Abel Idrice Adjieufack
email: adjieufack21@gmail.com

Maraf Mbah Bake

Nguemo Nguimkeu

Julien Pilmé

Ibrahim Mbouombouo Ndassa
email: ibrahim.mbouombouo@univ-yaounde1.cm

Exploring the Sequence of Electron Density along the Chemical Reactions between Carbonyl Oxides and Ammonia/Water Using Bond Evolution Theory

The molecular mechanism of the reactions between four carbonyl oxides and ammonia/water is investigated using the M06-2X functional together with 6-311++G(d,p) basis set. The analysis of activation and reaction enthalpy shows that the exothermicity of each process was increased with the substitution of electron donating substituents (methyl and ethenyl). Along each reaction pathway, two new chemical bonds C-N/C-O and O-H were expected to form. A detailed analysis of the flow of the electron density during their formation have been characterized from the perspective of bonding evolution theory (BET). For all reaction pathways, BET revealed that the process of C-N and O-H bond formation takes place within of four stability structural domains (SSD) which can be summarized as follows: the depopulation of V(N) basin with the formation of first C-N bond (appearance of V(C,N) basin), cleavage of N-H bond with the creation of V(N) and V(H) monosynaptic basin and finally the V(H,O) disynaptic basin related to H-O bond. On the other hand, in the case of water, the cleavage of O-H bond with the formation of V(O) and V(H) basins is the first stage, followed by the formation of the O-H bond as a second stage and finally the creation of C-O bond.

I. Introduction

One of the fundamental goals of chemistry is to describe matter at the microscopic scale in the perspective to understand its structure and to explore the mechanisms that ensure its stability. Until the beginning of the twentieth century, chemistry was essentially limited to the writing of chemical formulas for which the chemical bond was symbolized by a "trait".

However, the discovery of the electron in 1897 [1] and that of the nucleus in 1911 [2] have allowed the appearance of more detailed models of the atom and their mutual interactions.

Among these, first model by Lewis in 1916 [3] has proposed an organization of electrons around the nuclei. This vision has been completed later by the Valence Shell Electron Pair Repulsion (VSEPR) model of Gillespie [4,5] which gave a vision of the atom for which the notion of electronic pairs is central.

Quantum mechanics have brought a rigorous description of molecular systems but at the cost of a vision relocated from the material. However, the introduction of the concept of probability density have restored this more localized vision and that of the densities of pairs have given a theoretical seat to the concept of electron pairs. From the early 1970s, Richard Bader has shown that electronic density is an important source of key information, which is obtained experimentally by X-ray diffraction spectra, or by calculation. [6,7] Its mathematical properties make it possible to divide the molecular space of rigorous way in atomic volumes. This division introduced the notion of topological atom in a molecule through the theory "Quantum Theory of Atoms in Molecules" (QTAIM), [8,9] while remaining in a purely quantum context. The atom in the molecule then has a well-defined volume and specific properties as a charge and an energy. This theory paved the way for Quantum Chemical Topology (QCT) [10,[START_REF]Applications of Topological Methods in Molecular Chemistry[END_REF] and its formalism has been adapted to many density functions like the ELF electron localization function of Becke and Edgecombe published in 1990. [START_REF] Becke | [END_REF]13] The ELF function measures, from a physical point of view, the increase in the spatial localization of the electron known as localization basin. Moreover, it provides a mapping method for the probability of the presence of an electron pair for a multi-electron system.

Working in this sense, Kohout have proposed the Electron Localizability Indicator (ELI-D) [14,15] which is based on the motion of same spin electron density. ELI-D is an useful tool for the detection of chemical bonding signatures in molecule or crystal. [16,17] The combination of AIM and ELI-D approaches is also an issue to determine the Raub-Jansen Index (RJI) [18,19] which is an appropriate tool to describe the bond polarity. Recently, Silvi et al [20] have tried to unify the concept of secondary bond which describes the non-covalent interactions between an acceptor molecule and an electron donor one which generally yield to complexes. They have proposed and verified by quantum calculations a set of VSEPR-inspired rules allowing the construction of bound secondary isomers(AX 4-n E n ).

Although the ELF and ELI-D analyses are a well-established one-density descriptors for both covalent and non-covalent interactions, [21,22] the development of its chemical reactivity component remains a fascinating challenge. Using the electron density, Krokidis and Silvi have proposed the bonding evolution theory (BET) [23,24] which joins the electronic localization function (ELF) [24,25] with the Thom's catastrophe theory (CT). [START_REF] Thom | Structural Stability and Morphogenesis, an Outline of a General Theory of Models[END_REF] BET has become a powerful tool to monitor the electronic changes or reorganization of electron pairs occurring along the reaction paths. [START_REF]Applications of Topological Methods in Molecular Chemistry[END_REF][27][28][29][30][31][32] Research on chemical reactivity has moved from the analysis of the stationary points to the potential energy surface. In this context, a new mechanistic methodology designed to identify the favorable reaction route between several reactants was recently proposed. [33,34] However, the key questions that must be addressed to understand the nature of the molecular mechanisms remain related to fundamental chemistry: the distribution and the evolution of electron density along the reaction pathway. The bonding evolution theory, based on the changes described by Thom's catastrophe theory on the electron localization function topology along a given reaction path, is capable of unambiguously identifying the main chemical events happening throughout chemical reactions, [35][36][37][38][39][40][41] and generating the corresponding curly arrows to describe the reaction mechanism. [START_REF] Andrés | Quantum Chemical Topology Approach for Dissecting Chemical Structure and Reactivity[END_REF][START_REF] Andrés | [END_REF] Carbonyl oxides well known as Criegee intermediates (CIs) are strong oxidizing agents formed by ozonolysis of unsaturated hydrocarbons in the atmosphere. [44] Being among these Criegee intermediates, formaldehyde oxide (CH 2 OO), acetone oxide ((CH3) 2 COO) and methyl vinyl ketone oxide (cis and trans) due to their COO functional group can react with molecules such as H 2 O, CH 3 OH, NO 2 , NH 3 and CH 3 NH 2 to form a strong hydrogen bonds. [45] Criegee intermediates being more reactive molecules and difficult to study experimentally, computational chemists have been devoted to the exploration of their chemical reactivity. Sajadi et al. [46] have explored the potential energy surface of the reaction between CH 2 OO and (CH 3 ) 2 COO with dimethyl sulfide (DMS) at the M11-L/6-311++G(2d,2p) level. The authors found within of transition state theory (TST) and RRKM theory that both reactions occurred via a strong Van der Waals complexes leading to adducts with a new C-S single bond.

Mallick et al. [47] have studied the reaction of N 2 O with CH 2 OO using M06-2X/aug-cc-pVTZ computational level. They found that the chemical reaction mechanism could proceed through six different channels with a rate constant of 1.79x10 -30 cm 3 molecule -1 s -1 for a favorable reaction pathway.

The potential energy surface of the reaction between carbonyl oxides and ammonia/water explored by some authors is found in the literature. Jørgensen et al. [48] have investigated the reaction mechanism of the chemical reactions between ammonia and carbonyl oxides using B3LYP/6-311++G(2d,2p) as computational method. They have concluded that the structural rearrangement occurred with the formation of a C-N bond between nitrogen atom of ammonia and C3 carbon atom of carbonyl oxides, followed by the hydrogen atom transfer from ammonia to the terminal oxygen of carbonyl oxides (formation of O-H chemical bond). Anglada et al. [49] have studied the chemical reaction of fifteen carbonyl oxides and water in order to understand the substitution effect in the reactivity of carbonyl oxides using Density Functional Theory (B3LYP) and ab initio (CCSD(T), CASSCF and CASPT2) methods. For these authors, carbonyl oxides containing the electron withdrawing substituents presented a low energy barrier and reacted rapidly with water molecule, while those ones with electron donating substituents had high energy barriers and reacted slowly. In the perspective to confirm the O-H bond formation during the chemical reaction of carbonyl oxides, Chhantyal-Pun et al. [50] have performed the

II. Theory

In quantum chemical topology, the properties of one-density scalar functions are used for discussing the nature of chemical bonding in molecules and solids. [START_REF]Applications of Topological Methods in Molecular Chemistry[END_REF]51] This approach provides a partition of the function into non-overlapping atomic regions so-called basins. The gradient field is characterized by its critical points (where ρ(r) = 0) and their connectivity. Critical points can be either local maxima, minima or saddle points. In the QTAIM approach, the topological atom is defined as the union of a nucleus and of its basin. ) at the BCP. [52] When this ratio is smaller than 1, the kinetic energy density is the leading term and electrons are destabilized close to the BCP, thus no covalency is expected (for example pure ionic or van der Waals bonding).

Another scalar field is the Electron Localization Function (ELF) which is a measure of the possibility of finding an electron in the vicinity of a reference electron at a given point with the same spin. [START_REF] Becke | [END_REF] It was set up by the original formulation of Becke and Edgecombe and defined as follows:

(

The equation ( 1) lies on the conditional same-spin pair probability scaled by the homogeneous electron gas (HEG) kinetic energy density ,

ELF ranges from 0 to 1 where 1 is a situation void of Pauli repulsion, i.e. a high probability of finding electron localization. A few years later, this formulation has been generalized to the DFT theory by Savin and co-workers, [13] and rationalized in terms of the local excess kinetic energy due to the Pauli repulsion.

The usefulness of the electronic localization function is mainly due to the way it allows the analysis of the electronic localization in a very chemically intuitive way. During this ELF analysis, the reaction pathway is divided into domains (basins) where the presence is maximal to find an electron paring. [START_REF] Becke | [END_REF]22] Following the chemical intuition, two types of basins can be itemized: core and valence basins. The second one is classified into their synaptic order i.e. the number of core basins with which they share a common boundary [53] : Relative to the position changes of different structural stability domain (SSD) found during the BET analysis for a given reaction pathway, we have proposed to evaluate the synchronicity of chemical process by defining the synchronicity indicator. [54,55] Its expression is given by equation ( 4).

protonated basin denoted V(A,
In equation ( 4), n is the number of SSD found along the BET and and are the final and initial IRC coordinates values. In addition, and represent the different values of IRC between appeared SSDs. The maximum value of amounts to 1 and its minimum value can be expressed as followed:

The absolute synchronicity ( ) can be derived from equation ( 4) and ( 5) as:

III. Computational Method

Full geometries optimizations have been performed with the aid of Gaussian 16

program. [45] The electronic structures as well as the geometries of the reactants, products, and transition states have been determined at the M06-2X functional [46] using 6-311++G(d,p) basis set. The transition states TSs were further confirmed by vibrational analysis and characterized by the only one imaginary vibrational mode. The IRC curves [47,48] were calculated following the path of reactant to the product using a standard integration method. [49] To perform the ELF topological analysis within the BET theory, the wave function was obtained for each point of the IRCs. The ELF analysis was performed with the aid of TopMod package [50] by considering a cubic grid of 0.2 bohr. Finally, ELF basin positions were visualized using Drawmol [51] while electron population evolution of basins along the IRC was done with Drawprofile. [52] The QTAIM analysis was performed using the TopChem2 package [53] while the ELF basin isosurfaces were also displayed with Drawprofile [52] software. - are also reported in Table 1. As activation energy, the activation enthalpy and free Gibbs energy follow the same tendencies. For those of reaction, the enthalpy is negative, suggesting an exothermic process for these chemical reactions between carbonyl oxides with ammonia.

IV. Results and discussion

Energetical, structural and QTAIM analysis

Reaction carbonyl oxides 1-4 with ammonia

Furthermore, the substitution of the two hydrogen atoms by methyl and ethenyl groups increases the exothermicity with an enthalpy energy of product 1b is 11.5, 12.4 and 14.4

kcal/mol lower than 2b, 3b and 4a, respectively.

-Reaction between carbonyl oxides 1-4 with water.

The relative activation energies, relative enthalpies, entropies and Gibbs free energies of all stationary points engaged in the reaction between carbonyl oxides 1-4 and water are also reported in Table 1. TS1-b presents an activation energy which is 5.6, 5.5 and 7.0 kcal/mol lower than the TS2-b, TS3-b and TS4-b, respectively. This analysis of activation energy allows us to conclude that the substitution of carbonyl oxide 1 by the electron donating substituents (methyl and ethenyl) increases the activation barrier. This is also verified for the reaction energy with product 1b which is 11.9, 14.0 and 16.1 kcal/mol more than those of 2b, 3b and 4b, respectively.

Concerning the thermodynamics parameters, as for the case of ammonia, the enthalpy energy is also negative, suggesting an exothermic character in the formation of products.

-

Analysis of structural parameters

The structures of transition states (TSs) are represented in Figures 1 and2. The bond lengths of O1-H1 and C3-N4 at TS1-a are and , respectively (Figure 1). The substitution of hydrogen atoms of the dipole 1 by methyl groups, allows a decrease in O1-H1 and C3-N4 length at TS2-a by and , respectively. In the cases of TS3-a and TS4-a, the same phenomenon is observed when one methyl group of carbonyl oxide 2 is replaced by ethenyl. TS3-a and TS4-a present the shortest lengths of O1-H1 (1.924 at TS3a and at TS4-a) forming bonds compared to first two transitional states. This bond shortening can be explained based on steric hindrance around the carbon C3 of the carbonyl oxides.

Figure 2 shows the structures of TSs involved in the reactions between carbonyl oxides and water. Compared to Figure 1, Figure 2 reveals that the replacement of ammonia molecule by water has a great effect on the geometrical structure of transition state (TSs). In fact, the length of O1-H1 and C3-O4 forming bonds are shorter than those in the case of reactions between carbonyl oxides and ammonium. -

QTAIM analysis of TS structures

We now continue with a QTAIM analysis for the both TS series. As shown in Table 2, two new bond critical points have been identified, first between H1 and O1 and second, between C3 and N4 in the case of ammonia and C3 and O4 in the case of water, respectively. These results reveal a typical signature of the formation of the two emerging chemical bonds.

As observed in Table 2, the atomic charge of the H1 atom varies between 0.45e to 0.51e along the TS(1-4)a series while H1 displays a charge upper than 0.63e along the TS(1-4)b series. In addition, since some covalency is witnessed by the |V b |/G b ratio that is always greater than 1, (i.e. the potential energy density dominates and electrons are stabilized at the BCP), the O1-H1 emerging bond seems to be easier triggered in the TSb series (ratio upper than 1.0) than in the TSa series (ratio lower than 1.0). Therefater, the QTAIM analysis shows that, along the

BET Analysis

-BET Analysis for the reaction between carbonyl oxides 1-4 with ammonia.

The mechanism of the reaction between carbonyl oxide 1 and ammonia

To further clarify the bond formation/breaking process, the analysis of ELF topology (BET) along the reaction path between carbonyl oxide 1 and ammonia is studied. The BET study shows clearly a total of four structural stability domains (SSD) are recorded as displayed in Figure 3.

The first domain (SSD-I) displays the ELF basin populations of the two reactants as reported in Table S2(SI). In the framework of carbonyl oxide 1, these basins are highlighted as follows: two monosynaptic basin V(O1) and V(O2) with a total population of 6.23 and 3.41e, and two disynaptic basins V(O2,C3) and V(O1,O2) integrating respectively 2.78 and 1.01e. The ammonia shows four basins related to N4 lone pair [V(N4) with 2.05e] and N-H bonds [V(H1,N4) with 1.96e, V(H2,N4) and V(H3,N4) integrating 1.92e each one]. At the end of domain, some basins such V(N4), V(O1,O2) and V(O2,C3) present a decrease of 1.69, 0.81 and 2.09e in theirs electronic populations.

The decrease of V(N4) population continues along SSD-II domain with its disappearance in favor of the formation of new basin disynaptic basin V(C3,N4) as can be seen in Figure 4 and in Scheme 2. This new basin V(C3,N4) represents the first main topological change and illustrates the formation of C3-N4 bond. Its population of 1.73e corresponds almost to the electron population of the former monosynaptic basin V(N4) at the end of first domain. Contrary to SSD-II with formation of C3-N4 bond, the third domain (SSD-III) exhibits two new monosynaptic basins [V(H1) and V(N4)] (see Figure 4) coming from the cleavage of H1-N4 bond [disappearance of V(H1,N4) basin with a population of 2.09e at the end of SSD-III domain]. The two V(H1) and V(N4) fold catastrophes start with a population of 0.55 and 1.57e, respectively at the beginning of the domain. The appearance V(N4) around the N4 nitrogen atom represents the restoration of the lone pair N4 that was involved in the formation of the carbon-nitrogen bond (C3-N4) at the SSD-II.

SSD-I SSD-II SSD-III SSD-IV V(O1) V(H3,N4) V(H2,N4) V(H1,N4) V(N4) V(O2) V(O2,C3) V(C3,N4) V(C3,N4) V(N4) V(H1) V(H1,O1) V(O1) V(O1) V(O1) V(O2) V(O2) V(O2) V(H2,N4) V(H2,N4) V(H2,N4)
Finally, the last topological change occurs along the domain SSD-IV with the formation of last chemical bond H1-O1 via the creation of V(H1,O1) disynaptic basin. The V(H1,O1) basin begin with population of 1.61e and reaches 1.75e at the end of domain. Its population comes from the disappearance of V(H1) with 0.42e at the end of third domain and the reduction of V(O1) population by 1.14e at the beginning of domain.

The synchronicity ( ) and absolute synchronicity ( ) have been evaluated using the equations 4-6 and their values are: =0.79 and =0.65. This value of absolute synchronicity confers a synchronous character at 65% for topological changes taking place along the reaction path TS1-a.

-

The mechanism of the reaction between carbonyl oxides 2-4 and ammonia

Like the TS1-a reaction path, the BET analysis was also performed along the TS2a, TS3-a and TS4-a pathways associated with the reactions between carbonyl oxides 2-4 and bohr.amu 1/2 compared to TS1-a. For the cases of TS3-a and TS4-a, the same phenomenal is also observed with an advance of 1.27 and 1.27 bohr.amu 1/2 for V(C3,N4) appearance. The V(O1,H1) disynaptic basin takes place also with an advance of 1.31 bohr.amu 1/2 at TS3-a and 1.45 bohr.amu 1/2 at TS4-a. In spite of this difference in the reaction coordinate of appearance for the V(C3,N4) and V(O1,H1) basins, the electron population are similar at the beginning of topological domains.

The estimated value of synchronicity ( ) and absolute synchronicity ( ) are: 0.85, 0.83 and 0.87, and 0.76, 0.72 and 0.79 for TS2-a, TS3-a and TS4-a, respectively.

Compared to TS1-a, the topological changes take place along these TS2-a, TS3-a and TS4-a reactive pathways with a very synchronous character.

-BET Analysis for the reaction between carbonyl oxides 1-4 with water.

The mechanism of the reaction between carbonyl oxide 1 and water.

Like the reaction between carbonyl oxides 1-4 with ammonia, we have also explored the ones with water. Figure 5 displays the evolution of electron population along the reaction carbonyl oxide 1 and water. From this figure 5, the molecular chemical mechanism is divided into three domains (SSD) through the BET analysis. As in the case of ammonia, the first domain SSD-I contains the main basins of different atoms engaged in the formation of the two new O1-H1 and C3-O3 bonds. We have three basins for water molecule: V(H1,O3), V(H2,O3) and V(O3) integrating 4.51, 1.71 and 1.70e, respectively (Table SI7). The carbonyl oxide 1, in the case of ammonia presents the same basins (with almost the same populations) which are: V(O1), V(O2), V(O1,O2) and V(O2,C3) with 6.18, 3.34, 1.05 and 2.82e, respectively.

Contrary to TS1-a, the second domain SSD-II corresponds here to the cleavage of H1-O3 via the transformation into V(H1) and V'(O ) monosynaptic basins as we can see in

Conclusion

The reactions between four carbonyl oxides and ammonia/water yielding to alkylamines and hydroxyalkyl -hydroperoxides respectively, have been investigated using By substituting the hydrogen atoms by methyl and ethenyl groups, three reactions pathways were also possible and their molecular mechanism was also analyzed with BET theory. In the case of ammonia, the different stages of bond formation remain the same (four SSDs) along TS2-a, TS2-a and TS4-a reactive pathways compared to TS1-a. However, in the case of water, this substitution increases the number of SSDs to four compared to TS1-b where only three was found. This main difference comes from the double appearance of new basins V(O1,H1) and V(C3,O3) at the third domain along the TS1-b, and contrary to their consecutive appearance along TS2-b, TS3-b and TS4-b.

Associated Content

Tables containing electronic energies, enthalpies, entropies and Gibbs energies for the reaction of dipoles a-d with ammonia/water; Electron populations for each BET analysis for reaction of dipoles a-d with ammonia/water; Lewis structures for each SSD found along the reaction between dipole b-d and ammonia/water.

Corresponding Authors

Email: adjieufack21@gmail.com and ibrahim.mbouombouo@univ-yaounde1.cm

  experimental and computational studies on reaction of CH 2 OO and (CH3) 2 COO with NH 3 and CH 3 NH 2 . They found through photoionization mass spectra data that NH 2 CH 2 OOH and CH 3 N(H)CH 2 OOH were the main hydroperoxide adducts experimentally observed. For a deep understanding of the formation of C-N, C-O and O-H chemical bonds, a detailed chemical insight on the progress of the reaction between carbonyl oxides and ammonia/water (see Scheme 1) from the perspective of bonding evolution theory (BET) is examined with a focus on the determination of reaction mechanisms, and emphasizing the way of curly arrows meeting electron density transfers in chemical reaction mechanisms. In what follow, the motivation of this present work is given by answering the following questions during the BET analyzing: i) where and how charge transfer take place along the reaction progress? how electron density rearranges and how this rearrangement can be associated with chemical events such as the breaking/forming of chemical bonds, along the reaction progress? iii) how could the electronic reorganization proceed along the reaction path? which type of catastrophe appears along each reaction pathway during the BET? iv) how is the nature of the chemical mechanism along the reaction pathway, concerted or stepwise? v) which chemical bond between C-N/C-O and O-H is formed first along each reactive pathway?

  Among the saddle points, a bond critical point (BCP) has a crucial role because, the values of some descriptors at the BCP are related to the nature of the chemical bond. The electron density at the BCP, ρ b , is typically larger than 0.20 e bohr -3 in shared-shell interactions, in other words, covalent bonds, and smaller than 0.10 e bohr -3 in closed-shell interactions (e.g. ionic, van der Waals or hydrogen bonding). When the Laplacian of the density at the BCP,  2 ρ b , is negative, the local concentration of charge indicates a shared-shell interaction. In contrast, if  2 ρ b is positive there is a depletion of charge typically characterizing a closedshell interaction. Another descriptor often used in QTAIM to differentiate two shared-shell and closed-shell bonding scheme is the ratio |V b |/(2G b ), based on the potential energy density (V b ) and the kinetic energy density (G b

  H), monosynaptic basin for a lone pair [V(A)] and disynaptic basin [V(A,B)] for a bonding region between two (A and B) or several nuclei. The basin populations are then obtained from the usual condensation of the electron density computed over the ELF basin volumes.

Scheme 1 Scheme 1 .

 11 Scheme 1 displays the stereochemical structures of the different carbonyl oxides 1-4 and their corresponding reaction with ammonia and water, yielding to alkylamines, (1-4)a and hydroxyalkyl -hydroperoxides, (1-4)b respectively. The reaction channel-a represents the reaction of carbonyl oxides 1-4 with ammonia while channel-b describes the one of water.

Figure 1 .

 1 Figure 1. M06-2X/6-311++G(d,p) optimized geometries for the TSs involved in these reactions between carbonyl oxides and ammonia with indicated bond distances.

Figure 2 .

 2 Figure 2. M06-2X/6-311++G(d,p) optimized geometries for the TSs involved in these reactions between carbonyl oxides and water with indicated bond distances.

  TS2 series, by going from the TS1-b structure to the TS4-b ones, the covalent character of the H1-O1 and C3-N4/O4 bonds is reinforces since ρ b and the |V b |/G b values increase while the  2 ρ b ones tends to decreased. The same trends are observed for the TS(1-4)a series. In summary, these latter results are perfectly consistent with the increase of relative activation energies observed along the carbonyl oxides series from TS1-b to TS4-b structures.

Figure 3 .

 3 Figure 3. Evolution of electron population (in e) of selected basins engaged in the formation of O1-H1 and C3-N4 bonds along the TS1-a reactive pathway.

Figure 4 .

 4 Figure 4. ELF basin isosurfaces (=0.7) of each of the SSD recorded and engaged in the formation of O1-H1 and C3-N4 bonds along the TS1-a reactive pathway.

  ammonia. The number of SSD recorded along each pathway remains four as the case of TS1a. Figures SI1-3 represent the evolution of electron density of selected basins engaged in the formation of H1-O1 and C3-O3 bonds while figures SI4-6 contain the ELF basin isosurfaces representation found in each SSD and Scheme SI1 illustrates the Lewis structure for each SSD. The substitution of the hydrogen (carbonyl oxide 1) by the methyl and ethenyl group does not affect the number of SSD recorded but influences the reaction coordinate of appearance of the main topological changes. In the case of TS2-a, the appearance of V(C3,N4) and V(O1,H1) basins occur at the reaction coordinate Rx=0.42 and 3.13 bohr.amu 1/2 respectively, with an advance of 1.06 [V(C3,N4)] and 1.30 [V(O1,H1)]

Figure 6 .

 6 Figure 6. The V'(O ) monosynaptic basin with 1.49e corresponds to the creation of a lone pair around the O3 oxygen atom and conferring a negative charge on this atom (Scheme 3), while V(H1) monosynaptic basin with 0.41e belongs to hydrogen with positive charge. Furthermore, the last domain SSD-III corresponds to the double appearance of cusp catastrophe for the two main topological changes specifically O1-H1 and C3-O3 bonds. The population of the new basins V(H1,O1) and V(C3,O3) disynaptic basins starts with 1.67 and 0.76e and grows up to 1.75 and 1.39e. The V(H1,O1) population comes from the disappearance of V(H1) and the reduction of V(O1) monosynaptic basin population. At the beginning of the domain, the V(O1) recorded a loss of 1.27e for H1-O1 forming bond while the V(O3) disynaptic basin also recorded a loss of 0.44e for the second C3-O3 forming bond.The higher population contribution of V(C3,O3) basin comes from the disappearance of the second V'(O ) which possessed a population of e at the end of the second domain SSD-II. Furthermore, this disappearance of V'(O ) basin also increases the electron population of V(O3) which reaches at 4.62e.

Figure 5 .

 5 Figure 5. Evolution of electron population (in e) of selected basins engaged in the formation of O1-H1 and C3-O3 bonds along the TS2-a reactive pathway.

Figure 6 .Scheme 3 .between carbonyl oxides 2 - 4

 6324 Figure 6. ELF basin isosurfaces (=0.7) of each of the SSD recorded and engaged in the formation of O1-H1 and C3-N4 bonds along the TS2-a reactive pathway.

3 .

 3 the M06-2X functional together with 6-311++G(d,p) basis set. The analysis of activation energy and thermodynamic parameters clearly shows that substitution of the hydrogen atom (carbonyl oxide 1) by the electron donor groups (carbonyl oxides 2-4) affects the latter. The exothermicity along these chemical reactions increases from the carbonyl oxide 1 to 4 due to presence of methyl and ethenyl groups on C3 carbon atom of carbonyl oxides 2-One transitional state is found for each reaction pathway with an expected formation of two new chemical bonds, C3-N4 and H1-O1 in case of ammonia, C3-O3 and H1-O1 for water respectively. For the reaction pathway TS1-a, four structural stability domains are found and the molecular mechanism can be summarized as follows : formation of C3-N4 bond coming from the sharing of the initial population of V(N4) monosynaptic basin between C3 carbon atom of dipole a and N4 nitrogen of ammonia, and yielding V(C3,N4) basin, the transfer of the V(H1,N4) basin population to the H1[V(H1)] and N4 [V(N4)] atoms in order to restore the Lewis configuration around the N4 nitrogen atom) and finally, H1-O1 bondforming process via the creation of V(H1,O1). However, for TS1-b reaction pathway, only three structural stability domains are recorded and different steps of chemical rearrangements are: first step corresponds to the transfer of V(H1,O3) basin population to H1 and O3 atoms while the last step deals with the creation of two new basins V(H1,O1) and V(C3,O3) which are related to H1-O1 and C3-O3 bonds.

Table 1

 1 are substituted by two methyl groups in the perspective to form carbonyl oxide 2, the activation energy arising from this substitution is 7.1 kcal/mol. This activation barrier is 6.2 kcal/mol higher than TS1-a. Secondly, one methyl group of dipole b is replaced by ethenyl one, two carbonyl oxides namely 3 and 4 are possible with the different (Z and E) positions of ethenyl group. Compared to the reaction of carbonyl oxide 1, they present an activation energy gap of 5.9 and 7.7 kcal/mol. With respect to the activation energy, the reaction of carbonyl oxide 1-4 with ammonia are very much influenced by the presence of the substituents at the carbonyl oxide framework and playing a key role on the reactivity of carbonyl oxides. Furthermore, the reaction activation energies follow the same tendency with the activation energy of adduct 1a which is 11.9, 13.2 and 15.1 kcal/mol lower than the ones of adducts 2a, 3a and 4a, respectively.

displays the activation energies of the different species engaged in reaction of carbonyl oxides with ammonia and water. The activation barrier of TS1-a for the carbonyl oxide 1 with ammonia is 0.9 kcal/mol. Firstly, when the two hydrogen atoms of carbonyl oxide 1

Table 1 .

 1 The relative (E°, in kcal/mol) activation energies and thermodynamics parameters (H° and G°, in kcal/mol and S°, in cal/mol.K) of the different complexes engaged in reaction of carbonyl oxides with ammonia and water computed at 25°C with M06-2X/6-311++G(d,p) level of approximation.

	Species	E°	H°	S°	G°
	TS1-a	0.9	0.7	-6.3	2.6
	TS2-a	7.1(6.2)	6.9(6.2)	-9.8(-3.5)	9.8(7.2)
	TS3-a	6.7(5.9)	6.3(5.5)	-9.7(-3.4)	9.2(6.6)
	TS2-a	8.6(7.7)	8.2(7.5)	-10.4(-4.1)	11.4(8.8)
	TS1-b	7.4	6.7	-8.6	9.3
	TS2-b	12.9(5.6)	11.5(4.8)	-9.2(-0.6)	14.3(5.0)
	TS3-b	12.8(5.5)	11.0(4.3)	-10.1(-1.4)	14.0(4.7)
	TS4-b	14.4(7.0)	12.7(5.9)	-10.0(-1.4)	15.6(6.3)
	1a	-46.7	-46.6	-7.8	-42.3
	2a	-34.8(11.9)	-33.1(11.5)	-10.7(-2.9)	-29.9(12.4)
	3a	-33.5(13.2)	-32.(12.4)	-10.7(-2.9)	-29.1(13.2)
	4a	-31.6(15.1)	-30.2(14.4)	-10.8(-3.0)	-27.0(15.3)
	1b	-43.3	-41.0	-6.7	-39.0
	2b	-31.3(11.9)	-29.7(11.3)	-8.7(-2.0)	-27.1(11.9)
	3b	-29.3(14.0)	-28.0(13.0)	-8.8(-2.1)	-25.4(13.6)
	4b	-27.2(16.1)	-25.8(15.2)	-8.0(-1.3)	-23.4(15.6)
	Thermodynamics parameters (relative enthalpies, entropies and Gibbs free energies)

Table 2 .

 2 Selected QTAIM descriptors (in a.u.) obtained at M06-2X/6-311++G(d,p) level of theory for the O1-H1 and C3-N4/O4 interactions involved in the TS systems.

		QTAIM charges	Descriptors at the	Descriptors at the
					O1-H1 BCP			C3-N4/O4 BCP
		q(H1) q(N4) or q(O4)
	TS1-a	0.45	-1.07	0.019	0.069	0.901	0.051	0.102	1.205
	TS2-a	0.48	-1.06	0.029	0.101	0.962	0.077	0.088	1.488
	TS3-a	0.51	-1.06	0.031	0.108	0.968	0.076	0.089	1.471
	TS4-a	0.49	-1.05	0.032	0.108	0.982	0.083	0.080	1.551
	TS1-b	0.65	-1.16	0.069	0.128	1.395	0.069	0.144	1.256
	TS2-b	0.64	-1.15	0.099	0.105	1.625	0.070	0.133	1.286
	TS3-b	0.63	-1.14	0.106	0.095	1.679	0.072	0.132	1.304
	TS4-b	0.63	-1.14	0.109	0.085	1.720	0.071	0.132	1.296

Acknowledgements A.A.I. thanks the University of Namur (Belgium) for his UNamur-CERUNA Ph.D. Mobility Fellowship. Authors are gratefull to the Consortium des Équipements de Calcul Intensif (CÉCI, http://www.ceci-hpc.be) and particularly the Technological Platform of High-Performance Computing for all calculations through financial support from the FNRS-FRFC, of the Walloon Region, and of the University of Namur (Conventions No. 2.5020.11, GEQ U.G006.15,1610468, and RW/GEQ2016).

Author Information

Abel Idrice Adjieufack: 0000-0001-8769-6036 Ibrahim Mbouombouo Ndassa :0000-0003-0714-4982

Notes

The authors declare no competing financial interest.


Nanna Zhou Hagström

Rahul Jangid

Meera Madhavi

Diego Turenne

Jerey Brock

Erik S Lamb

Boyan Stoychev

Justine Schlappa

Natalia Gerasimova

Benjamin Van Kuiken

Rafael Gort

Laurent Mercadier

Loïc Le Guyader

Andrey Samartsev

Andreas Scherz

Giuseppe Mercurio

Hermann A Dürr

Alexander H Reid

Monika Arora

Hans T Nembach

Justin M Shaw

Emmanuelle Jal

Eric E Fullerton

Mark W Keller

Roopali Kukreja

Stefano Bonetti

Thomas J Silva

Ezio Iacocca

Symmetry-dependent ultrafast manipulation of nanoscale magnetic domains

Femtosecond optical pumping of magnetic materials has been used to achieve ultrafast switching and recently to nucleate symmetry-broken magnetic states. However, when the magnetic order parameter already presents a broken-symmetry state, such as a domain pattern, the dynamics are poorly understood and consensus remains elusive. Here, we resolve the controversies in the literature by studying the ultrafast response of magnetic domain patterns with varying degrees of translation symmetry with ultrafast X-ray resonant scattering. A novel data analysis technique is introduced to disentangle the isotropic and anisotropic components of the x-ray scattering. We nd that the scattered intensity exhibits a radial shift restricted to the isotropic component, indicating that the far-from-equilibrium magnetization dynamics are intrinsically related to the spatial features of the domain pattern. Our results suggest novel pathways for the spatiotemporal manipulation of magnetism via far-from-equilibrium dynamics and by carefully tuning the ground state magnetic textures.

I. INTRODUCTION

Ultrafast manipulation of symmetry is achievable in a wide variety of physical systems that rely on nonequilibrium pathways to access hidden states in their energy landscape. Far-from-equilibrium transitions from symmetric to symmetry-broken states have been observed in a variety of material systems, e.g. photo-

induced superconductivity [1], structural modication of alloys [2], manipulation of topological phases in Weyl semimetals [3], vibrational dynamics following melting of atomic charge order in nickelates [4], hidden states during spontaneous symmetry breaking of charge density waves [5], and charge separation of chiral organic molecules [6]. Symmetry can be also manipulated in magnetic materials because of the interplay between their local and nonlocal order parameters. Recent studies have * To whom correspondence should be addressed; E-mail: eia- cocca@uccs.edu.

indeed demonstrated that the homogeneously magnetized ferrimagnet GdFeCo undergoes phase-ordering kinetics through the ultrafast formation of localized defects [7]. Moreover, topological phases could be accessed in ferromagnetic materials biased with an external magnetic eld, demonstrating the picosecond emergence and subsequent stabilization of skyrmion lattices [8] and in ferrimagnets showing the transition from helical to skyrmion phases [9]. In these works, the manipulation of symmetry occurred within the magnetic or spin degree of freedom. However, ultrafast excitation of metallic magnetic materials [10] also induces non-equilibrium spin currents [11] producing torques [12] that aect the picosecond dynamics of the spin degree of freedom [13 20] and induce phonon modes coupled to the magnetic system by magnetostriction to form nanosized spin-wave solitons [21].

A clear manifestation of spin-current-induced ultrafast magnetization dynamics is found in materials exhibiting magnetization textures. It was recognized that materials stabilized in a stripe domain pattern could be demagnetized more quickly than the uniformly magnetized sample [17]. This nding provided a rst indication that the non-local magnetic texture aects the ultrafast behavior of the material. It is then natural to inquire how the dierent magnetic textures with distinct translation symmetries aect the picosecond magnetization dynamics and how can the textures themselves be manipulated by optical excitations? Such control of magnetism at the femtosecond timescales is particularly important for proposals of energy ecient and fast magnetic storage devices [22] where the information is encoded by magnetic domains along tracks, eectively imposing a randomized magnetic texture that should not to be corrupted by external stimuli.

To study the far-from-equilibrium dynamics of magnetic textures occurring at the nanoscale, time-resolved

x-ray scattering from free electron lasers remains the preferred method to achieve the necessary combined temporal and spatial resolution [7,11,16,17,19,20,23]. The detected scattered intensity pattern directly correlates to the symmetry of the magnetization texture. Stripe domains exhibit spatial translation symmetry along one dimension, leading to a distinctive anisotropic scattering pattern. In contrast, labyrinth domains can be regarded as stripe domains where the underlying anisotropy is lifted, thus causing a randomized long-range order that gives rise to an isotropic scattering ring. This behavior is well-known in the broader eld of pattern formation, particularly of Turing patterns [24]. In the context of x-ray Bragg diraction, labyrinth domains would be similar to polycrystalline or powder sample which consists of randomly-oriented crystallites (or grains) leading to formation of well-known Debye rings. An extreme case are amorphous materials which have short range order but no long range order resulting in broad rings. On the other hand, stripe domains would be akin to diraction from a single crystal which exhibits spatial translation symmetry across the entire sample resulting in a Bragg spot.

Studies in both stripe and labyrinth domain patterns have provided a wealth of observations that to date remain disparate and controversial. Initial studies on labyrinth domains in Co/Pt multilayers reported a ring contraction of ≈ 4 % that was interpreted as a result of spin-current induced domain-wall broadening [16]. This conjecture followed from the impossibility of a fractional expansion of the domain pattern, i.e., a change in periodicity in the probed section, that would imply domain-wall speeds over the speed of light. Later studies in CoFe/Ni multilayers that could access higher order diraction rings were able to disentangle domain-wall broadening from the spectral periodicity, demonstrating that the observed shift was related to nanoscopic variations in the domain pattern [20]. In particular, domain-wall broadening of 31 % was identied from the relative harmonic amplitudes of the scattering rings, while the observed harmonic shift of 6 % accompanied by a 15 % linewidth broadening was consistent with domain-wall motion on the order of 2 km/s. More recently, a similar shift of ≈ 2.6 % in the scattered ring was observed for chiral labyrinth domains which was explained by dierent demagnetization rates for homogeneous (domains) and inhomogeneous (domain-walls) regions of the sample [25].

Interestingly, this shift has not been observed in stripe domains [17,19] contradicting earlier works which propose modications of the domain-wall prole as the key eect. However, domain-wall broadening of 41 % after 20 ps was clearly observed in Ref. [19].

We clarify these controversies by studying the timeresolved x-ray scattering from a magnetic multilayer sample which can exhibit stripe, labyrinth and mixed domain characteristics. By isolating the dierent symmetries in the observed scattering pattern, we demonstrate symmetry-dependent ultrafast dynamics. In particular, only the isotropic component exhibits a shift of its peak position in reciprocal space, even when both symmetry components are present at the same time. This result alone conclusively proves that domain-wall broadening, which is presumably operative in both stripe and labyrinth samples, cannot explain the peak shift of the diraction ring in labyrinth samples, as was previously proposed in Ref. [16].

For the mixed states, our data suggests that recovery time for the isotropic and anisotropic components are different. Our studies indicate that the labyrinth magnetic textures are more prone to be spatially manipulated at ultrafast timescales and opens new routes for the spatial ultrafast manipulation of magnetism. x 8 / Co 90 Fe 10 (0.25 nm)/Cu(5 nm)/Ta(3 nm)) were fabricated by sputter-deposition on polycrystalline Si membranes embedded in a Si substrate enabling X-ray transmission measurements [26]. Our multilayer composition informs that the saturation magnetization is [26]. We have also measured damping to be α = 0.015 and a g-factor of g = 2.18. Before the beamtime, MFM measurements showed the presence of out-of-plane labyrinth domains with an average size of 110 nm at remanence. EuXFEL [27]. The XFEL generates linearly polarized Xray pulses with 25 fs duration. In this experiment, we use a pulse-to-pulse separation of 18 µs, with 26 pulses per train, with the 468 µs trains having a repetition rate of 10 Hz. This eectively results in 260 pulses per second impinging on the samples. While the EuXFEL is capable of a much higher pulse frequency with even more pulses per train, longer pulse trains with shorter pulseto-pulse separation resulted in readily apparent sample damage. Even with these conditions, the sample is at an elevated temperature during the time-resolved measurements (see Appendix A). The incoming X-ray intensity (I 0 ) is monitored shot-by-shot using a X-ray gas monitor (XGM) [28].

II. EXPERIMENTAL METHODS

A. X-ray scattering measurements

M s = 616 kA/m
The pump laser is synchronized with the FEL at half of the X-ray probe frequency to collect both the scattering data from ultrafast dynamics (pumped) and in quasi-equilibrium (unpumped) within the same measurement run. The X-ray scattering is collected on the DSSC 2D detector, able to match the repetition rate of the XFEL [29]. The DSSC records data at twice the Xray pulse rate in order to collect so-called dark data frames in between pulses for the best background correction [30]. The sample to detector distance is xed at 3 m.

We note that the white regions in the scattering correspond to non-active or faulty areas of the DSSC detector.

The samples are probed resonantly with linearly polarized X-rays tuned to the L 3 absorption edge of Ni (852 eV , 1.45 nm). The samples are pumped with a YAG-whitelight-seeded laser with central wavelength λ = 800 nm and 35 fs pulse duration [31]. The X-ray spot size is estimated to be 20×20 µm 2 , and the pump laser has a Gaussian prole with 40×40 µm 2 at full-width half-maximum.

Magnetic domains act as a grating for X-rays at a resonant magnetic edge so that their scattering mathematically represents the two-dimensional Fourier transform of the grating [32]. By means of dichroic absorption and scattering [33,[START_REF] Stöhr | Magnetism: From Fundamentals to Nanoscale Dynamics[END_REF], due to dierent cross-section of oppositely-aligned up/down domains, small-angle X-ray scattering (SAXS) is collected on the DSSC 2D detector.

For further information on resonant magnetic scattering we refer the readers to references [20,33,[START_REF] Kortright | [END_REF]36] Representative examples of static scattering patterns are shown in Figure 1 We note that we do not use an external in-plane eld to induce a stripe-like domain pattern [17,19,32], but instead nd this preferential orientation in sample areas that are subject to strain. As further elaborated upon in Appendix B, this strain is the result of irreversible and slow plastic deformation due to sample heating that occurred during the course of the experiment.

B. Data analysis method

Traditionally, time-dependent 2D scattering has been analyzed within a 1D representation achieved by azimuthal integration, either over the isotropic ring [16,20,25,37] or the anisotropic lobes [17,19]. However, this simple analysis hides valuable spatial information obtained by scattering experiments. In particular, our scattering data exhibits both mixed isotropic-anisotropic scattering and has a signicant amount of data lost to the non-active regions of the detector. For example, azimuthal averaging of the data results in the apparent development of a bimodal distribution due to missing pixels, discussed in Appendix C. Such artifacts can result in an erroneous quantication of the far-from-equilibrium physics at play. For these reasons, we developed a 2D tting procedure that accurately models the varying degree of domain symmetry in our samples and allows us to reconstruct the full scattering pattern, therefore providing an accurate picture of the far-from-equilibrium magnetization dynamics in our samples.

Motivated by the MFM images, we utilize a tting function for the scattered intensity given by I(q, φ) = I 0 + I iso (q) + I aniso (q, φ),

where q is the wavevector, φ is the azimuthal angle at which the anisotropic lobes are oriented, I 0 is a uniform background, I iso (q) is the isotropic component that is a function of the wavevector, and I aniso (q, φ) is the anisotropic component which is a function of both the wavevector and the azimuthal angle. The scattered intensities are proportional to the modulus square of the magnetic scattering amplitude resulting from the magnetic texture. Because intensities are photon counts, the square root of the intensity is a measure of the relative contrast due to the amplitude of the magnetization modulus within the magnetic texture.

We mathematically assume that both scattering patterns arise from an intermixed spatial pattern, resulting in no coherent interference contribution to the scattering, as elaborated in the Appendix D. This implies that the domain patterns exhibit a highly varying spatial periodicity that precludes any possibility of long-range phase coherence of the resultant scattering, in agreement with the MFM images of Figure 1(e)-(g).

Based on previous works [7,20], we dene the isotropic scattering intensity as where A 0 is the amplitude, q 0 is the radius of the isotropic peak position, and Γ 0 is the linewidth.

I iso (q) = A 0 (q -q 0 ) 2 /Γ 2 0 + 1 2 (2)
The anisotropic scattering can be phenomenologically represented by a Fourier series

A n sin 2 (n(φ -θ)). The intensity is thus dened to second order as

I aniso (q, φ) = |A 1 | sin 2 (φ -θ) + A 2 sin 2 (2(φ -θ)) (q -q 1 ) 2 /Γ 2 1 + 1 2 (3)
This functional form considers that the anisotropic scattering is aligned at an angle θ, has an anisotropic peak position q 1 , and a linewidth Γ 1 . The two amplitude coefcients correspond to the dominant scattering amplitude A 1 and a deviation from a sinusoidal azimuthal prole, A 2 . We nd that A 2 is typically two orders of magnitude smaller than A 1 .

Fitting a 2D function with eight tting parameters requires a detailed and robust protocol. We utilize the following procedure: 1) the center of the scattering intensity q = 0 is determined at the beginning of each run. Indeed, even a one-pixel oset of the center can generate artefacts such as asymmetries in the radial peak position. For this reason, the correct determination of q = 0 is critical.

2)

The anisotropic component alignment θ is determined.

3) The tting parameters are included sequentially with the goal of determining a good initial guess in an automated way. 4) Fitting of Eq. ( 1) is performed with all eight parameters adjustable. By use of this procedure, we obtain high-delity ts with small and spatially randomized residuals, discussed in Appendix C. From the tted parameters, we focus on the magnetization quench and the radial peak position of each component. The average magnetization for each symmetry component is proportional to the amplitudes A 0 and A 1 , insofar as A 2 ≪ A 1 for the anisotropic component. The peak position is directly obtained from the tted parameters q 0 and q 1 . The parameters extracted from all available data sets are presented in the Table F.

An example of a 2D scattering t using Eq. ( 1) is shown in Fig.

f quench = (1 + B -A) + Ae -(t-t0)/τm -Be -(t-t0)/τ R (5)
where t 0 is the time zero of the dynamics, τ m is the quench constant, τ R is the recovery constant, A and B are dimensionless constants related to the quench and recovery of the magnetization. A more complete form of this equation was derived in Ref. [38]. Here, we use a simplied form that disregards the longer algebraic recovery constant that could not be tted accurately within the 20 ps traces. From this equation we can obtain the quench time t min and the maximum quench of the magnetization ∆M/M as derived parameters from the tted variables. Further details on how to obtain such variables and their errors are given in the Appendix F. The quantities extracted from the ts are summarized in Table I.

III. ULTRAFAST MAGNETIZATION DYNAMICS

The normalized modulus of the magnetic diraction amplitude at t < 0 is shown in Fig. 3 scattering patterns and (b) for anisotropic scattering patterns. We observe a similar amount of quenching for both types of scattering using the same uence. In particular, we determine 38.5 ± 0.8 % and 37.6 ± 0.4 % for the isotropic and anisotropic components, respectively.

The solid gray curves in Fig. 3(a) correspond to the unpumped data and serve to conrm the negligible evolution of the magnetization in a quasi-equilibrium state.

It is worth pointing out that the samples do not return to thermal equilibrium at the repetition rate of the experiment, as further discussed in the Appendix A. We also studied the uence dependence and relative position We distinguish between full overlap (FO) and partial overlap (PO) between pump and probe. We report the extracted domain size π/qi (nm) from the pre-pumped signal, maximum shift in radial peak position (%), maximum demagnetization (%), and demagnetization recovery time (ps) for all measurements.

between the pump and the probe, labelled as FO (full overlap) and PO (partial overlap, shifted by 50 µm) in Table I. We discuss this dependence in section IV. The eect of pump uence was investigated and found to be in agreement with previous works [14,16,17,3941]. These results are shown in the Appendix G and further validate the 2D tting approach.

In addition to the quench, we also detect an ultrafast contraction of the ring radius of 0.84 ± 0.06 % for the isotropic component, shown in Fig. 3(c). On the contrary, we do not observe conclusive evidence of a peak position shift for the anisotropic scattering, as shown in Fig. 3(d). While it can be argued that a small shift of 0.5 % is apparent, we cannot conclusively quantify this shift within our signal-to-noise ratio. In addition, other data sets do not exhibit any apparent shift, as shown in Table . I.

These observations are consistent with previous works where a shift in the radial wavevector q was only observed when measuring isotropic scattering from labyrinth domains [16,20,25] and no shift reported for anisotropic scattering from stripe domains [17,19]. We note that the detected shifts (1-4 %) are smaller than those observed in earlier works (5-6 %) [16,20]. This dierence in our data relative to earlier reports could be a consequence of the lower uences and/or the elevated temperature of our samples due to the large eective repetition rate of the instrument (see Appendix A). An elevated temperature before time zero reduces the local magnetic moment such that the net electron-spin scattering that drives the magnetization dynamics is weaker. Regardless, we conclude based on our analysis of the isotropic and anisotropic scattering that the dierence in the shifts in q for stripe and labyrinth domains is not related to varying sample properties or experimental details since our measurements are obtained from the same sample and experiment.

To investigate this further, we turn to the mixed scattering patterns where both isotropic and anisotropic contributions exhibit a similar photon count. A representative example is shown in Fig. 4 for the time-dependent magnetization, (a) and (b), and peak shift, (c) and (d) of the isotropic (blue) and anisotropic (orange) components, respectively. As in Fig. 3, the solid gray curves represent the unpumped data. We note that the signalto-noise ratio of the mixed states is lower than that of pure states, but the main features can be recovered from the 2D ts with good accuracy. There are two main observations from Fig. 4. First, the quench of both symmetry components is similar, estimated to be 38.2 ± 1.4 % and 34.9 ± 1.8% for the isotropic and anisotropic components, respectively. Second, there is a distinct shift in the peak position of the isotropic component while no shift can be conclusively detected from the anisotropic component. The same trend is observed in all the mixed scattering patterns measured and analyzed, as summarized in Table I. Notably, the ultrafast shift of the isotropic component is consistently larger than any shift in the anisotropic component.

Our observation of a peak shift only, or at least predominantly, for the isotropic ring suggests either a systematic eect arising from modication of x-ray scattering or a symmetry-dependent eect arising from the magnetic order itself. Modication of x-ray scattering, whereby ultrafast domain-wall broadening could lead to an apparent shift in the scattering ring was proposed in Ref. [16]. This approach is analogous to invoking a non-equilibrium Debye-Waller factor that mimics thermal uctuations in crystal lattices. This approach was largely disproved in Ref. [20] based on the simultaneous quantication of domain-wall broadening and peak shifts.

While our current experiments have limited dynamic range compared to previous works [19,20], our twodimensional experimental results based on the rst order scattering can further demonstrate that invoking a Debye-Waller-like factor in this context is not appropriate. Under the assumption that such a factor originates from a stochastic eect, it stands to reason that it must be isotropic in q, with a characteristic form e -|q| 2 /2σ 2 [START_REF] Rühle | Physical Metallurgy (Fourth Edition)[END_REF]. In Ref. [20], a relationship between σ(t) and the measured linewidth Γ and equilibrium domain size q was obtained, and can be algebraically manipulated as

∆q q = Γ 2 2σ 2 + Γ 2 (6)

Isotropic component

Anisotropic component From this formula, we nd two relevant limits.

If σ is small, the attenuation results in ∆q/q ≈ 1 for both isotropic and anisotropic components. This eectively implies a case in which the scattering is fully attenuated and instead dominated by thermal vibrations. In the limit of σ > Γ, which represents the experimental condition by which thermal vibrations are shorter than the domain-pattern correlation length, then ∆q/q ∝ Γ 2 .

Consequently, the ratio between the anisotropic and isotropic shifts should be proportional to Γ 2 1 /Γ 2 0 . From our data analysis (see Tables IV andV), we nd that the squared linewidth ratio is ≈ 0.25 in all cases. This relative shift was not observed even though our experimental data provided such accuracy. For example, the mixed state pumped at 15 mJ/cm 2 in partial overlap (run 6 in Table I) was found to exhibit a ∆q/q of 1.88 ± 0.07

IV. DISCUSSION

The main observation of our work is the unequivocal shift in the isotropic component of the scattering, even when simultaneously observed with an anisotropic component. We substantiate this statement by statistical analysis of the tted parameters presented in Table I. We use weighted averages and weighted standard deviations to minimize the impact of data points determined with lower accuracy. An average peak shift of 1.09±0.51 % for the isotropic component was calculated. We emphasize that this statistically signicant peak shift is a general behavior and thus independent of the particular uence conditions, day of the experiment, and regions illuminated. Contrarily, an average peak shift of 0.22 ± 0.22 % was found for the anisitropic component. This means that the peak shift in the anisotropic component is statistically insignicant within the signal-to-noise ratio of the measurement.

Our data analysis procedure also allows us to extract information on the quench time, t min , and recovery time constant, τ R , for each scattering component. The average quench time was determined to be 0.78 ± 0.11 ps and 0.70 ± 0.10 ps for the isotropic and anisotropic components, respectively. This is clear indication that the quench time is independent of the symmetry component. Magnetic quench occurs in textured magnetic materials due to both the increase of the magnon population and domain-wall broadening. Therefore, the similar quench time is in agreement with the notion that ultrafast demagnetization occurs at the atomic scale, driven by the coupling between light and electrons at femtosec-ond timescales. It was found that the recovery time constant τ R scales linearly with quench amplitude ∆M/M for both the isotropic and anisotropic components of the diraction ring. We consider a recovery speed from a weighted linear t of the recovery time constant and the quench amplitude, where the recovery rate is dened as d(∆M/M )/d(τ R ). This allows us to compare the overall dependence of τ R on quenching for both components of diraction when they have dierent quench amplitudes.

The average reciprocal recovery speed were determined to be 5.2 ± 0.8 ps and 4.8 ± 0.3 ps for the isotropic and anisotropic components, respectively. As for the quench time, the recovery constant is statistically indistinguishable for both the cases.

A microscopic mechanism explaining the observed behavior is still lacking. Numerical modeling is challenging because reasonably large magnetic volumes are required to stabilize magnetic domains and are currently only accessible with micromagnetic models. As shown elsewhere, micromagnetic models are not appropriate to model the ultrafast increase in the magnon thermal population, specically for short-wavelength magnons [7], but progress is being made from the point of view of Landau-Lifshitz-Bloch models [START_REF] Raposo | [END_REF]. Theoretical models are also constrained to date to macroscopic averages [44] and superdiusive spin currents [12,45]. Therefore, we can only conjecture based on the experimental data.

The presented results strongly suggests that regions of the sample with domains preferentially parallel to one another are eectively inert to spatial modication. Conversely, regions of the sample with disordered domains can be spatially modied. Considering that the domain sizes are on the order of 80 nm, this eect must be localized within the domain walls which are the only objects in the system capable to exchange information between neighboring domains. We speculate that torque exerted from angular momentum is a possible mechanism acting on the domain-walls in the form of e.g., superdiusive spin currents [12] or magnons with wavelengths under 100 nm. It would be also interesting to consider the possibility of spin-wave dispersive shock waves [46] that have been numerically [4751] and experimentally [52] observed in magnetic materials. While these mechanisms remain to be demonstrated, it would be interesting to consider them in future studies in a far-from-equilibrium regime.

V. CONCLUSION

We reveal the dependence of the ultrafast spin dynamics on the nanoscale conguration of magnetic domains. Our results dispel the apparent inconsistency of previously reported values for the radial peak shift after ultrafast pumping, given that all the presented data were taken with (1) the same sample, (2) was measured in the same manner, and (3) at the same facility. We now clarify that the previous inconsistency in detection of the peak shift is because signicant shift only occurs in samples with labyrinth domain patterns, characterized by an isotropic SAXS scattering ring. Moreover, our results strongly suggest that this behavior is intrinsically related to the magnetization textures and their symmetries. Surprisingly, these distinct dynamics arise from the symmetry of long-range ordered magnetic domains with sizes ranging between 70 and 90 nm, and extending for several microns, both dimensions longer than the typical mean free path of electrons in metallic multilayers [START_REF] Stöhr | Magnetism: From Fundamentals to Nanoscale Dynamics[END_REF].

The mechanism of the observed shift is still debatable.

While a uniform domain expansion can be excluded by impossibly large domain-wall motion speeds [16], it remains plausible that domains spatially rearrange [20] or locally demagnetize at dierent rates. Our results invite further experimental and theoretical research to clarify the impact of symmetries on the transfer of angular momentum between the electronic and spin degrees of freedom for far-from-equilibrium phenomena. In particular, the possible excitation of short-wavelength spin waves from the domain walls could be envisioned as a mechanism to exchange angular momentum [53] between domains and exert torque [12] on domain walls.

Δ" (ps) Δ" (ps) Δ" (ps) Δ" (ps) We observe an oset between the pumped (blue) and unpumped (orange) data before ∆t ≤ 0 attributed to the data not having fully returned to equilibrium between pulses.

15mJ/cm 2 10mJ/cm 2 5mJ/cm 2 20mJ/cm 2 (a) (b) (c) (d)
Our pumped-probe measurements were performed raw image from the DSSC. For example, we consider the scattering data shown in Figure 7(a). Taking q = 0 as the center of the scattering intensity, we azimuthally integrate the left and right modules of the scattering. As a result, we observe in Figure 7(b) that the azimuthal prole of the left and right features do not fully overlap at the same wavenumber. This is not physical because the lobe pattern must be mirror symmetric about the origin.

By including the center of the scattering as a tting parameter, we are able to determine the true q = 0 as a function of pixels in the DSSC detector. The azimuthal prole of the features then overlap perfectly as shown in Fig. 7(c). We nd that the oset is typically within 2 pixels, which is less than 0.5 mm. We keep the center as a oating parameter when tting the time-resolved measurements.

Another important aspect of the data analysis is found in the inactive areas of the DSSC detector, that sometimes cover substantial parts of the diraction pattern as seen in Fig. 8(a). The 2D t extrapolates the intensity in areas with missing pixels and allows us to reconstruct the full scattering, as shown in Fig. 8

(b).

A natural question is whether this reconstruction is accurate and physically meaningful. An eective means of testing the importance of our 2D tting procedure is to compare the azimuthal averages of both the raw data, Since the raw data spans a substantial fraction of the DSSC detector without any active pixels, it is not surprising that the azimuthal averages of the raw data and the full 2D tting function, which analytically accounts the missing pixels, would exhibit very dierent azimuthal averages. If we take the full 2D tting function and mask out the detector area that does not have any active pixels, Fig. 8(c) before performing an azimuthal average (orange dotted curve), we nd that the azimuthal averages of the raw data and the masked 2D t are nearly identical. But we also see that both these azimuthal averages exhibit an articial shoulder at q ≈ 0.045 nm -1 that suggests a bimodal distribution. On the other hand, the anomalous shoulder is no longer present when we azimuthally integrate the tted 2D function for the diraction. In other words, azimuthal integration/averaging of raw SAXS scattering data that spans large portions of a detector without active pixels can easily introduce artifacts that are eliminated when performing a proper t with our continuous 2D tting function.

This shows that the intensity mismatch, asymmetry, and shoulder in the raw data are only due to the inactive areas of the DSSC and indicates that great care must be taken when extracting information from azimuthally averaged or integrated data from a modular DSSC. These artefacts can suggest nonexistent physics in the ultrafast regime.

The quality of the t is determined from the residual between the raw data and the t. In Fig. 9 we show an example scattering pattern, its t, and the residual.

There is a clear nite residue at the location of the ring that indicates that our phenomenological function fails to correctly account for the detailed scattering prole.

However, the residual is on the order of 10 %. In addition, the residual uctuates rapidly, which can be also a consequence of shot noise, proportional to the intensity, and the speckle pattern. We then conclude that our ts are suciently accurate to extract physically meaningful information.

We further validate our results by showing that the 2D t is able to track small changes in the anisotropic peak position q 1 . We generate a mock scattering pattern with added shot noise, shown in Fig. 10(a) and a shift in q 1 of approximately 1%. The shift in q 1 is recovered by the tting procedure within error. The parameters are given in Table II 10(a)). In the initial guess, the anisotropic scattering radius q1 is slightly shifted. This change is recovered by the 2D t as shown in the last column.

Appendix D: Absence of coherent interference in the scattering pattern

We demonstrate that the tting function for the diraction pattern can be approximated by the addition of two intensities without any cross-correlation. For simplicity, we consider a 1D model with regions of spatial width a, each with a phase ϕ and a periodicity q n . The 1D real space function f 1 can be expressed as an innite sum of such regions, given by

f 1 (x) = ∞ n sin(q n x + ϕ n )Π((x + n/2)/a), (D1)
where Π(x/a) is the pulse function of width a.

By use of standard Fourier transform properties, we obtain the Fourier transform F 1 (q) = F{f 1 (x)} as The intensity colorbar is shared between all four panels and can be interpreted as photons/pixel/pulse.

F 1 (q) = a 2
∞ n cos(ϕ n )e iπ/2 + sin(ϕ 0 ) e ina(q+qn)/2 sinc a(q + q n ) 2π

+ cos(ϕ n )e -iπ/2 + sin(ϕ 0 ) e ina(q-qn)/2 sinc a(q -q n ) 2π .

(D2)

Therefore, the intensity is

|F 1 (q)| 2 = a 2 4 ∞ n sinc 2 a(q + q n ) 2π + sinc 2 a(q -q n ) 2π + 2a 2 4 ∞ n,m cos ϕ m -ϕ n - a 2 (nq n -mq m ) sinc a(q + q n ) 2π sinc a(q + q m ) 2π (D3)
The third term in Eq. (D3) represents a cross-term from the overlap of the sinc functions. If we assume that the periods between neighboring regions are similar, q n ≈ q m , then only those regions where n ≈ m will contribute to the spectrum. If this were not the case, then overlap between the sinc functions would be small, and the cross-term would be negligible. It follows that the cross-term is dominantly proportional to cos (ϕ m -ϕ n ).

If the phase between neighboring regions is randomized, then the innite sum over ϕ m -ϕ n will span every phase between 0 and 2π. Therefore the cross-term is zero.

This situation is equivalent to consider that each region has a nite correlation length.

This toy model illustrates that the cross-term can be neglected for the mixed state, where the spatial periodicity is randomized.

Appendix E: Pulse-resolved photon count in the scattering data

The DSSC is a soft X-ray detector with single-photon sensitivity and a frame rate that is able to match the pulse rate of the FEL [29]. The resulting scattering image, e.g., Fig. 11(a), is the average of all frames collected within a run. After multiplying by the gain of the detector, the intensity can be directly related to the average number of photons hitting each pixel per pulse. In our experiment, the gain was set to 0.5 photon/bin. III summarizes the photon count from all the pump-probe measurements.

In section II B we explain how we categorize our data according to Eq. ( 4). Figure 12 shows the distribution of ratios for the collected data. We can clearly distinguish three groups that correspond to the three types of scattering that we observe. Due to the non-zero background, the isotropic scattering has a ratio close to 1

while anisotropic scattering has a ratio close to -0.5. We note that, because the anisotropic photons are more localized in space, fewer photons are required to observe anisotropic scattering, while more photons are needed in order to clearly observe isotropic scattering above the noise.

Appendix F: Fitting the demagnetization constants

The tting procedure for extracting demagnetization values and time constants is explained in detail by Unikandanunni et al.. We use the equation from [38]:

∆M M 0 = ( A 1 τ R -A 2 τ m τ R -τ m e -(t-t0)/τm - τ R (A 1 -A 2 ) τ R -τ m e -(t-t0)/τ R - A 2 (t -t 0 )/τ R2 + 1 ) ⊛ Γ(t) (F1)
where τ m is the quench constant, τ R is the recovery constant, A 1 and A 2 are dimensionless constants related to the quench and recovery amplitudes. We t t 0 , the time zero of the dynamics, in order to account for jitter in the pump arrival. The expression within parentheses is convoluted with a Gaussian Γ(t) to account for the nite duration of the X-ray and IR pulses.

We disregard the second, algebraic recovery time Unikandanunni et al. used in their work since we did not collected data at long enough times to properly t this value. Therefore, we use a simpler expression given by

f quench = 1 + Ae -(t-t0)/τm -Be -(t-t0)/τ R + B -A, (F2)
where the new coecients, A and B, are related to the coecients in Eq. (F1) by

A 1 = A -B τ m τ R , (F3a)
A 2 = A -B. (F3b)
From this equation, we can analytically obtain the quench time by nding the minimum of f quench , given by

t min = t 0 - τ m τ R τ R -τ m ln Bτ m Aτ R . (F4)
It follows that the error δt min can be computed by standard propagation of uncertainty for each tted variable. TABLE III. The average photon count is quantied per pulse and per second. For all the runs the XFEL delivered 26 pulses per train with a train repetition rate of 10 Hz. The average photon count in the full scattering is decomposed into the anisotropic and isotropic components as well as the uniform non-magnetic background.

The partial derivatives are:

∂t min ∂t 0 = δt 0 , (F5a)
∂t min ∂A = τ m τ R τ R -τ m δA A ≈ τ m A δA, (F5b)
∂t min ∂B = τ m τ R τ R -τ m δB B ≈ τ m B δB, (F5c)
∂t min ∂τ m = τ R τ R -τ m 1 + τ R τ R -τ m ln Bτ m Aτ R δτ m ≈ 1 + ln Bτ m Aτ R δτ m , (F5d)
∂t min ∂τ R = τ m τ R -τ m 1 + τ m τ R -τ m ln Bτ m Aτ R δτ R ≈ τ m τ R δτ R . (F5e)
The approximations consider that τ m ≪ τ R , but this leads to an innite error in τ m , so the approximation must be taken with care.

The quench is then f quench evaluated at t min ,

A q = A 1 - Bτ m Aτ R τ R /(τ R -τm) -B 1 - Bτ m Aτ R τm/(τ R -τm) , (F6)
and the error δA q can be obtained from propagation of uncertainty of the following quantities

∂A q ∂A = 1 - Bτ m Aτ R τ R /(τ R -τm) δA ≈ δA, (F7a)
∂A The ratio is computed according to Eq. ( 4). Isotropic scattering has a ratio close to 1 while anisotropic scattering has a ratio close to -0.5. For a ratio between -0.3 and 0.9 we categorize the scattering pattern as mixed. component, even when pumped with a large uence.

q ∂B = 1 - Bτ m Aτ R τm/(τ R -τm) δB, (F7b)
∂A q ∂τ R = 1 τ R (τ R -τ m ) 2 Aτ R Bτ m Aτ R τ R /(τ R -τm) τ R -τ m + τ m ln Bτ m Aτ R -Bτ m Bτ m Aτ R τm/(τ R -τm) τ R -τ m + τ R ln Bτ m Aτ R δτ R , ≈ Bτ m τ 2 R δτ R (F7c) ∂A q ∂τ m = 1 τ m (τ R -τ m ) 2 Bτ m Bτ m Aτ R τm/(τ R -τm) τ R -τ m + τ R ln Bτ m Aτ R -Aτ R Bτ m Aτ R τ R /(τ R -τm) τ R -τ m + τ m ln Bτ m Aτ R δτ m ≈ B τ R δτ m . (F7d)

Pre-pumped Pumped

Run ID Fluence SAXS pattern size (nm) Γ (µm -1 ) ∆q/q (%) ∆M/M (%) tmin (ps) quench speed (ps -1 ) τR (ps) We distinguish between full overlap (FO) and partial overlap (PO) between probe and probe. Extracted domain size π/q (nm) and linewidth Γ (nm -1 ) from the pre-pumped signal, maximum shift in radial peak position (%), maximum demagnetization (%), quench time (ps) and quench speed (ps -1 ) and demagnetization recovery time (ps) for all measurements. We distinguish between full overlap (FO) and partial overlap (PO) between probe and probe. Extracted domain size π/q (nm) and linewidth Γ (nm -1 ) from the pre-pumped signal, maximum shift in radial peak position (%), maximum demagnetization (%), quench time (ps) and quench speed (ps -1 ) and demagnetization recovery time (ps) for all measurements.

Figure 1 (

 1 Figure 1(a) shows the pump-probe schematic of the experimental setup. The experiments are performed at the Soft x-ray Coherent Scattering (SCS) beamline at the

  (b)-(d). This data is collected within the same membrane but with the X-ray beam illuminating dierent areas of it. Each of these patterns illustrate distinct long-range symmetries present in the membrane which were characterized in real space after the pump-probe experiments using magnetic force microscopy (MFM) as shown in panels (e)-(g). The isotropic ring (b) results from labyrinth domains (e) while the anisotropic, lobed pattern (d) results from domains with a similar translation symmetry to stripe domains (g). We also observe a mixed state where both symmetry features are visible in the scattering (c). The corresponding MFM image (f ) indicates that this pattern arises from a varying degree of randomness in the spatial periodicity of the domain structure. In other words, the preferentially labyrinth and striped areas are spatially intermixed rather than clustered.

FIG. 1 .

 1 FIG. 1. (a) Schematic of the experimental pump-probe setup. The sample is excited by IR pulses and probed by linearly polarized X-rays tuned to the L3-edge of Ni. The scattered photons are collected on the DSSC detector. Representative diraction patterns are shown for (b) labyrinth, (c) mixed, and (d) stripe domain patterns. All three patterns are found on the same sample membrane. Translation of the pump/probe beams across the sample allows access to dierent regions of the sample that exhibit the dierent scattering patterns. Corresponding 10 × 10 µm 2 MFM real-space images are shown in (d)-(f), illustrating the varying degree of randomness for each domain pattern.

2 . 4 )FIG. 2 .

 242 FIG. 2. Two dimensional t of the scattering data. (a) The raw experimental data suers from missing intensity in the non-active regions of the DSSC. (b) Two dimensional t with Eq. (1). The t allows to fully separate the (c) isotropic and the (d) anisotropic components of the scattering.

FIG. 3 .

 3 FIG.3. Two-dimensional tting of the full diraction pattern is used to extract the time traces of both the magnetization and the peak position for the isotropic and anisotropic scattering patterns with 10 mJ cm -2 FO pump uence (runs 8 and 9 in TableI, respectively). The normalized magnetization of isotropic and anisotropic scattering patterns is shown in (a) and (b), respectively. The magnetization is shown by the solid colored solid curves and corresponding unpumped data by solid gray curves. In both cases, the amount of quenching is similar. The corresponding time traces for the scattering peak position is shown in (c) and (d) for the isotropic and anisotropic scattering patterns, respectively. by colored solid curves. The gray solid curves corresponds to the unpumped scattering peak position in quasi-equilibrium. Only the isotropic component exhibits a signicant peak shift estimated to 0.84 ± 0.06 %. The peak shift of the anisotropic component, while noticeable by eye in Fig.3d, is not statistically signicant to within error bars. The shaded area in all cases corresponds to the standard deviation of the tted quantities. Because of the high accuracy in the tting for each time instance, the standard deviation of the tted variables is not visible to the eye. Noise is instead dominated by measurement uctuations.

FIG. 4 .

 4 FIG.4. Two-dimensional tting results for the mixed state diraction pattern with 25 mJ cm -2 pump uence (run 3 in TableI). The time traces of both the magnetization and the radial peak position for the isotropic and anisotropic components are extracted from tting the full diraction pattern. In all panels, the gray curves represent the unpumped data. The normalized magnetization of isotropic and anisotropic components is shown in (a) and (b), respectively. We observe a similar amount of quenching in both cases. The scattering peak position is shown in (c) and (d) for the isotropic and anisotropic components, respectively. Only the isotropic component of the mixed SAXS pattern exhibits a signicant shift in the radial peak position. This is similar to what was found for the purely isotropic or purely anisotropic SAXS scattering patterns, where only data with purely isotropic scattering exhibited signicant radial peak shifts. Here we detect a shift of 3.5 ± 2.3 %. The black curve is the ve-point moving average of the data. The shaded area corresponds to the standard deviation of the tted quantities.

  and 0.23 ± 0.05 for the isotropic and anisotropic components, respectively. The prediction based solely on the Debye-Waller factor suggests instead an isotropic shift of 0.72 ± 0.03, well within accuracy. Therefore, the argument of an attenuation factor producing an apparent shift in the scattering fails to explain our observations. Summarizing this section, we studied the timedependent ultrafast response of a magnetic sample with dierent domain patterns with distinct translation symmetry. In cases where the quenching was similar, we observe a shift in the scattering peak position only for the isotropic component for both the preferential labyrinth domain and the mixed state. Our attempt to t the observed peak shift with a Gaussian Debye-Waller factor (DWF), ostensibly capturing spectral attenuation eects due to domain-wall broadening, was not successful. The failure of this model stems from a signicant quantitative mismatch between the observed peak shift and broadening of the diraction ring: If we attribute the ring broadening to a DWF, then the amount of peak shift that we observe is actually ≈ 3 times smaller than expected. Instead, this experimental observation is consistent with a symmetry-dependent eect arising from the magnetic order itself.

FIG. 5 .

 5 FIG. 5. Fitted lobe amplitude for a series of pump-uences (a) 20 mJ cm -2 , (b) 15 mJ cm -2 , (c) 10 mJ cm -2 , (d) 5 mJ cm -2 .We observe an oset between the pumped (blue) and unpumped (orange) data before ∆t ≤ 0 attributed to the data not having fully returned to equilibrium between pulses.

  with 26 x-ray pulses per train at a pulse rate of 56 kHz and 13 IR pulses at half this rate. Within one measurement, we record both a pumped and an unpumped signal separated by 18 µs. While we observe no apparent damage in the unpumped signal we do observe an oset before time-zero in the tted amplitudes of pumped and unpumped signals.

Figure 5 FIG. 6 .FIG. 7 .

 567 Fig. 6(b), while others showed partially oriented domains at dierent angles, see Fig.6(c) and (d). The dimpling of the membrane results from thermallyinduced buckling during the pulse train utilized at the Eu-XFEL. The pulse trains are 0.5 ms in duration, with 26 pulses in each train. As such, the time interval between each pulse in the pulse train is insucient to achieve thermal relaxation. Estimating an optical absorption coecient of 10 %, we obtain a temperature

Fig. 8 (

 8 Fig.8(a), and the full 2D tting function, Fig.8(b). The comparison is shown in Fig.8(d), where the averaged raw data is represented by the solid green curve and the averaged full t is represented by the dotted blue curve.

  . The residual Fig. 10(b) shows the dierence between the mock data and the t. There is a clear residue with uctuating values due to the shot noise of the mock data.

FIG. 8 .FIG. 9 .

 89 FIG. 8. (a) The raw scattering collected on the DSSC. (b) The result of the 2D t. (c) 2D t masked with the inactive areas of the detector. (d) Azimuthally integrated scattering of the raw data (solid green curve), 2D t (dashed blue curve) and masked 2D t (dotted orange curve). The 2D t only matches the raw data once it has been masked with the DSSC inactive areas.

FIG. 10 .FIG. 11 .

 1011 FIG. 10. (a) A mock scattering pattern. (b) The resulting residual of the 2D t. Note that the intensity scale of the residual is one order of magnitude smaller than that of the raw scattering pattern.

The 2D t allows

  photons to the anisotropic contribution. The background contributes with an average of 0.5 photon/pulse. With 26 pulses per train at a train repetition rate of 10 Hz, we get 4550 photons per second in the full scattering with 2470 photons in the isotropic component, 1950 photons in the anisotropic component, and 130 photons in the background. Table III summarizes the photon count from all

  FIG.13. Two-dimensional tting of the full diraction pattern is used to extract the time traces of both the magnetization and the peak position for the isotropic and anisotropic scattering patterns. In all panels, the gray curves represent the unpumped data. The normalized magnetization of isotropic and anisotropic scattering patterns is shown in (a) and (b), respectively. At a uence of 10 mJ cm -2 , the magnetization is shown by the solid curves, also shown in the main text. We observe a similar amount of quenching in both cases. For the isotropic scattering, the eect of a spatial shift between the pump and probe spots exhibits a weaker amount of quenching due to a weaker optical pumping, shown by the light-colored curve named partial overlap (PO). For the anisotropic component, a higher uence of 15 mJ cm -2 results in a higher quench, shown by the light-colored curve. The corresponding temporal evolution of the scattering peak position is shown in (c) and (d) for the isotropic and anisotropic scattering patterns, respectively. Similar to the results shown in the main text, only the isotropic component exhibits a shift. The shaded areas in all cases correspond to the standard deviation of the tted parameters.

  ± 2.3 38.2 ± 1.4 0.970 ± 0.049 2.05 ± 0.17 74.06 ± 0.14 0.5 ± 0.5 34.9 ± 1.8 0.836 ± 0.049 1.68 ± 0.15 4 15 mJ cm -2 FO (day 4) mixed 71.86 ± 0.28 2.7 ± 2.7 21.2 ± 1.0 0.738 ± 0.012 1.73 ± 0.18 74.98 ± 0.

											Anisotropic component
				Pre-pumped		Pumped		Pre-pumped		Pumped
	Run ID	Fluence	SAXS pattern size (nm) ∆q0/q0 (%) ∆M/M (%)	tmin (ps)	τR (ps)	size (nm) ∆q1/q1 (%) ∆M/M (%)	tmin (ps)	τR (ps)
	1	20 mJ cm -2 FO (day 3)	mixed	78.30 ± 0.11 1.0 ± 0.4 43.5 ± 0.9 0.714 ± 0.033 3.16 ± 0.27 74.55 ± 0.04	0	35.4 ± 1.0 0.592 ± 0.003 2.25 ± 0.14
	2	15 mJ cm -2 FO (day 3)	mixed	77.27 ± 0.14 1.5 ± 0.6 29.3 ± 1.0 0.660 ± 0.039 2.80 ± 0.23 73.58 ± 0.04	0	20.9 ± 1.4 0.549 ± 0.043 1.52 ± 0.14
	3	25 mJ cm -2 FO (day 4)	mixed	71.96 ± 0.34 3.5 14	0	18.0 ± 1.8 0.679 ± 0.021 0.99 ± 0.15
	5	15 mJ cm -2 FO (day 5) anisotropic	-	-	-	-	-	73.08 ± 0.01	0	56.7 ± 0.7 0.58 ± 0.02 2.86 ± 0.08
	6	15 mJ cm -2 PO (day 5)	mixed	77.56 ± 0.04 1.88 ± 0.07 21.9 ± 0.7 0.610 ± 0.02 1.29 ± 0.07 74.16 ± 0.03 0.23 ± 0.05 24.1 ± 0.2 0.61 ± 0.02 1.04 ± 0.05
	7	10 mJ cm -2 PO (day 5)	isotropic	89.38 ± 0.02 0.70 ± 0.05 12.6 ± 0.5 0.840 ± 0.02 0.87 ± 0.05	-	-	-	-	-
	8	10 mJ cm -2 FO (day 5)	isotropic	76.66 ± 0.01 0.84 ± 0.06 38.5 ± 0.8 0.964 ± 0.02 1.57 ± 0.03	-	-	-	-	-
	9	10 mJ cm -2 FO (day 5) anisotropic	-	-	-	-	-	72.3 ± 0.01	0	37.6 ± 0.4 0.813 ± 0.013 1.65 ± 0.02

TABLE I .

 I Extracted parameters from the 2D and temporal double-exponential ts. Measurements are listed by pump uence and scattering pattern.

TABLE II .

 II Parameters of the phenomenological 2D t function used to generate the mock data before adding shot noise (Fig.

Table IV and

 IV Table V show the extracted tting pa-rameters discussed in our work for all measurements per-

	Isotropic photons -anisotropic photons	total photons	Isotropic Mixed Anisotropic

FIG. 12. Distribution of data according to categorization.

  1 20 mJ cm -2 FO (day 3) mixed 78.30 ± 0.11 1.482 ± 0.012 1.0 ± 0.4 43.5 ± 0.9 0.714 ± 0.033 60.9 ± 3.1 3.16 ± 0.27 2 15 mJ cm -2 FO (day 3) mixed 77.27 ± 0.14 1.679 ± 0.017 1.5 ± 0.6 29.3 ± 1.0 0.660 ± 0.039 44.4 ± 3.0 2.80 ± 0.23 3 25 mJ cm -2 FO (day 4) mixed 71.96 ± 0.34 2.572 ± 0.055 3.5 ± 2.3 38.2 ± 1.4 0.970 ± 0.049 39.4 ± 2.5 2.05 ± 0.17 4 15 mJ cm -2 FO (day 4) mixed 71.86 ± 0.28 2.534 ± 0.045 2.7 ± 2.7 21.2 ± 1.0 0.738 ± 0.012 28.7 ± 1.4 1.73 ± 0.18 6 15 mJ cm -2 PO (day 5) mixed 77.56 ± 0.04 1.086 ± 0.004 1.88 ± 0.07 21.9 ± 0.7 0.610 ± 0.02 35.8 ±1.8 1.29 ± 0.07 7 10 mJ cm -2 PO (day 5) isotropic 89.38 ± 0.02 1.249 ± 0.002 0.70 ± 0.05 12.6 ± 0.5 0.840 ± 0.02 14.9 ± 0.7 0.87 ± 0.05 8 10 mJ cm -2 FO (day 5) isotropic 76.66 ± 0.01 0.953 ± 0.002 0.84 ± 0.06 38.5 ± 0.8 0.964 ± 0.02 39.9 ± 1.2 1.57 ± 0.03

TABLE IV .

 IV Measurements are listed by pump uence and scattering pattern.

  FO (day 5) anisotropic 73.08 ± 0.01 0.729 ± 0.001 0 56.7 ± 0.7 0.58 ± 0.02 96.7 ± 3.7 2.86 ± 0.08 6 15 mJ cm -2 PO (day 5) mixed 74.16 ± 0.03 0.671 ± 0.002 0.23 ± 0.05 24.1 ± 0.2 0.61 ± 0.02 39.6 ± 1.5 1.04 ± 0.05 9 10 mJ cm -2 FO (day 5) anisotropic 72.3 ± 0.01 0.646 ± 0.001 0 37.6 ± 0.4 0.813 ± 0.013 46.2 ± 0.9 1.65 ± 0.02

							Anisotropic component
				Pre-pumped		Pumped
	Run ID	Fluence	SAXS pattern size (nm)	Γ (µm -1 )	∆q/q (%) ∆M/M (%) tmin (ps) quench speed (ps -1 ) τR (ps)
	1	20 mJ cm -2 FO (day 3)	mixed	74.55 ± 0.04 0.711 ± 0.004	0	35.4 ± 1.0 0.549 ± 0.043	59.8 ± 3.5	2.25 ± 0.14
	2	15 mJ cm -2 FO (day 3)	mixed	73.58 ± 0.04 0.715 ± 0.005	0	20.9 ± 1.4 0.549 ± 0.043	38.1 ± 3.9	1.52 ± 0.14
	3	25 mJ cm -2 FO (day 4)	mixed	74.06 ± 0.14 0.808 ± 0.016 0.5 ± 0.5 34.9 ± 1.8 0.836 ± 0.049	41.7 ± 3.3	1.68 ± 0.15
	4	15 mJ cm -2 FO (day 4)	mixed	74.98 ± 0.14 0.794 ± 0.015	0	18.0 ± 1.8 0.679 ± 0.021	26.5 ± 2.8	0.99 ± 0.15
	5	15 mJ cm -2

TABLE V .

 V Measurements are listed by pump uence and scattering pattern.

ACKNOWLEDGMENTS

The authors acknowledge the European XFEL in Schenefeld, Germany, for provision of X-ray free-electron laser beamtime at Scientic Instrument SCS and thank the instrument group and facility sta for their assistance. The authors thank Andrea Castoldi (DSSC consortium) for contributing to the generation of DSSC gain les and Carsten Deiter for the confocal microscopy images. R.J., M.M. and R.K. acknowledge support from AFOSR Grant. No. FA9550-19-1-0019. D.T., and H.A.D. acknowledge support from the Swedish Research

The EuXFEL can achieve MHz repetition rates with the SCS beamline being able to deliver up to 150 x-ray pulses per train. However we observed that higher repetition rate and shorter pulse-to-pulse separation lead to visible damage in our samples.

Appendix C: 2D tting procedures and quality estimation

We highlight the importance of tting the center of the scattering data instead of relying on the center of the