phys. Express 15 (2022) 023001.
  * [13] S. Watanabe, Y. Yamada, and S. Sasaki, Physica C 426–431 (2005) 473.
  * [14] F. Ishikawa, K. Fukuda, Y.i Hiura, Y. Onda, A. Nakayama, Y.h Yamada, A. Matsushita and T. Kagayama, J.of Phys.: Conf. Ser. 15 (2009) 052079.
  * [15] M. Kuwabara, M. Matsukawa, K. Sugawara,H. Taniguchi, A. Matsushita, M. Hagiwara, K. Sano, Y. Ono, and T. Sasaki, J. Phys. Soc. Jpn. 85 (2016) 124704.
  * [16] H. Taniguchi, Y. Nakarokkaku, R. Takahashi, M. Murakami, A. Nakayama, M. Matsukawa, S.i Nakano, M. Hagiwara, and T. Sasaki, J. Phys. Soc. Jpn. 90 (2021) 015001.
  * [17] T. Senzaki, M. Matsukawa, T. Yonai, H.Taniguchi, A. Matsushita, T. Sasaki and M. Hagiwara, Functional Materials Synthesis and Physical , Properties, Recent Perspectives in Pyrolysis Research, chapter - DOI 10.5772/intechopen.95170 (18 pages) PUBLISHED13 April 2022.
  * [18] T. Teramura, M. Matsukawa, T. Senzaki, H. Taniguchi, K. Sano, Y. Ono, M. Hagiwara, Physica B 646 (2022) 414226.
  * [19] F. Izumi and K. Momma, Solid State Phenom., 130 (2007) 15-20.
  * [20] P. Bordet, C. Chaillout, J. Chenavas, J. L. Hodeau, M. Marezio,J. Karpinski, and E. Kaldis, Nature 334 (1988) 596.
  * [21] Y. Yamada, J. Ye, S. Horii, A. Matsushita, , S. Kubo, Journal of Physics and Chemistry of Solids 62 (2001) 191.
  * [22] M. E. Lopez-Morales, D. Rios-Jara, J. Taguena, R. Escudero, S. La Placa, A. Bezinge, V. Y. Lee, E. M. Engler, P. M. Grant, Phys. Rev. B41 (1990) 6655.
  * [23] .B. Song, J.K. Liang, L.T. Yang, J.R. Chen, G.Y. Liu, H.F. Yang, G.H. Rao, Appl. Phys. A 77 (2003) 915.
  * [24] K. Honnami, M. Matsukawa, T. Senzaki, T.Toyama, H. Taniguchi, K. Ui, T. Sasaki, K. Takahashi, M. Hagiwara, Physica C585 (2021) 1353869.
  * [25] T. Chiba, M. Matsukawa, J. Tada, S. Kobayashi, M. Hagiwara, T. Miyazaki, K. Sano, Y. Ono, T. Sasaki, and J.Echigoya, J. Phys. Soc. Jpn. 82 (2013) 074706.
  * [26] T. Nakanishi, N. Motoyama, H. Mitamura, N. Takeshita, H. Takahashi, H. Eisaki, S. Uchida, and N. Môri, Phys. Rev. B 72 (2005) 054520.
  * [27] K. Kitazawa, S. Kambe, M. Naito, I. Tanaka and H. Kojima, Jpn. J. Appl. Phys. 28 (1989) L555.
  * [28] N. R. Werthamer, E. Helfand, and P. C. Hohenberg, Phys. Rev. 147, (1966) 295.
  * [29] M. Suzuki and M. Hikita, Phys. Rev. B44, (1991) 249.
  * [30] M. Hagiwara, S. Tanaka, T. Shima, K. Gotoh, S. Kanda, T. Saito, and K. Koyama, Physica C 468 (2008) 1217.
  * [31] M. Kato, M. Nakanishi, T. Miyano, T. Shimizu, M. Kakihana, K. Yoshimura, K. Kosuge , Journal of Solid State Chemistry 139 (1998) 266-273.
  * [32] Y. X. Guo, R. Hoier, T. Graf, and J. -Y. Genoud, Phil. Mag. B72 (1995) 383.


# A Rate Model of Electron Populations for Non-linear High-Fluence X-ray
Absorption Near-Edge Spectra

Robin Y. Engel  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607
Hamburg, Germany  Physics Department, Universität Hamburg, Luruper Chaussee
149, 22761 Hamburg, Germany  Markus Scholz  Deutsches Elektronen-Synchrotron
DESY, Notkestr. 85, 22607 Hamburg, Germany  European XFEL, Holzkoppel 4, 22869
Schenefeld, Germany  Jan O. Schunck  Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  Physics Department, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany  Martin Beye  Deutsches
Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  Physics
Department, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany

###### Abstract

Absorbing a focused, femtosecond X-ray pulse from a Free-Electron Laser (FEL)
can lead to extreme electronic excitations in solids. This excitation drives
changes of the electronic system over the course of the pulse duration and the
overall absorption of the pulse becomes fluence-dependent. Thus, fluence-
dependent non-linear X-ray Absorption Near Edge Spectroscopy (XANES) is
sensitive to the valence excitation dynamics around the Fermi level on the
few-femtosecond timescale. Here we present a simplified rate model based on
well-established physical mechanisms to describe the evolution of the
electronic system.

We construct temporal and spatial differentials for the processes of resonant
absorption, stimulated emission, non-resonant absorption, Auger decay, valence
band thermalization and scattering cascades of free electrons.

The phenomenological rate model approach provides a direct understanding how
each physical process contributes to the fluence-dependent changes observed in
XANES measurements. Without accounting for fluence-dependent changes to the
density of states, the model shows good agreement with experimental results on
metallic nickel over more than three orders of magnitude in fluence,
establishing electron redistribution as the main driver of non-linear
absorption changes at high fluences. Although in the closest vicinity of the
resonance, more complex approaches are necessary to describe remaining
discrepancies of the fluence-dependence changes, the demonstrated capability
to describe spectral changes up to extreme fluences yields fundamental
insights into the complex dynamics after intense core excitation and provides
an important tool for the design and evaluation of future FEL experiments, in
particular for the development of non-linear X-ray spectroscopy.

##  I  Introduction

Translating non-linear spectroscopy methods that are well established in the
longer wavelength ranges [1, 2, 3] to the X-ray regime is particularly
attractive, since the strong localization of transitions involving core
electrons as well as element-specific absorption edges promise additional
selectivity to these already potent analytical tools [4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. A prerequisite for non-linear
X-ray spectroscopy is a high density of X-ray photons, such as those found in
the tightly focused X-ray radiation from a Free-Electron Laser (FEL). The
absorption of such pulses drastically modifies the electronic structure of any
investigated material, even on the timescale of femtosecond-short pulses.
Absorption around material resonances directly probes transitions between the
core-levels and states around the Fermi level and is thus sensitive to the
electron dynamics in the valence band of materials. Electronic structure
changes in this region affect the degree of absorption experienced by later
parts of the same pulse and the fluence-dependent absorption can be used to
derive information on the excited state and its evolution [23, 24, 25, 26, 27,
28, 29, 15, 30].

Several models have been put forward to describe aspects of this interplay
between photon absorption and electronic system [31, 25, 29, 15, 30]. Away
from material resonances, fluence-dependent X-ray absorption has been
successfully modeled using rate models in three-level systems [31, 25]
modeling a ground, core-excited and intermediate valence excited state. When
probing the valence bands around material resonances however, three population
parameters representing the state of the material at a given point in time
become insufficient to represent the non-thermal electron energy distribution
around the Fermi level in an extended solid that is relevant for the rates of
excitation and relaxation. This makes rate models using only three levels
unsuitable to model non-linear absorption near resonances.

In this work, we present an expanded rate model to describe the evolution of
the electronic system in terms of an energy-resolved population of the valence
band within a constant Density of States (DoS). The valence electrons are
heated through energy transfer from the scattering of free electrons from
Auger decays and non-resonant absorption. The model uses material parameters
known or calculated for the ground-state and scales these parameters in
accordance with changes in electronic populations. Only the time constants for
the valence band thermalization and the scattering cascades of free electrons
are treated as free parameters. We present calculations matching measurements
of X-ray absorption spectra recorded with monochromatic X-rays in transmission
through metallic nickel foils around the nickel 2$p_{3/2}$ ($L_{3}$) edge
[32]; our model reproduces the main fluence-dependent changes in the measured
spectra over more than three orders of magnitude. While the measurements are
discussed in detail in a separate publication [32], this paper illustrates the
framework of the rate model and strives to provide an intuitive understanding
of the mechanisms that drive non-linear changes.

The following text is structured as follows. In section (II), we give a
qualitative overview of the rate model: Section II.1 describes the algorithm
and approximations made in the interest of computational viability, Section
II.2 outlines the mathematical formulation for the rates of all relevant
processes, which are then assembled into differentials of the photon and
electron populations in section II.3. Section II.4 elaborates on relevant
input parameters. Finally, we discuss the use and implications of our model in
section III.

##  II  The rate model

We describe the propagation of X-ray photons through the sample as well as the
dynamics of electron populations within the sample using a set of ordinary
differential equations. These rate equations describe the evolution of photon
and electron densities and are assembled from terms that each describe a
specific physical process. The rate of each process is based on ground-state
rates, scaled with the appropriate fractional occupation (the number of
electrons divided by the number of states). Each process rate is described in
detail in section II.2. Such scaling inherently prevents any state from
exceeding its physically meaningful population (between zero and the number of
available states) and also enforces that the number of electrons in the sample
is conserved over the simulated time.

The model allows for an arbitrary number of incident resonant photon energies
$E_{i}$, for each of which a Gaussian temporal profile of incident intensity
is assumed. For the presented calculations, only a single resonant photon
energy was used, representing measurements with monochromatic X-rays. Incident
photons are the only source of energy flow into the system, and all energy
eventually contributes to the thermal energy of the valence system.

Figure 1:  Photon-, electron- and energy-densities and their interactions. A
photon density $N_{phot}$ drives resonant interactions between the core
electrons $R_{C}$ and specific valence electrons $\rho_{j}$. It also drives
non-resonant excitations from the entire valence electron system
$R_{V}=\sum_{j}\rho_{j}$ to free electrons $R_{free}$, which have a total
energy of $E_{free}$. Auger decays transfer electrons from the valence system
to both core states and free electrons; scattering cascades transfer electrons
and energy from the free states to the valence system; thermalization drives
the valence system towards a thermalized Fermi-Dirac distribution.
$M_{C},M_{V}$ and $m_{j}$ represent the number of available states and are
pictured as bars to represent the energy bins of the numerical calculation.

Figure 1 schematically summarizes the modeling of the electronic system: The
electron populations $R_{C}$ and $R_{V}$ describe the total number of
electrons bound in the core and valence system, respectively, for a single
atom in the sample. Their values are limited by the number of available
states, $M_{C}$ and $M_{V}$. In the presented example on the nickel L3-edge
spectra, the ground-state populations are $R_{C}=4$, representing the
2p3/2-electrons and $R_{V}=10$, representing electrons from the 3$d$ and 4$s$
states. Electrons in intermediate shells (in the given example the 3$p$ and
3$s$ electrons) contribute to the non-resonant absorption but are not
explicitly simulated. We describe the electronic population of the valence
system in an energy-resolved manner, splitting it up into a discrete number of
densities $\rho_{j}$, where $j$ represents the index along the valence energy
axis. The number of available states $m_{j}$ for each energy bin in the
valence system is derived from the calculated ground-state DoS [33, 34] up to
30 eV above the Fermi level $E_{F}$. Beyond this value, the free electron gas
model is used to calculate the density of states [35]. We set the energy
cutoff of the energy-resolved valence band at 800 eV above the Fermi level.
All electrons with even higher energies, such as photo-electrons created via
non-resonant absorption and Auger-electrons from the decay of core-holes, are
described in a separate pool of electrons $R_{free}$ without energy
resolution, although the total energy in this pool is tracked by the parameter
$E_{free}$.

Within our model, the DoS does not change over time since the DoS is dominated
by the atomic lattice potentials and thus mainly affected by nuclear motion
[36]. Since the coupling of electronic excitations to nuclear motion is much
slower than the purely electronic dynamics, observations that are temporally
confined to the duration of a single FEL pulse (on the order of tens of
femtoseconds) are usually dominated by electronic processes.

###  II.1 Algorithm

In contrast to a Monte-Carlo approach, where interaction pathways of many
individual particles are computed and averaged, our rate model describes
processes in terms of non-quantized, average quantities (densities) in a
finite-element analysis.

The problem is formulated as an idealized X-ray pulse, Gaussian in time and
space, traveling at the speed of light through a three-dimensional sample. The
photon density interacts non-linearly with the sample. Each voxel of the
sample is characterized by a complex electronic state as described above and
in Figure 1, which itself evolves in time after excitation. To make this
problem tractable, we apply additional key approximations that allow for a
drastic reduction of the computational effort that would come with a naive
implementation of this problem in a four-dimensional (three spatial dimensions
and time) finite elements algorithm: First, we describe electronic excitations
as strictly localized and neglect their propagation; instead, only photons
propagate through the sample, and only in forward direction111We neglect
spontaneous X-ray emission, which would constitute up to 0.9 % of nickel
$L_{3}$ core-hole decays [37].. Considering that a sample of 20 nm thickness
is traversed by light in less than 100 as, the propagation of photons is
calculated as if it happened instantaneous in between the time-steps performed
to depict the evolution of the electronic system. This de-coupling of
dimensional dependence effectively simplifies the problem into two sets of
separately solvable, one-dimensional initial value problems: the photon
propagation in space and the electronic evolution in time.

To solve these, we formulate the time-differential of all electronic states
(see section II.3), depending on the photon density incident at a certain
time. The temporal evolution is solved using the fourth-order Runge-Kutta
method with adaptive time-stepping based on a fifth-order local approximation
[38]. In the meantime, whenever a time-differential is computed, the current
incident photon density is first propagated through the sample using the
explicit fourth-order Runge-Kutta method (in space) to retrieve the photon
density at each depth of the sample.

This way, the time-dependent transmission for a Gaussian temporal pulse
profile is calculated for a range of different overall pulse energies incident
per sample area. This treatment yet neglects the transversal profile of the
beam. Because the absorption is non-linearly dependent on the incident
fluence, the transmission of a transversally inhomogeneous beam must be
integrated over a specific spot profile with spatially varying transmission.
Here, a two-dimensional Gaussian profile of the FEL spot is then accounted for
by integrating the transmitted intensity based on previously calculated
fluence-dependent transmission. The Figures 2, 3, 4 and 5 in the following
section show intermediate results from simulations calculated for a specific
fluence, as opposed to integrating over the transmission in a Gaussian spot-
profile. With these simplifications, the overall computational complexity is
drastically reduced.

###  II.2 Processes

The following describes the rates at which the physical processes occur for
each atom. For a better overview, we introduce the processes as individual
terms and assemble them into differential equations in the next section. In
principle, each process is described using an absorption length or lifetime
which is known from ground-state measurements and then scaled linearly with
the changing electron populations with respect to the ground state. The
normalization is such that the ground-state rate is reproduced for an
undisturbed electron system and the rate vanishes when the corresponding
transition cannot happen due to a lack of electrons or holes. We use the
indices $j$ and $i$ to refer to specific energies, where the index $i$ is used
for photon energies of X-rays and the index $j$ for the energy of electronic
states in the valence band.

####  II.2.1 Resonant interaction

The resonant interaction describes both resonant absorption (core-valence
transitions) and stimulated emission (valence-core transitions) as a single
process. It is calculated for each energy $E_{j}$ in the valence system that
is resonant with a given photon energy $E_{i}$.

$P^{res}_{j}=\left(\frac{R_{C}}{M_{C}}-\frac{\rho_{j}}{m_{j}}\right)\frac{N_{i}}{\lambda_{i}^{res}}\delta_{ij}$ $\lambda^{res}_{i}$: | Resonant absorption length
---|---
$R_{C}$: | Number of core electrons
$M_{C}$: | Number of core states
$\rho_{j}$: | Valence electrons at $E_{j}$
$m_{j}$: | Valence states at $E_{j}$
$N_{i}$: | Number of photons per $\mathrm{nm^{2}}$ at $E_{i}$
$\delta_{ij}$: | Kronecker-delta

The first terms (in brackets) represent the difference in the occupation of
core states $R_{C}/M_{C}$ and resonant valence states $\rho_{j}/m_{j}$. The
dominance of absorption over stimulated emission or vice-versa is determined
solely by this difference, as they represent an optically driven two-level
system in the incoherent limit. If the core level population is smaller than
the valence population, the resonant interaction process becomes negative,
representing the dominance of stimulated emission. The second term on the
right is the number of irradiated photons divided by the penetration length.
The Kronecker delta ensures that only photons and electrons in corresponding
energy bins interact.

####  II.2.2 Non-Resonant absorption

The non-resonant absorption summarizes photon absorption from other electronic
states than the resonant core-level, especially from the valence electrons.
Photon densities $N_{i}$ at all incident energies reduce each population
$\rho_{j}$.

$P^{\mathrm{non-
res}}_{i,j}=\frac{\rho_{j}}{R_{V}^{0}}\frac{N_{i}}{\lambda^{non-res}}$

$\lambda^{non-res}$: | Non-resonant absorption length
---|---
$R_{V}^{0}$: | Total number of valence electrons
| in the ground state

The interaction is normalized by the total valence band population in the
ground state $R_{V}^{0}$, so that the sum of the first term over all $j$
becomes unity if all $\rho_{j}=\rho_{j}^{0}$ (since
$R_{V}^{0}\equiv\sum_{j}\rho_{j}^{0}$). The second term represents the non-
resonant absorption in the ground state as can be experimentally determined
sufficiently before the resonance in the spectrum.

Figure 2:  Instantaneous transmission (including resonant and non-resonant
absorption) over time for a pulse at 857.5 eV with a pulse energy of 30
$\mathrm{J/cm^{2}}$ (blue line, left axis), as well as the temporal profile of
the incident photon density (orange dots, right axis).

This treatment does not explicitly differentiate the non-resonant absorption
from core energy levels other than the one treated by $R_{C}$. In the given
example with photons resonant to the nickel 2$p$-absorption, the 3$s$ and 3$p$
core electrons only contribute to a minority of the non-resonant absorption
events. In this model, we choose for simplicity to scale this contribution
together with the non-resonant scattering from the valence electrons.

An exemplary incidence profile and the resulting transmission over time are
shown in Figure 2.

####  II.2.3 Auger decay

The model explicitly treats Auger decays that involve one core-hole and two
electrons from the valence band. The rate at which an electron in density
$\rho_{j}$ would decay via an Auger process is calculated as:

$P^{\mathrm{Auger}}_{j}=(M_{C}-R_{C})\frac{\rho_{j}}{R_{V}^{0}}\frac{R_{V}}{R_{V}^{0}}\frac{1}{\tau_{C}}$ $\tau^{C}$: | Core-hole lifetime
---|---
$R_{V}$: | Total number of valence electrons

The first factor (in brackets) is the number of unoccupied core states, i.e.
core-holes. The second factor describes the relative population of electrons
at the energy $E_{j}$ and the third term is the relative population of the
entire valence band to which the electron could transfer its energy. The
latter two are normalized by the respective ground state population. The last
term is the decay rate in the ground state, where $\tau_{C}$ represents the
ground-state lifetime of a single core-hole. Altogether, this describes Auger
decays as interactions between two valence electrons, one emitted and one
filling the $2p_{3/2}$ core-hole. In reality, some fraction of Auger decays
will emit electrons from the 3$s$ or 3$p$ core levels instead, followed by
further Auger processes which emit electrons with the remaining energy of the
original core-hole. These are not treated separately in our description, since
the indirect decay is, on the one hand, a minority contribution and on the
other hand, ultimately results in the same energy transfer to the valence
band, albeit with a slightly longer time delay due to the intermediate steps.

####  II.2.4 Free-electron scattering

Inspired by earlier approaches to a simplified solution of the Boltzmann
equation [39], we approximate the scattering rates of electrons in terms of
characteristic time constants $\tau_{scatt}$ and $\tau_{th}$ for the free
electrons and valence electrons, respectively.

The lifetime of free electrons $\tau_{scatt}$ represents the inverse rate at
which free electrons $R_{Free}$ scatter and decay to the valence system. While
this parameter is ultimately empirical, it depicts a cascade of individual
scattering events between electrons. In such a cascade, each free electron
eventually transfers all its kinetic energy to the valence system.

$P^{\mathrm{scatt}}=R_{Free}\frac{1}{\tau_{scatt}}$ $\tau_{scatt}$: | Free electron scattering time constant
---|---
$R_{Free}$: | Number of free electrons

####  II.2.5 Electron thermalization

Similarly, $\tau_{th}$ characterizes the time with which the valence system
approaches an internal thermal equilibrium.

$P^{\mathrm{therm}}_{j}=\left[r_{j}(T,\mu)-\rho_{j}\right]\frac{1}{\tau_{th}}$

$\tau_{th}$: | Valence thermalization time constant
---|---
$T$: | Equivalent electronic temperature
$\mu$: | Chemical potential

To this end, the chemical potential and equivalent electronic temperature are
calculated in each time-step based on the current internal energy $U$ and
number of valence electrons $R_{V}$. The Fermi distribution for the calculated
chemical potential and temperature then yields a momentary target electron
distribution $r_{j}(T,\mu)$, which is approached with the electron
thermalization constant $\tau_{th}$.

$\displaystyle r_{j}(T,\mu)=m_{j}\frac{1}{e^{(E_{j}-\mu)/k_{B}T}}$
$\displaystyle U=\sum_{j}\rho_{j}E_{j}/\sum_{j}\rho_{j}$ $\displaystyle
R_{V}=\sum_{j}\rho_{j}$

$U$: | Total energy of the valence system
---|---
$R_{V}$: | Current total population of the valence system
$r_{j}(T,\mu)$: | Electron density expected for
| a fully thermalized valence system
$k_{B}$: | Boltzmann constant

While the calculation of $r_{j}$ from $T$ and $\mu$ is straightforward,
determining $T$ and $\mu$ from $U$ and $R_{V}$ is an inverse problem. This is
solved by iterative optimization using the Levenberg-Marquardt method222First,
the Levenberg-Marquardt root-finding algorithm is applied with a maximum of
400 iterations. If the required residual is still exceeded, the algorithm
switches to least-squares optimization to refine the root-finding result. This
combination proved a good compromise between computation speed and stability
in regions with small gradients in the loss function..
Note that, because by definition both electron densities: the momentary
$\rho_{j}$ and the thermalized goal distribution $r_{j}$ hold the same amount
of electrons and internal energy, the sum over the valence band of all
electron-distributing thermalization rates is always zero and the change in
overall energy is also zero, i.e. $\sum_{j}P^{\mathrm{therm}}_{j}=0$ and
$\sum_{j}E_{j}P^{\mathrm{therm}}_{j}=0$.

Figure 3 shows how the temperature (given as an energy $k_{B}T$ in units of
eV) and the chemical potential develop over time for a high fluence of 30
$\mathrm{J/cm^{2}}$. Although reaching higher temperatures, these results are
in agreement with studies treating the nickel valence system heated with
optical lasers [40, 41].

Figure 3:  Temperature and chemical potential over time for a pulse at 857.5
eV with a pulse energy of 30 $\mathrm{J/cm^{2}}$. The solid line represents
the properties at the sample surface and the thin lines represent the deeper
layers that are exposed to less X-ray fluence, as indicated by arrows and
increasing transparency.

###  II.3 Differentials

From these process terms, we can assemble the time-differential of the various
pools of electrons, photons, and energy. Because the simulation conserves the
number of electrons in the sample, in the sum over all $j$, every process term
describing electronic transitions appears equally often with positive and
negative sign, representing a transition of electrons from one state to
another. For example, the term for Auger decay appears twice with a negative
sign in the valence electron differential, and each once with a positive sign
in the differential for core- and free electrons.

####  II.3.1 Valence electrons

$\begin{split}\frac{d\rho_{j}}{dt}&=P^{res}_{i=j}\\\ &-\sum_{i}P^{\mathrm{non-
res}}_{i,j}\\\
&-P^{\mathrm{Auger}}_{j}-\frac{\sum_{j}\rho_{j}P^{\mathrm{Auger}}_{j}}{R_{V}}\\\
&+P^{\mathrm{therm}}_{j}\\\
&+\frac{h_{j}}{H_{V}}P^{\mathrm{scatt}}+P_{red}\left(\frac{-\rho_{j}}{R_{V}}+\frac{h_{j}}{H_{V}}\right)\end{split}$

$h_{j}$: | Number of valence holes, $h_{j}=m_{j}-\rho_{j}$
---|---
$H_{V}$: | Total number of valence holes, $H_{V}=\sum_{j}h_{j}$
$P_{red}$: | Electron re-distribution rate due to
| scattering cascades from free electrons

The valence system interacts via all modeled processes, therefore we comment
on the equation above line by line. The resonant absorption rate $P^{res}_{j}$
changes the valence electron densities $\rho_{j}$ at all incident photon
energies $E_{i}$ where $i=j$. Via non-resonant absorption, incident photon
energies $E_{i}$ can remove electrons from $\rho_{j}$. The third line shows
the primary and secondary Auger electrons. Note that in the sum over all
energies $E_{j}$, terms 3 and 4 each remove one electron from the valence band
for each Auger process occurring. The thermalization term in line four drives
electrons towards the thermal distribution based on the current internal
energy and population of the valence band, without changing the total valence
occupation, as discussed in section II.2.5. The last line describes the effect
of electron scattering. The first term represents electrons from the free
electron pool that are re-joining the valence system into a random unoccupied
state $h_{j}$.

The second term describes the electron redistribution inside the valence
system in order to take up the energy released by the re-joining electron.
This redistribution is calculated as a function of the rate of electronic
scattering $P^{scatt}$ and represents the effect of electronic scattering
cascades. In such a cascade, a number of electrons is moved from occupied
states $\rho_{j}$ to unoccupied states $h_{j}$. The total rate of electrons
redistributed in this time step through scattering $P_{red}$ is given by the
ratio of the energy that needs to be taken up by the valence system to the
energy that the valence system can additionally accommodate.

$P_{red}=\frac{S_{scatt}-S_{joining}}{U_{h}-U_{e}}$

The denominator of $P_{red}$ represents the energy that can maximally be
redistributed to the valence system as the difference between the energy that
could be contained in the unoccupied states

$U_{h}=\frac{\sum_{j}h_{j}E_{j}}{H_{V}}$

and the one already contained in the occupied states

$U_{e}=\frac{\sum_{j}\rho_{j}E_{j}}{R_{V}}.$

The numerator is given by the difference between the rates at which energy is
released from the free electron energy pool

$S_{scatt}=P^{\mathrm{scatt}}\frac{E_{free}}{R_{free}}$

and the rate at which energy is gained in the valence system due to the
formerly free electrons occupying random unoccupied valence states:

$S_{joining}=\sum_{j}\frac{h_{j}}{H_{V}}P^{\mathrm{scatt}}E_{j}$

Figure 4 shows an example for the development of the valence system within a
single voxel at the sample surface, exposed to a pulse of 30 J/cm2 fluence,
resonant to states 7 eV above $E_{F}$.

Figure 4:  Density of states. Panel a) shows the total DoS as used by the
algorithm. Panel b) shows the relative population of the valence band over
time at the sample surface for a pulse of 857.5 eV with a pulse energy of 30
$\mathrm{J/cm^{2}}$. The non-thermal saturation of the valence states
(sometimes called spectral bleaching or hole burning) is clearly visible at
the resonant energy 7 eV above $E_{F}$ and highlighted with a blue ellipse.
Apart from that, the shifting and broadening of the thermalizing valence
distribution is apparent, together with the rising electronic temperature $T$
and shifting chemical potential $\mu$. The valence thermalization time is
evident here in the visible lag of the thermalizing valence occupation behind
the chemical potential.

####  II.3.2 Core electrons

$\frac{dR_{C}}{dt}=-\sum_{i,j}P^{res}_{i,j}+\sum_{j}P^{\mathrm{Auger}}_{j}$

The population of core electrons is reduced (or increased, depending on the
sign of $P^{res}_{i,j}$) by resonant transitions of all incident photon
energies $E_{i}$ to states at all energies $\rho_{j}$ (although this
contribution is only non-zero at $i=j$), and is increased by Auger decay from
electrons of all energies $j$ in the valence system. Note that the spontaneous
emission channel is neglected in our model as it is designed for soft X-ray
energies where Auger emission accounts for most core-hole decays (here
specifically, 99.1% of the nickel L3 core-hole decays [37, 42]). In another
concession to the specific experiment simulated here, we further neglect fast
electrons leaving the sample, since the electron mean free path is much
shorter than the sample thickness (about 1.3 nm [43] compared to a 20 nm thick
sample). While a loss process for free electrons would be trivial to
implement, the total number of electrons in the system being strictly constant
over time is a valuable indicator for the self-consistency of the calculation.

####  II.3.3 Free electrons

$\frac{dR_{Free}}{dt}=\sum_{i,j}P^{\mathrm{non-
res}}_{i,j}+\sum_{j}P^{\mathrm{Auger}}_{j}-P^{\mathrm{scatt}}$

Unbound or free electrons are generated by non-resonant absorption from all
incident photon energies $E_{i}$ as well as Auger-decays from all energies in
the valence band. The population is reduced by the free electron scattering
rate $P^{\mathrm{scatt}}$.

####  II.3.4 Photon absorption and emission

$\frac{dN_{i}}{dz}=-P^{res}_{i=j}-\sum_{j}P^{\mathrm{non-res}}_{i,j}$

The number of photons is reduced or increased by resonant interaction and
reduced by non-resonant absorption. Note that this is a purely spatial
differential that depicts the instantaneous transmission of a certain number
of photons through the entire sample in each time-step.

####  II.3.5 Energy of free electrons

$\begin{split}\frac{dE_{Free}}{dt}&=\sum_{i,j}P^{\mathrm{non-
res}}_{i,j}(E_{F}+E_{i}-E_{j})\\\
&+\left(\frac{\sum_{j}E_{j}(\rho_{j}-P^{\mathrm{Auger}}_{j})}{\sum_{j}(\rho_{j}-P^{\mathrm{Auger}}_{j})}+E_{F}\right)\sum_{j}P^{\mathrm{Auger}}_{j}\\\
&-P^{\mathrm{scatt}}\frac{E_{Free}}{R_{Free}}\end{split}$

$E_{F}$: | Energy of the Fermi level
---|---

Because photons can ”kick out” electrons from arbitrary states in the valence
system, it becomes necessary to track the total kinetic energy of the free
electrons, even though the distribution of energy among these electrons is not
tracked. The rate of energy transfer to the free electron bath is described as
the sum of non-resonant absorption and Auger decays, each multiplied by their
respective energies (first and second line, respectively). The energy of the
secondary Auger electrons is calculated as the average energy of all electrons
other than the primary Auger electrons, as those drop to the core level.
Finally, each electron that leaves $R_{Free}$ reduces the energy of the bath
by the average energy, which is $\frac{E_{Free}}{R_{Free}}$.

Explicitly tracking this energy also enables us to demonstrate the
conservation of energy within the simulation; since there is no channel that
allows energy to leave the sample, the energy held in the electronic sub-
systems matches that of the absorbed photons at all times. Figure 5 shows the
internal energy of the electronic subsystems over time, integrated over an
area of 1 nm2 and the full 20 nm thickness of the sample. The comparison of
the energy in various sub-systems demonstrates how quickly the energy of the
photon pulse is transferred to valence excitations. Furthermore, observing the
energy conservation has proven to be an invaluable tool to select sufficiently
fine binning in time, space and energy, as it is particularly sensitive to the
accumulation of numerical errors. For example, the calculation for the
homogeneous illumination with 30 $\mathrm{J/cm^{2}}$ shows a cumulative error
in energy of 0.35%. This is in contrast to the electron conservation, which is
strictly kept to machine precision level due to the symmetric way the process
terms are arranged to form the time differentials.

Figure 5:  Energy in the sample system over time for a pulse at 857.5 eV,
integrated over the full 20 nm thickness of the sample and a 1 nm2 area
illuminated with a fluence of 30 $\mathrm{J/cm^{2}}$. The absorbed energy is
calculated from the difference between incident and transmitted photons, while
the total energy is a sum of the energy held in the electronic sub-systems of
core-holes, free electrons, and valence excitation. Due to the fast process
rates in comparison to the pulse duration, the energy held in core excitations
and free electrons remains small, which is why a 10 times scaled curve is also
shown.

###  II.4 Parameters

There are four categories of parameters: First, resolution parameters for the
number of steps in time, space, and energy are chosen as a compromise between
calculation time and numerical error; second, experimental parameters such as
the pulse duration, peak fluence, and bandwidth of the interacting photons
reflect experimental conditions; third, ground-state properties such as the
atomic density and number of states and electrons, as well as Auger-decay
limited core-hole lifetime are drawn from published literature, while the
resonant and non-resonant absorption lengths are derived from the ground-state
spectrum as described below. Fourth, the model-inherent phenomenological
parameters are the valence thermalization time $\tau_{th}$ and electron
scattering time $\tau_{scatt}$, which are varied to achieve the best match to
the experimental results. A list of the relevant parameters and the chosen
values for the present calculations, selected to match the experiment
presented in [32], is shown in the appendix Table LABEL:tab:_parameters.

The parameterization of several ground-state properties deserves further
comment. Firstly, the DoS (motivated at the start of this section) was
subdivided into bins $m_{j}$ of varying size, favoring a fine resolution for
the bound states. The size of the energy bin that is resonantly coupled to the
core level by the incident photons is chosen such that it represents the
interaction bandwidth of the photons. The interaction bandwidth is calculated
as the convolution of the bandwidth of incident photons (i.e. the resolution
of the experiment) and the natural line width of the core excitation. We
further account for the final state broadening of excitations into less
tightly bound states by enlarging the interaction bandwidth by 0.1 eV per eV
above the Fermi level. Furthermore, the resonant and non-resonant absorption
lengths are derived from the ground-state spectrum. We treat the non-resonant
absorption length as constant, i.e. independent of photon energy, and derive
it from the pre-edge absorption level. The transition matrix element of a
core-valence transition exhibits a resonant enhancement close to the
absorption edge, which translates into an energy dependence of the resonant
absorption length. Above the Fermi level, where the DoS is unoccupied in the
ground state, the resonant absorption length is encoded in the ground-state
absorption spectrum. As the transition matrix element from the core level to
states below the Fermi level is experimentally not straightforward to access,
we use the approximation that the energy-dependence of the transition matrix
element is symmetric around the Fermi energy. To derive the resonant
absorption length, the non-resonant absorption level is subtracted from the
spectrum and line broadening is accounted for by deconvolution with a pseudo-
Voigt-profile of 50% Gaussian and Lorentzian share and a width of 640 meV
Full-Width Half-Maximum (FWHM), representing 420 meV broadening from the
experimental resolution and 480 meV from the core-hole lifetime [42]. The
deconvolved resonant absorption spectrum above the Fermi level is then
mirrored around the Fermi energy and the discontinuity within 320 meV around
it is reconstructed with cubic interpolation. This results in the mirrored
resonant absorption spectrum shown in Figure 6, which is used as the resonant
absorption length parameter $\lambda_{i}^{res}$. Note that the results of
simulated spectra were finally re-convolved with the same pseudo-Voigt-profile
to simulate the same experiment.

Figure 6:  Derivation of absorption lengths as input parameters, reconstructed
from a measured ground-state spectrum (blue line). The non-resonant absorption
level (blue dots) was determined from the pre-edge region. The measured
resonant absorption length was deconvolved with the experimental resolution
(orange dashes) and mirrored around the rising edge to retrieve a symmetric
resonant absorption length around the resonance (green dot-dashed line). See
main text for details.

##  III  Discussion

###  III.1 Parameter Study

Figure 7: Fluence-dependent Ni $L_{3}$-edge spectra, simulated with different
parameters and compared to the measurements presented in [32] For the spectra
(III) to (VI), one parameter was varied with respect to the best batch (II).
Each set of spectra is offset by another 250 mOD as indicated by the
horizontal lines. The error bars of the experimental data represent the 95%
confidence intervals for each bin of 102 meV width; the solid lines represent
smoothed spectra using a Savitzky-Golay filter using windows of 21 bins and
4th-order polynomials. The average fluence of events contributing to each
spectrum is given in the legend. Dashed simulated spectra do not have a
corresponding measurement.

In Figure 7 we show the measured non-linear X-ray absorption spectra labeled
(I) presented and discussed in [32] together with sets of simulated spectra
computed for different sets of parameters.

The first set of simulated spectra (also presented in [32]) represents the
best match with the experimental conditions and the parameters shown in table
LABEL:tab:_parameters and is labeled (II), while the consecutive sets, labeled
(III) to (VI), demonstrate how the results change when individual parameters
are modified. We present this set of simulations to showcase how our model may
be used to understand the relation between the non-linear changes and various
parameters. To fit the experimental results, however, only the parameters
$\tau_{th}$ and $\tau_{scatt}$ are treated as unknowns, while all other
parameters are known experimental or ground-state parameters.

In the best matching simulation (II), the experimental observations of a red-
shifted rising edge, increased pre-edge absorption, as well as reduced
absorption at and beyond the absorption peak, are reproduced. However, the
saturation effect at the resonance is over-estimated and the lack of measured
saturation around 852 eV cannot be explained by our model. See [32] for a more
detailed discussion of these deviations.

The next simulation (III) was performed without non-resonant absorption. While
this eliminates the pre-edge absorption rise, some shift of the absorption
onset is still predicted within the original peak, while the spectra above the
resonance onset behave similarly to the best-matching simulation.

With simulations (IV) and (V), we demonstrate the effect of prolonging the
thermalization time or shortening the pulse duration by a factor of ten,
respectively. Both have the similar effect of strongly reducing the peak shift
and a moderate decrease of saturation beyond the resonance.

Finally, we show non-linear spectra (VI) where the free electron scattering
process was eliminated ($\tau_{scatt}=\infty$). This prevents the majority of
the excitation energy from entering the valence system and thus drastically
reduces valence heating. This causes the pre-edge absorption rise and rising
edge shift to vanish nearly entirely, and absorption-decrease due to
saturation at and above the resonance is also strongly reduced. These
differences underscore the importance of electronic scattering cascades to
these phenomena.

While an extensive study of the correlations between the various model
parameters and the observed effects is beyond the scope of this work, this
brief parameter study allows an interpretation of how red-shift and pre-edge
absorption rise occur: Let us consider a case where the incident photon energy
is slightly below the absorption edge. Initially, only non-resonant absorption
transfers energy to the sample, specifically by creating free electrons in
form of photo-electrons. This energy is then transferred to the valence band
due to electronic scattering cascades, where it causes many small, non-thermal
excitations. These are homogeneously distributed over all valence energies and
thus only lead to a small increase in pre-edge absorption (see the simulation
with slow thermalization). However, this distribution of excitations develops
towards the shape of a Fermi distribution over time, specifically with the
thermalization time constant. If the pulses are long in comparison to the
thermalization time, a significant number of states below the Fermi level
becomes available for core-valence transitions during the pulse duration. This
causes the observed increase in pre-edge absorption. Once the first empty
valence states become available at the current photon energy, resonant
absorption begins to occur in addition to the non-resonant absorption. The
additional resonant absorption leads to more free electrons from Auger-decay,
which in the same way as the photo-electrons further contribute to secondary
electron scattering and thermalization. Because thermalization mostly creates
free states just below the Fermi level, this process of self-enhancing rise of
overall absorption manifests in a fluence-dependent shift of the rising edge
to lower energies.

###  III.2 Limitations and Potential

As demonstrated in the parameter study above, this kind of analysis can
provide a straightforward interpretation of how non-linear changes to the
absorption spectrum emerge, and what each change says about the development of
the electronic populations.

While the model already demonstrates good agreement with experimental results
[32], below we discuss the limitations of our approach and possibilities for
expanding or refining it.

In the model as described here, the DoS is not only assumed constant but also
does not differentiate between spin-up and spin-down states, although we are
treating a magnetic material. Splitting up the DoS in spin-up and spin-down
states would allow for the inclusion of angular momentum conservation and
transfer in the various scattering rates.

For thin samples where the electron mean free path becomes similar to the
layer thickness, a loss process for free electrons should be introduced to
account for electrons leaving the sample.

Fluorescent decay should be accounted for when moving to harder X-rays [37].
While introducing the decay channel itself would be simple, accurately
accounting for reabsorption may be less straightforward, since the model only
propagates light in one direction.

Furthermore, the thermalization time $\tau_{th}$ is used in this work as a
global fitting parameter, although electron thermalization times have been
suggested to depend on electronic temperature [44]. Since the electronic
temperature and target distribution are calculated every time-step, an
arbitrary dependence could be easily introduced, albeit with the necessity of
additional fitting parameters.

The DoS is dominated by the crystal lattice, which is typically stable on the
sub-100 fs timescale. However, recent Time-Dependent Density Functional Theory
(TDDFT) calculations show that electronic processes (i.e. sub 100 fs) can also
lead to modifications of the DoS via spin-orbit coupling modeled by
introducing an onsite Hubbard correlation $U$ to the mean-field Hamiltonian
[45]. Since the rate model approach is generally not suited to calculate the
DoS, incorporating modifications to the DoS would require a close interplay of
the rate model with (TD)DFT calculations, leading to an ultimately much more
complex approach reminiscent of models developed for the study of radiation-
induced damage mechanisms [46, 47].

Furthermore, we derive the interaction bandwidth (the valence energy range to
which core states can be resonantly coupled by incident photons) as a
convolution of instrumental resolution and the lifetime of the core
excitation, i.e. the Auger lifetime. The final state lifetime broadening of
excitations into continuum states is described as a continuous broadening of
0.1 eV per eV above the Fermi energy [48]. It is however reasonable to expect
that the final state lifetime is further shortened at higher fluences, due to
increased rates of both electronic scattering and stimulated emission; the
latter would be particularly relevant at the resonance peak. Such further
broadening would cause more valence states to be available for resonant
interaction and reduce the observed saturation effect while increasing the
number of core-holes that may be created by high fluences333When considering
only resonant absorption, the core level, and resonant valence states
constitute a classic two-level system. In the extreme fluence limit, the
relative populations of such a system are given by the state degeneracy ratio,
i.e. the ratio between the number of involved core and valence states..
Accounting for a fluence-dependent broadening of the interaction bandwidth in
a refined rate model might help to remedy the overestimation of the saturation
effect at the absorption peak in the presented calculations.

Another candidate for further refinement is an energy dependence of the
electronic scattering rates. The presented model uses fixed rates for
thermalization and scattering cascades, which both act on all valence states
indiscriminately. This description would be especially inadequate when applied
without modifications to a bandgap material. An advanced model could describe
both the thermalization rate of the valence band and the energy of excitations
from scattering cascades in an energy-resolved manner444It might be non-
trivial to normalize this energy dependence such that the electronic
scattering processes do not violate the conservation of energy..

While such refinements may seem attractive, a core strength of the rate model
approach is its relative simplicity and computational tractability, as well as
the use of known ground-state parameters, which supports a straightforward
physical interpretation. It is ultimately a mostly classical, phenomenological
model which offers a complementary approach to ab-initio calculations. Every
added complexity should therefore be weighed against its relevance, as a
simpler model facilitates a meaningful interpretation and avoids introducing
ambiguity in the results due to correlations between redundant input
parameters.

Since the model operates on widely applicable principles, we expect that it
may be applied to a wide range of materials with some predictive power, while
the limitations described above apply. The results will especially deviate
from observations wherever multi-particle effects, such as electron
correlations or quasiparticles become relevant.

As presented, the model enables an understanding of the electronic population
history under strong X-ray fluences and characterization of the resulting non-
linear absorption near a core resonance. Non-linear absorption studies like
the one analyzed here [32] allow one to characterize the excitation dynamics
of the electronic system under study. Furthermore, consideration of these
dynamics is, due to the extreme fluences required, particularly relevant for
methods in the emerging field of non-linear X-ray spectroscopy [5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 4, 16, 17, 18, 19, 20, 21].

##  IV  Conclusion

In this work, we propose a model of differential rate equations describing the
various excitation and decay processes that connect core- and valence
electronic states to quantitatively describe the non-linear changes to X-ray
absorption around a core resonance that occur when employing increasingly high
FEL pulse energies. We present the framework of the model in detail, applying
it to the case of recently measured non-linear absorption spectra of nickel
films at the $L_{3}$ edge, recorded with monochromatic X-rays [32].

We demonstrate how the rate model reaches good agreement with the experimental
results while disentangling the contributions of the relevant physical
processes: resonant and non-resonant absorption, Auger decay, electron
thermalization, and electronic scattering cascades. Our rate model contains
numerous simplifications in order to approximate the result of complex
interactions between many particles in terms of average transition rates.
Nevertheless, the resulting description enables a quantitative understanding
of the evolution of the system and the processes responsible for spectral
changes. In the presented calculations, the most relevant effect is valence
system heating due to secondary electron cascades from free electrons. Our
results allow quantifying the absorbance changes caused by fundamental
electron population dynamics, which are crucial to disentangle from collective
quantum effects studied with currently evolving non-linear X-ray spectroscopy
techniques.

###### Acknowledgements.

Funding by the Helmholtz Association (grant VH-NG-1105) is gratefully
acknowledged. This research was supported in part through the Maxwell
computational resources operated at Deutsches Elektronen-Synchrotron DESY,
Hamburg, Germany
Author Contributions: R.Y.E. and M.B. developed the model. R.Y.E. wrote the
manuscript with contributions from all authors.

##  Appendix A Run-Time and Code Availability

Using adequate multiprocessing on a single node of a computing cluster allows
computation of non-linear spectra such as presented here within many minutes
to several hours, depending on the required number of steps in space, time,
and energy. The simulation code is publicly available [49].

##  Appendix B Parameter table

Table 1: Parameters for the presented simulation results. The first block lists parameters that define the resolution of the simulation, the second block shows experimental conditions, the third phenomenological fitting parameters and the last block contains physical ground-state properties of the sample. Symbol | Code | Description | Unit | Value
---|---|---|---|---
$N_{z}$ | Nsteps_z | Steps in sample depth | - | 50
$N_{E_{j}}$ | N_j | Steps in energies considered in valence system | - | 90
$N_{E_{i}}$ | N_points_E | Number of photon energies / points in the spectrum | - | 69
- |  | N_local_fluences
---
_to_calculate
Number of fixed fluences that are directly simulated | - | 30
- | N_pulse_energies | Number of final pulse energies with a Gaussian spot profile | - | 20
- | Nsteps_r | Number of steps in the radial integration of the Gaussian spot | - | 100
$dt_{min}$ | timestep_min | Minimum allowed time-step | fs | 0.15
- | Energy_axis_max | Maximum energy in the valence system | eV | 800
- | Energy_axis_fine_until | Finer sampling for energies lower than this | eV | 30
- | Energy_axis_min | Valence band origin | eV | -10
- | DoS_band_origin | Energy minimum from where to use the loaded DOS | eV | -10
- | DoS_band_dd_end | Energy maximum from where to use the loaded DOS | eV | 30
$\sigma_{tj}$ | tdur_sig | Rms pulse duration of photons | fs | 13
$E_{i}$ | E_i | Photon energy of incident photons | eV | 848-856
$I_{0}$ | I_0 | Number of photons irradiated |  photons nm-2 | $0-1.4\text{e}4$
$N_{E_{i}}$ | N_photens | Number of different photon energies irradiated | - | 1
$T_{0}$ | temperature | Initial sample temperature | K | 300
$\sigma_{BW}$ | interaction_bandwidth |  Bandwidth of resonant interaction at $E_{j}$ | eV | 0.638
$\tau_{scatt}$ | tau_scattering | Scattering time of free electrons | fs | 1.5
$\tau_{th}$ | tau_th | Thermalization time of non-thermal valence states | fs | 6
- | DoS_shapefile | Filename of the total DOS from DFT-calculation | - |  from [33]
$Z$ | Z | Total sample thickness | nm | 20
$\rho$ | atomic_density | Atomic density |  atoms nm-3 | 91.4
$R_{V}^{0}$ | valence_GS_occupation | Valence electrons per atom in the ground state |  states atom-1 | 10
$M_{C}$ | core_states |  Core electrons/states per atom in the L3 core level |  states atom-1 | 4
$E_{f}$ | E_f |  Fermi level; used as zero for energy axes of $E_{j}$ and $E_{i}$ | eV | 850.5
$\tau_{C}$ | tau_core_hole |  core-hole lifetime, from [42] | fs | 1.4
$\lambda^{non-res}$ | lambd_nonres | Absorption length due to non-resonant absorption | nm | 248
$\lambda_{Ej}^{res}$ | lambd_res_Ej | Absorption length due to resonant absorption | nm | 20-83

## References

  * Boyd [2008] R. W. Boyd,  _Nonlinear optics_ , 3rd ed. (Academic press, 2008).
  * Demtröder [2015] W. Demtröder,  _Laser Spectroscopy 2: Experimental Techniques, Fifth Edition_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 2015) pp. 1–757.
  * Levenson and Kano [1987] M. D. Levenson and S. S. Kano,  _Introduction to nonlinear laser spectroscopy_ (Academic Press, Inc. (London) LTD., 1987).
  * Beye  _et al._ [2019] M. Beye, R. Y. Engel, J. O. Schunck, S. Dziarzhytski, G. Brenner, and P. S. Miedema, Non-linear soft x-ray methods on solids with MUSIX—the multi-dimensional spectroscopy and inelastic x-ray scattering endstation, Journal of Physics: Condensed Matter 31, 014003 (2019).
  * Mukamel [2005] S. Mukamel, Multiple core-hole coherence in x-ray four-wave-mixing spectroscopies, Phys. Rev. B 72, 235110 (2005).
  * Glover  _et al._ [2012] T. E. Glover, D. M. Fritz, M. Cammarata, T. K. Allison, S. Coh, J. M. Feldkamp, H. Lemke, D. Zhu, Y. Feng, R. N. Coffee, M. Fuchs, S. Ghimire, J. Chen, S. Shwartz, D. A. Reis, S. E. Harris, and J. B. Hastings, X-ray and optical wave mixing, Nature 488, 603 (2012).
  * Beye  _et al._ [2013] M. Beye, S. Schreck, F. Sorgenfrei, C. Trabant, N. Pontius, C. Schüßler-Langeheine, W. Wurth, and A. Föhlisch, Stimulated x-ray emission for materials science, Nature 501, 191 (2013).
  * Weninger  _et al._ [2013] C. Weninger, M. Purvis, D. Ryan, R. A. London, J. D. Bozek, C. Bostedt, A. Graf, G. Brown, J. J. Rocca, and N. Rohringer, Stimulated electronic x-ray raman scattering, Phys. Rev. Lett. 111, 233902 (2013).
  * Shwartz  _et al._ [2014] S. Shwartz, M. Fuchs, J. B. Hastings, Y. Inubushi, T. Ishikawa, T. Katayama, D. A. Reis, T. Sato, K. Tono, M. Yabashi, S. Yudovich, and S. E. Harris, X-ray second harmonic generation, Physical Review Letters 112, 163901 (2014).
  * Tamasaku  _et al._ [2014] K. Tamasaku, E. Shigemasa, Y. Inubushi, T. Katayama, K. Sawada, H. Yumoto, H. Ohashi, H. Mimura, M. Yabashi, K. Yamauchi, and T. Ishikawa, X-ray two-photon absorption competing against single and sequential multiphoton processes, Nature Photonics 8, 313 (2014).
  * Bencivenga  _et al._ [2015] F. Bencivenga, R. Cucini, F. Capotondi, A. Battistoni, R. Mincigrucci, E. Giangrisostomi, A. Gessini, M. Manfredda, I. P. Nikolov, E. Pedersoli, E. Principi, C. Svetina, P. Parisse, F. Casolari, M. B. Danailov, M. Kiskinova, and C. Masciovecchio, Four-wave mixing experiments with extreme ultraviolet transient gratings, Nature 520, 205 (2015).
  * Schreck  _et al._ [2015] S. Schreck, M. Beye, and A. Föhlisch, Implications of stimulated resonant x-ray scattering for spectroscopy, imaging, and diffraction in the regime from soft to hard x-rays, Journal of Modern Optics 62, S34 (2015).
  * Lam  _et al._ [2018] R. K. Lam, S. L. Raj, T. A. Pascal, C. D. Pemmaraju, L. Foglia, A. Simoncig, N. Fabris, P. Miotti, C. J. Hull, A. M. Rizzuto, J. W. Smith, R. Mincigrucci, C. Masciovecchio, A. Gessini, E. Allaria, G. De Ninno, B. Diviacco, E. Roussel, S. Spampinati, G. Penco, S. Di Mitri, M. Trovò, M. Danailov, S. T. Christensen, D. Sokaras, T. C. Weng, M. Coreno, L. Poletto, W. S. Drisdell, D. Prendergast, L. Giannessi, E. Principi, D. Nordlund, R. J. Saykally, and C. P. Schwartz, Soft x-ray second harmonic generation as an interfacial probe, Physical Review Letters 120, 023901 (2018).
  * Tamasaku  _et al._ [2018] K. Tamasaku, E. Shigemasa, Y. Inubushi, I. Inoue, T. Osaka, T. Katayama, M. Yabashi, A. Koide, T. Yokoyama, and T. Ishikawa, Nonlinear spectroscopy with x-ray two-photon absorption in metallic copper, Physical Review Letters 121, 083901 (2018).
  * Higley  _et al._ [2019] D. J. Higley, A. H. Reid, Z. Chen, A. A. L. Loïc Le Guyader and, T. Liu, T. Chase, G. L. Dakovski, A. Mitra, E. Yuan, H. A. Dürr, W. F. Schlotter, and J. Stöhr, Femtosecond x-ray induced changes of the electronic and magnetic response of solids from electron redistribution, Nature communications 10, 5289 (2019).
  * Rottke  _et al._ [2022] H. Rottke, R. Y. Engel, D. Schick, J. O. Schunck, P. S. Miedema, M. C. Borchert, M. Kuhlmann, N. Ekanayake, S. Dziarzhytski, G. Brenner, U. Eichmann, C. von Korff Schmising, M. Beye, and S. Eisebitt, Probing electron and hole colocalization by resonant four-wave mixing spectroscopy in the extreme ultraviolet, Science Advances 8, eabn5127 (2022).
  * Rouxel  _et al._ [2021] J. R. Rouxel, D. Fainozzi, R. Mankowsky, B. Rösner, G. Seniutinas, R. Mincigrucci, S. Catalini, L. Foglia, R. Cucini, F. Döring, A. Kubec, F. Koch, F. Bencivenga, A. A. Haddad, A. Gessini, A. A. Maznev, C. Cirelli, S. Gerber, B. Pedrini, G. F. Mancini, E. Razzoli, M. Burian, H. Ueda, G. Pamfilidis, E. Ferrari, Y. Deng, A. Mozzanica, P. J. M. Johnson, D. Ozerov, M. G. Izzo, C. Bottari, C. Arrell, E. J. Divall, S. Zerdane, M. Sander, G. Knopp, P. Beaud, H. T. Lemke, C. J. Milne, C. David, R. Torre, M. Chergui, K. A. Nelson, C. Masciovecchio, U. Staub, L. Patthey, and C. Svetina, Hard X-ray transient grating spectroscopy on bismuth germanate, Nature Photonics 15, 499 (2021).
  * Bencivenga  _et al._ [2021] F. Bencivenga, R. Mincigrucci, F. Capotondi, A. Calvi, R. Cucini, L. Foglia, E. Pedersoli, E. Principi, A. Simoncig, P. Cinquegrana, M. B. Danailov, G. De Ninno, S. Di Mitri, G. Gaio, A. Gessini, L. Giannessi, N. Mahne, M. Manfredda, I. P. Nikolov, G. M. Penco, L. Raimondi, P. R. Ribic, C. Svetina, M. Trovo, M. Zangrando, and C. Masciovecchio, An approach for realizing four-wave-mixing experiments stimulated by two-color extreme ultraviolet pulses, in  _International Conference on X-Ray Lasers 2020_, July 2021, edited by D. Bleiner (SPIE, 2021) p. 28.
  * Wirok-Stoletow  _et al._ [2022] S. Wirok-Stoletow, R. Jin, D. Kolbasova, S.-K. Son, A. Aquila, and R. Santra, Nonsequential two-photon absorption in solid Ge irradiated by an intense x-ray free-electron-laser pulse, Physical Review A 106, 023118 (2022).
  * Boemer  _et al._ [2021] C. Boemer, D. Krebs, A. Benediktovitch, E. Rossi, S. Huotari, and N. Rohringer, Towards novel probes for valence charges via X-ray optical wave mixing, Faraday Discussions 228, 451 (2021), arXiv:2103.14710 .
  * Serrat [2021] C. Serrat, Localized core four-wave mixing buildup in the x-ray spectrum of chemical species, The Journal of Physical Chemistry Letters 12, 1093 (2021).
  * Schwickert  _et al._ [2022] D. Schwickert, M. Ruberti, P. Kolorenč, S. Usenko, A. Przystawik, K. Baev, I. Baev, M. Braune, L. Bocklage, M. K. Czwalinna, S. Deinert, S. Düsterer, A. Hans, G. Hartmann, C. Haunhorst, M. Kuhlmann, S. Palutke, R. Röhlsberger, J. Rönsch-Schulenburg, P. Schmidt, S. Toleikis, J. Viefhaus, M. Martins, A. Knie, D. Kip, V. Averbukh, J. P. Marangos, and T. Laarmann, Electronic quantum coherence in glycine molecules probed with ultrashort x-ray pulses in real time, Science Advances 8, 1 (2022).
  * Nagler  _et al._ [2009] B. Nagler, U. Zastrau, R. R. Fäustlin, S. M. Vinko, T. Whitcher, A. J. Nelson, R. Sobierajski, J. Krzywinski, J. Chalupsky, E. Abreu, S. Bajt, T. Bornath, T. Burian, H. Chapman, J. Cihelka, T. Döppner, S. Düsterer, T. Dzelzainis, M. Fajardo, E. Förster, C. Fortmann, E. Galtier, S. H. Glenzer, S. Göde, G. Gregori, V. Hajkova, P. Heimann, L. Juha, M. Jurek, F. Y. Khattak, A. R. Khorsand, D. Klinger, M. Kozlova, T. Laarmann, H. J. Lee, R. W. Lee, K.-H. Meiwes-Broer, P. Mercere, W. J. Murphy, A. Przystawik, R. Redmer, H. Reinholz, D. Riley, G. Röpke, F. Rosmej, K. Saksl, R. Schott, R. Thiele, J. Tiggesbäumker, S. Toleikis, T. Tschentscher, I. Uschmann, H. J. Vollmer, and J. S. Wark, Turning solid aluminium transparent by intense soft x-ray photoionization, Nature Physics 5, 693 (2009).
  * Vinko  _et al._ [2010] S. M. Vinko, U. Zastrau, S. Mazevet, J. Andreasson, S. Bajt, T. Burian, J. Chalupsky, H. N. Chapman, J. Cihelka, D. Doria, T. Döppner, S. Düsterer, T. Dzelzainis, R. R. Fäustlin, C. Fortmann, E. Förster, E. Galtier, S. H. Glenzer, S. Göde, G. Gregori, J. Hajdu, V. Hajkova, P. A. Heimann, R. Irsig, L. Juha, M. Jurek, J. Krzywinski, T. Laarmann, H. J. Lee, R. W. Lee, B. Li, K.-H. Meiwes-Broer, J. P. Mithen, B. Nagler, A. J. Nelson, A. Przystawik, R. Redmer, D. Riley, F. Rosmej, R. Sobierajski, F. Tavella, R. Thiele, J. Tiggesbäumker, S. Toleikis, T. Tschentscher, L. Vysin, T. J. Whitcher, S. White, and J. S. Wark, Electronic structure of an XUV photogenerated solid-density aluminum plasma, Phys. Rev. Lett. 104, 225001 (2010).
  * Di Cicco  _et al._ [2014] A. Di Cicco, K. Hatada, E. Giangrisostomi, R. Gunnella, F. Bencivenga, E. Principi, C. Masciovecchio, and A. Filipponi, Interplay of electron heating and saturable absorption in ultrafast extreme ultraviolet transmission of condensed matter, Phys. Rev. B 90, 220303(R) (2014).
  * Yoneda  _et al._ [2014] H. Yoneda, Y. Inubushi, M. Yabashi, T. Katayama, T. Ishikawa, H. Ohashi, H. Yumoto, K. Yamauchi, H. Mimura, and H. Kitamura, Saturable absorption of intense hard x-rays in iron, Nature communications 5, 5080 (2014).
  * Rackstraw  _et al._ [2015] D. S. Rackstraw, O. Ciricosta, S. M. Vinko, B. Barbrel, T. Burian, J. Chalupský, B. I. Cho, H.-K. Chung, G. L. Dakovski, K. Engelhorn, V. Hájková, P. Heimann, M. Holmes, L. Juha, J. Krzywinski, R. W. Lee, S. Toleikis, J. J. Turner, U. Zastrau, and J. S. Wark, Saturable absorption of an x-ray free-electron-laser heated solid-density aluminum plasma, Phys. Rev. Lett. 114, 015003 (2015).
  * Principi  _et al._ [2016] E. Principi, E. Giangrisostomi, R. Cucini, F. Bencivenga, A. Battistoni, A. Gessini, R. Mincigrucci, M. Saito, S. Di Fonzo, F. D’Amico, A. Di Cicco, R. Gunnella, A. Filipponi, A. Giglia, S. Nannarone, and C. Masciovecchio, Free electron laser-driven ultrafast rearrangement of the electronic structure in Ti, Structural Dynamics 3, 023604 (2016).
  * Hantschmann and Föhlisch [2022] M. Hantschmann and A. Föhlisch, A rate model approach for FEL pulse induced transmissions changes, saturable absorption, x-ray transparency and stimulated emission, Journal of Electron Spectroscopy and Related Phenomena 256, 147139 (2022).
  * Higley  _et al._ [2022] D. J. Higley, Z. Chen, M. Beye, M. Hantschmann, A. H. Reid, V. Mehta, O. Hellwig, G. L. Dakovski, A. Mitra, R. Y. Engel, T. Maxwell, Y. Ding, S. Bonetti, M. Bucher, S. Carron, T. Chase, E. Jal, R. Kukreja, T. Liu, A. Föhlisch, H. A. Dürr, W. F. Schlotter, and J. Stöhr, Stimulated resonant inelastic x-ray scattering in a solid, Communications Physics 5, 1 (2022).
  * Hatada and Di Cicco [2014] K. Hatada and A. Di Cicco, Modeling saturable absorption for ultra short x-ray pulses, Journal of Electron Spectroscopy and Related Phenomena 196, 177 (2014).
  * Engel and et al [2022] R. Y. Engel and et al, High-fluence changes to resonant nonlinear x-ray absorption spectra are dominated by electron dynamics, Jointly submitted to Phys. Rev. Lett. (2022).
  * Persson [2016] K. Persson, Materials data on Ni (sg:225) by materials project (2016).
  * Jain  _et al._ [2013] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K. a. Persson, The Materials Project: A materials genome approach to accelerating materials innovation, APL Materials 1, 011002 (2013).
  * Williams and Fajardo [2020] G. O. Williams and M. Fajardo, Collisional ionization and recombination in degenerate plasmas beyond the free-electron-gas approximation, Phys. Rev. E 102, 063204 (2020).
  * Chen  _et al._ [2006] J. Chen, D. Tzou, and J. Beraun, A semiclassical two-temperature model for ultrafast laser heating, International Journal of Heat and Mass Transfer 49, 307 (2006).
  * Krause [1979] M. O. Krause, Atomic radiative and radiationless yields for k and l shells, Journal of Physical and Chemical Reference Data 8, 307 (1979).
  * Virtanen  _et al._ [2020] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and SciPy 1.0 Contributors, SciPy 1.0: Fundamental algorithms for scientific computing in python, Nature Methods 17, 261 (2020).
  * Bhatnagar  _et al._ [1954] P. L. Bhatnagar, E. P. Gross, and M. Krook, A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems, Physical Review 94, 511 (1954).
  * Bévillon  _et al._ [2014] E. Bévillon, J. P. Colombier, V. Recoules, and R. Stoian, Free-electron properties of metals under ultrafast laser-induced electron-phonon nonequilibrium: A first-principles study, Phys. Rev. B 89, 115117 (2014).
  * Lin  _et al._ [2008] Z. Lin, L. V. Zhigilei, and V. Celli, Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium, Phys. Rev. B 77, 075133 (2008).
  * Krause and Oliver [1979] M. O. Krause and J. H. Oliver, Natural widths of atomic K and L levels, K$\alpha$ x‐ray lines and several KLL auger lines, Journal of Physical and Chemical Reference Data 8, 329 (1979).
  * Powell and Jablonski [2000] C. Powell and A. Jablonski, NIST electron inelastic-mean-free-path database 71, version 1.1 (2000).
  * Mueller and Rethfeld [2013] B. Y. Mueller and B. Rethfeld, Relaxation dynamics in laser-excited metals under nonequilibrium conditions, Phys. Rev. B 87, 035139 (2013).
  * Lojewski  _et al._ [2022] T. Lojewski, M. F. Elhanoty, L. L. Guyader, O. Grånäs, N. Agarwal, C. Boeglin, R. Carley, A. Castoldi, C. David, C. Deiter, F. Döring, R. Y. Engel, F. Erdinger, H. Fangohr, C. Fiorini, P. Fischer, N. Gerasimova, R. Gort, F. de Groot, K. Hansen, S. Hauf, D. Hickin, M. Izquierdo, B. E. Van Kuiken, Y. Kvashnin, C.-H. Lambert, D. Lomidze, S. Maffessanti, L. Mercadier, G. Mercurio, P. S. Miedema, K. Ollefs, M. Pace, M. Porro, J. Rezvani, B. Rösner, N. Rothenbach, A. Samartsev, A. Scherz, J. Schlappa, C. Stamm, M. Teichmann, P. Thunstrom, M. Turcato, A. Yaroslavtsev, J. Zhu, M. Beye, H. Wende, U. Bovensiepen, O. Eriksson, and A. Eschenlohr, The interplay of local electron correlations and ultrafast spin dynamics in fcc Ni (2022), arXiv:2210.13162 .
  * Medvedev  _et al._ [2018] N. Medvedev, V. Tkachenko, V. Lipp, Z. Li, and B. Ziaja, Various damage mechanisms in carbon and silicon materials under femtosecond x-ray irradiation, 4open 1, 3 (2018).
  * Lipp  _et al._ [2022] V. Lipp, V. Tkachenko, M. Stransky, B. Aradi, T. Frauenheim, and B. Ziaja, Density functional tight binding approach utilized to study x-ray-induced transitions in solid materials, Scientific reports 12, 1551 (2022).
  * Stöhr [1992] J. Stöhr,  _NEXAFS spectroscopy_, Vol. 25 (Springer Science & Business Media, 1992).
  * Engel [2022] R. Engel, x-ray non-linear dynamics simulation code [yet private], https://github.com/F00lWise/XNLdyn (2022).