Robert Pełka
email: robert.pelka@ifj.edu.pl

Olaf Stefańczyk
email: olaf.stefanczyk@uj.edu.pl

Anna Małgorzata Majcher-Fitas

Corine Mathonière

Barbara Sieklucka
email: barbara.sieklucka@uj.edu.pl

Anna Małgorzata Majcher
email: anna.majcher@uj.edu.pl

Towards rationalizing photoswitchable behavior of Cu II 2 Mo IV cyanido-bridged molecule

Keywords: photomagnetism, octacyanidometallates, susceptibility, magnetization, relaxation

Cu II (enpnen)] 2 [Mo IV (CN) 8 ]•7H 2 O (enpnen = N,N ′ -bis(2-aminoethyl)-1,3 -propanediamine) molecular cluster compound (1) was subject to a series of irradiations with the light of 405 nm. On irradiation isothermal magnetization at 1.8 and 5 K in the field range 0 -70 kOe as well as magnetic susceptibility in the temperature range of 2 -300 K were subsequently detected. Both types of magnetic signals were next analyzed assuming that the irradiation triggers two independent processes: the metal to metal charge transfer (MMCT) leading to a state with the Arrhenius-type relaxation and the spin crossover (SC) transition ending in a state whose relaxation displays a threshold behavior. The first mechanism leads to an electron from the spinless Mo(IV) configuration being transferred to one of the Cu(II) ions transforming the trimer into the state Cu(II)-N-C-Mo(V)-C-N-Cu(I), with spin 1/2 on the Mo(V) ion and the spinless Cu(I) ion. The other mechanism gives rise to an excited paramagnetic Mo(IV) * linked to two paramagnetic Cu(II) centers with a possible superexchange interaction.

The spin of the excited Mo(IV) * species is equal to 1 and associated to a disruption of

Introduction

Spins and magnetization of molecular materials can be uniquely controlled by photons, which is the feature absent for conventional magnets [START_REF] Sato | Dynamic molecular crystals with switchable physical properties[END_REF][START_REF] Muñoz | Thermo-, piezo-, photo-and chemo-switchable spin crossover iron(II)-metallocyanate based coordination polymers[END_REF][START_REF] Bleuzen | Photomagnetism in clusters and extended molecule-based magnets[END_REF][START_REF] Sieklucka | Molecular Magnetic Materials: Concepts and Applications[END_REF][START_REF] Ohkoshi | Lightinduced spin-crossover magnet[END_REF][START_REF] Funck | Light-induced excited spin state trapping and charge transfer in trigonal bipyramidal cyanide-bridged complexes[END_REF]. The spin state of a molecular magnet can be changed either through photo-induced electron transfer (metal-to-metal charge-transfer MMCT process), where the redox states of two different metal centers are modulated [START_REF] Aguilà | Switchable Fe/Co Prussian blue networks and molecular analogues[END_REF][START_REF] Zhang | Reversible thermally and photoinduced electron transfer in a cyano-bridged {Fe 2 Co 2 } square complex[END_REF][START_REF] Dei | Photomagnetic effects in polycyanometallate compounds: an intriguing future chemically based technology?[END_REF][START_REF] Maxim | New photomagnetic cyanidobridged Cu II -Mo IV oligonuclear complexes: slight modification of the blocking ligands induces different structures[END_REF], or through the light-induced excited spin state trapping (LIESST) effect (photo induced transformation from low spin LS to high spin HS state) [START_REF] Guionneau | A reversible metal-ligand bond break associated to a spin-crossover[END_REF][START_REF] Létard | Photomagnetism of iron(II) spin crossover complexes-the T(LIESST) approach[END_REF]. The studied cluster compound [Cu II (enpnen)] 2 [Mo IV (CN) 8 ]

•7H 2 O reveals both mechanisms, which was reported in detail in [START_REF] Stefańczyk | Irradiation temperature dependence of the photomagnetic mechanisms in a cyanidobridged Cu(II) 2 Mo(IV) trinuclear molecule[END_REF]. In the latter work a model was put forward which enabled the calculation of the temperature dependence of the photo-induced susceptibility and the field-dependence of the isothermal magnetization detected just upon irradiation. The present work aims basically at repeating this approach. However, two crucial modifications are introduced. Firstly, unlike in [START_REF] Stefańczyk | Irradiation temperature dependence of the photomagnetic mechanisms in a cyanidobridged Cu(II) 2 Mo(IV) trinuclear molecule[END_REF],

where only the susceptibility data were taken into account in the fitting procedure, the full set of magnetic data (both susceptibility and magnetization) are incorporated into the test function to be minimized. Secondly, here the magnetic properties of the compound are calculated assuming separate spectroscopic factors for the MMCT and SC excited states of the Mo ion. This feature was absent from the previous model, where a single averaged spectroscopic factor was used to describe the field-dependence of both excited states of the cluster compound. In addition, while this approach to rationalizing the photomagnetic properties of a molecular compound is relatively novel (it has been sketched for the first time in [START_REF] Korzeniak | Photoswitchable Cu II 4 Mo IV and Cu II 2 Mo IV cyanidobridged molecules[END_REF], its crucial details were hidden to a large extent in the Supporting Information file of [START_REF] Stefańczyk | Irradiation temperature dependence of the photomagnetic mechanisms in a cyanidobridged Cu(II) 2 Mo(IV) trinuclear molecule[END_REF] and might therefore be easily overlooked by a less diligent reader. We thus thought that a complete presentation in a dedicated article may come in right and useful.

Experimental

A preliminary step in the investigations of the magnetic properties of compound 1 was the measurement of the isothermal magnetization and susceptibility of a bulk sample. The mass of 10.291 mg of the powder sample of 1 was mounted in a plastic draw in the Quantum Design SQUID probe. The magnetization was detected at temperatures 1.8, 3, 5, 8 K in the field range of 0-70 kOe and at 100 K in the field range of 0-40 kOe.

The susceptibility was measured using the dc field of 10 kOe in the temperature range 1.8-300 K [START_REF] Stefańczyk | Irradiation temperature dependence of the photomagnetic mechanisms in a cyanidobridged Cu(II) 2 Mo(IV) trinuclear molecule[END_REF].

Solid-state UV-vis-NIR absorption spectrum of 1 diluted in BaSO 4 was recorded with a PerkinElmer Lambda 35 UV/vis spectrophotometer equipped with a PerkinElmer Labsphere RSAPE-20 Reflectance Spectroscopy Accessory and converted with the Kubelka-Munk function. The deconvolution of the solid-state UV-vis-NIR absorption spectrum was performed by means of the Origin 8.0 software. Fitting peak functions to the experimental solid-state UV-vis-NIR absorption spectrum delivered six Gaussian components with maxima at 250, 309, 381, 432, 568, and 803 nm, see Fig. 1. Three bands at 250, 309, and 381 nm are assigned to the MLCT transition and two LF transitions of the [Mo(CN) 8 ] 4-anion, respectively [START_REF] Bleuzen | Photomagnetism in clusters and extended molecule-based magnets[END_REF]. Next two bands at 432 and 568 nm are attributed to the MMCT optical transition from the two highest valence bands of molybdenum(IV) to the two lowest conduction bands of copper(II) centers [START_REF] Umeta | Synthesis of the singlecrystalline form and first-principles calculations of photomagnetic copper(II) octacyanidomolybdate(IV)[END_REF]. The last band at 803 nm is related to the d-d transition of [Cu II (enpnen)] 2+ entities with vacant octahedral/square pyramidal geometry [START_REF] Maxim | New photomagnetic cyanidobridged Cu II -Mo IV oligonuclear complexes: slight modification of the blocking ligands induces different structures[END_REF][START_REF] Korzeniak | Photoswitchable Cu II 4 Mo IV and Cu II 2 Mo IV cyanidobridged molecules[END_REF][START_REF] Stefańczyk | Photo-induced magnetic properties of the [Cu II (bapa)[END_REF][START_REF] Zhang | Synthesis, characterization, and photoresponsive properties of a series of Mo(IV)-Cu(II) complexes[END_REF][START_REF] Zhang | A one-dimensional homochiral Mo(IV)-Cu(II) coordination polymer: spontaneous resolution and photoresponsive properties[END_REF][START_REF] Mathonière | Reversible photomagnetic properties of the molecular compound [{Cu II (bipy) 2 } 2 {Mo IV (CN) 8 }]•9H 2[END_REF][START_REF] Herrera | Math-310 onière, Reversible photoinduced magnetic properties in the heptanuclear complex [Mo IV (CN) 2 (CN-CuL) 6 ] 8+ : A photomagnetic high-spin molecule[END_REF][START_REF] Lever | Inorganic Electronic Spectroscopy[END_REF]. The 405 nm excitation line for photomagnetic studies was selected based on the above interpretation of the reflectance spectrum. Further details of its analysis and discussion may be found in [START_REF] Stefańczyk | Irradiation temperature dependence of the photomagnetic mechanisms in a cyanidobridged Cu(II) 2 Mo(IV) trinuclear molecule[END_REF].

The main step of investigations consisted in irradiation of the sample with the visible light. The photomagnetic studies were performed on a thin film of the powder sample of 1 (ca. 2 mg) spread on a double-sided adhesive Scotch tape, blocked tightly between two transparent polypropylene films and mounted in the Quantum De- sign SQUID probe equipped with an optical fiber entry enabling the transmission of laser light of 405 nm line into the sample space. The laser power was adjusted to 5 mW•cm -2 , which ensures the perfect balance between a very efficient generation of the photomagnetic effect in addition to a sufficient depth of light penetration into the thin film and a minimized thermalization effect that can lead to photodegradation of the sample during a very long measurement. The first measurement of the sample was carried out prior to irradiation in two modes: for isothermal magnetization at 1.8 and 5 K in the field range of 0-70 kOe (denoted by BFI in Fig. 3) and for susceptibility in the applied field of 10 kOe within temperature range of 1.8-300 K in the heating direction (denoted by BFI in Fig. 2). Next followed a series of subsequent runs in which the sample was first cooled down to 10 K (run A), 50 K (run B), 100 K (run C), 150 K (run D), and 200 K (run E) in the liquid helium bath and stabilized at these temperatures.

Then the sample was irradiated with the purple light of 405 nm for as long as 33.38 h (run A), 24.50 h (run B), 24.56 h (run C), 24.43 h (run D), and 17.94 h (run E). During irradiation the χT value was steadily increasing attaining the values of 1.133 emu K mol -1 (run A), 1.003 emu K mol -1 (run B), 0.905 emu K mol -1 (run C), 0.892 emu K mol -1 (run D), and 0.777 emu K mol -1 (run E) just before switching off the light source. The relative increase in the signal between the initial state before irradiation and the final state after irradiation amounts to 50.9 % (run A), 30.6 % (run B), 17.7 % (run C), 16.9 % (run D), and 1.70 % (run E). Just after stopping the irradiation the temperature of the sample was lowered down to 1.8 K and first the isothermal magnetization at 1.8 K and 5 K was measured in the field range of 0-70 kOe (increasing field mode) and immediately thereafter the field was fixed at the value of 10 kOe and the evolution of χT was detected in the temperature range of 2-300 K. The temperature change during the susceptibility measurement mode was checked to be perfectly linear with a steady rate of 0.4 K min -1 . After each run the sample was cooled down to the initial temperature of the subsequent run. On finishing the final run with irradiation the sample was cooled down to liquid 4 He, the magnetization was measured at 1.8 and 5 K, and finally the susceptibility was detected in the dc field of 10 kOe in the temperature range of 2-300 K (AFI& H in Fig. 2 and3).

The crystallinity of the sample after a few hours of exposure was not checked, because in the photomagnetic experiment the minimal amount of sample was used to obtain a homogeneous thin film. In addition, the sample was blocked between the polymer layers preventing the measurement of low-intensity diffraction peaks and leading to the observation of an amorphous peak. However, after several days of measurement inside the magnetometer we did not notice any disturbing changes in the appearance of the sample or in the UV-vis measurements for the thin film before and after the photomagnetic measurements. Similarly, in the IR measurements of the scratched sample we did not notice any changes apart from a slight contamination with the Scotch tape adhesive. 0 100 200 300 0.7 0.8 0.9

1.0 1.1 1.2 BFI A B C D E AFI&H FIT χT [cm 3 K mol -1 ] T [K]

Theory

To get a more quantitative insight into the details of possible interactions in the unexcited state model "0" is assumed, where the intermolecular interaction between the trimer entities Cu 2 Mo is treated within the molecular field theory. The corresponding Hamiltonian accounts only for the Zeeman coupling of the spins of the Cu(II) ions with the applied magnetic field (it is standard practice to use the cgs-emu system of units for molecular magnetism, cf. [START_REF] Kahn | Molecular Magnetism[END_REF])

Ĥ0 = g Cu µ B ŜCu 1 + ŜCu 2 • H (1)
It is straightforward to derive the field-and temperature dependent partition function Z 0 related to that Hamiltonian: where β = 1/k B T . The magnetization and susceptibility is obtained as appropriate 95 derivatives of the partition function, i.e.

Z 0 (H, T ) = 2 [1 + cosh(β µ B g Cu H)] , (2)
BFI 1.8 K AFI&H 1.8 K BFI 5 K AFI&H 5 K M [N A µ B ] series A 1.8 K series A 5 K FIT series B 1.8 K series B 5 K FIT series C 1.8 K series C 5 K FIT M [N A µ B ] µ 0 H [T] series D 1.8 K series D 5 K FIT µ 0 H [T] series E 1.8 K series E 5 K FIT µ 0 H [T]
M 0 (H, T ) = N A β ∂ ln Z 0 (H, T ) ∂ H χ 0 (H, T ) = N A β ∂ 2 ln Z 0 (H, T ) ∂ H 2 . ( 3
)
It is easy to demonstrate that the magnetization function reduces to a sum of two Brillouin functions corresponding to a couple of the Cu(II) spins. The molar magnetic susceptibility was calculated according to the following approximate formula

χ 0C = χ 0 1 -zJ ′ χ 0 N A µ 2 B g 2 Cu , (4)
where χ 0C denotes the susceptibility corresponding to the interacting (coupled) clus- ters. The susceptibility calculated along the lines sketched above was fitted to the experimental points yielding ferromagnetic intermolecular coupling zJ ′ = 0.13(1) cm -1 , and the Landé factor g Cu = 2.0168 [START_REF] Zhang | Reversible thermally and photoinduced electron transfer in a cyano-bridged {Fe 2 Co 2 } square complex[END_REF]. The corresponding agreement quotient for sus-

ceptibility is R χT = ∑ [(χT ) obs -(χT ) calc ] 2 / ∑ [(χT ) obs ] 2 = 4.1 • 10 -5 . Next, using
the Brillouin function and relaxing the value of the copper Landé factor g Cu the magnetization was calculated and fed into the analogous magnetization agreement quotient Hamiltonian corresponding to model I acquires the following form

R M = ∑ [M obs -M calc ] 2 / ∑ [M obs ] 2 incorporating

SC state

Ĥ1 = -J 1 ŜMo 1 • ŜCu + µ B g Mo 1 ŜMo 1 + g Cu ŜCu • H, (5)
where different Landé factors for the Mo(V) and Cu(II) ions are assumed and J 1 is the superexchange coupling constant between spin 1/2 of the Mo(V) ion and spin 1/2 of the unperturbed Cu(II) ion. It is straightforward to derive the field-and temperature dependent partition function Z 1 related to Hamiltonian Ĥ1 :

Z 1 (β , H) = 2e 1 4 β J 1 cosh 1 2 β µ B (g Cu + g Mo 1 )H + 2e -1 4 β J 1 cosh 1 2 β µ 2 B (g Cu -g Mo 1 ) 2 H 2 + J 2 1 ( 6
)
The Hamiltonian corresponding to model II reads

Ĥ2 = -J 2 ŜMo 2 • ŜCu 1 + ŜCu 2 + µ B g Cu ŜCu 1 + ŜCu 2 + g Mo 2 ŜMo 2 • H (7)
where similarly different Landé factors for the Mo(IV) * and Cu(II) ions were assumed and J 2 is the superexchange coupling constant between spin 1 of the Mo(IV) * ion and spins 1/2 of both the Cu(II) ions. In this case the corresponding partition function Z 2 cannot be given in a closed form and must be calculated numerically through the diagonalization of the Hamiltonian in Eq. 7 for each required value of the couple (H, T ).

The molar magnetization and susceptibility in both models is obtained as appropriate derivatives of the partition functions. We deliberately decide not to take into account the intermolecular interactions in both models in order to simplify the overall description and reduce the number of free parameters. Let us finally note that the two Landé factors g Mo 1 and g Mo 2 correspond to two distinct excited states of the Mo ion, i.e.

Mo 1 =Mo(V) and Mo 2 =Mo(IV) * , and as such they are in general not equal.

As mentioned above only a fraction f 1 of the sample undergoes the MMCT process giving rise to the enhanced susceptibility signal. The excited MMCT state is a metastable one, so the fraction f 1 diminishes in the course of time. Let us assume that the number of the MMCT excited trimer units and concomitantly with it the fraction f 1 decreases with time according to the exponential (Arrhenius) law given by

f 1 = f 10 exp - t τ , (8)
where f 10 is the initial fraction at moment t = 0 and τ is the mean lifetime of the metastable state revealing a temperature dependence associated with the presence of the energy barrier ∆ 1 preventing the state from relaxing back to the ground state described by an exponential rule

τ = τ 0 exp ∆ 1 k B T , (9)
where τ 0 defines a high-temperature limit of the lifetime. In the measurement per- formed on laser light irradiation the temperature of the sample increases linearly from

T min = 2 K to T max = 300 K within the period of ∆t m =12 h 25 min (the rate of increase is 0.4 K min -1 ), so one can write a simple relation between the current time t (where t = 0 coincides with the start of the measurement) and the current temperature of the sample

t = T -T min T max -T min ∆t m . (10)
Thus the fraction of the MMCT excited trimer units in the sample f 1 is the following function of temperature

f 1 = f 10 exp -e - ∆ 1 k B T T -T min T max -T min ρ , (11)
where ρ = ∆t m /τ 0 .

Similarly, only a fraction f 2 of the sample undergoes the SC process. In this case we expect that the deexcitation of the SC state and its return to the ground state configuration is a threshold process, where up to a certain temperature T 0 the population of the SC state remains roughly on the same level. It is only above the threshold temperature T 0 that the deexcitation process starts to be effective and the fraction f 2 starts to diminish noticeably. For the precise shape of temperature dependence of this process we draw on the formula describing the reverse SC transition [START_REF] Kahn | Molecular Magnetism[END_REF], i.e.

f 2 = f 20 1 + exp ∆ 2 1 T 0 -1 T , (12)
where f 20 denotes the initial (at the start of the measurement) fraction of the SC excited trimer units and ∆ 2 is a parameter accounting for the energy distance between the excited and ground state. Finally, the fraction of the ground state f 0 obeys at each temperature the following relation f 0 = 1f 1f 2 . Now, the susceptibility signal detected just upon irradiation denoted with the subscript "UI" can be described by an appropriate combination of signals inferred from model "0" (not excited trimer units)

and model I (MMCT excited trimer units) and model II (SC excited trimer units):

χ UI (H, T ) = [1 -f 1 ( f 10 , ∆ 1 , ρ, T ) -f 2 ( f 20 , ∆ 2 , T 0 , T )] χ 0C zJ ′ , g Cu , H, T + f 1 ( f 10 , ∆ 1 , ρ, T ) χ 1 (J 1 , g Cu , g Mo 1 , H, T ) + f 2 ( f 20 , ∆ 2 , T 0 , T ) χ 2 (J 2 , g Cu , g Mo 2 , H, T ) (13)
where all the parameters and variables are given explicitly. The total magnetization 160 upon irradiation was measured in each instance prior to the susceptibility run, hence one should calculate it as an analogous combination of signals using the initial values of the fractions, i.e.

M UI (H, T ) = [1 -f 10 -f 20 ] M 0 zJ ′ , g Cu , H, T + f 10 M 1 (J 1 , g Cu , g Mo 1 , H, T ) + f 20 M 2 (J 2 , g Cu , g Mo 2 , H, T ) . (14)

Results and Discussion

The main step of the analysis is the modeling of the susceptibility and magnetization signals observed upon irradiation in the five subsequent runs from A to E. To this end we defined a test function Q being the sum of two agreement quotients Q χT and

Q M (Q = Q χT + Q M )
taking into account all the measurement runs, where

Q χT = ∑ ω∈Ω ∑ i ω (χT ) i ω -χ UI (H, T i ω ) T i ω 2 ∑ ω∈Ω ∑ i ω (χT ) i ω 2 (15)
and

Q M = 2 ∑ i=1 ∑ ω∈Ω ∑ j ω [M i, j ω -M UI (H j ω , T i )] 2 ∑ ω∈Ω ∑ j ω [M i, j ω ] 2 (16)
where Ω = {A, B, C, D, E} is a set of the measurement runs, indices i ω and j ω enumer-165 ate the consecutive measurement points in a given run, and index i = 1, 2 corresponds to the two temperatures (T 1 = 1.8 K and T 2 = 5 K) at which the magnetization was measured in each run. Let us note that each run is characterized by an independent set of the initial fractions { f ω00 , f ω10 , f ω20 } of which only two are independent variables ( f ω00 + f ω10 + f ω20 = 1). Parameters zJ ′ and g Cu were fixed at the values found in the analysis of the bulk data. This leaves us with a set of 18 independent parameters:

∆ 1 , ρ, ∆ 2 , T 0 , J 1 , J 2 , g Mo 1 , g Mo 2 f ω00 , f ω10 (ω ∈ Ω).
Let us note here that the data set to be analyzed is quite large with 94 experimental points falling on one parameter. Using the implementation of the above definition in the Mathematica8.0 environment function Q was minimized by the principal axis method. Unlike in [START_REF] Stefańczyk | Irradiation temperature dependence of the photomagnetic mechanisms in a cyanidobridged Cu(II) 2 Mo(IV) trinuclear molecule[END_REF], where only after fixing ) and corresponds to the set of parameters listed in Table 1. This result should be compared to that reported in [START_REF] Stefańczyk | Irradiation temperature dependence of the photomagnetic mechanisms in a cyanidobridged Cu(II) 2 Mo(IV) trinuclear molecule[END_REF], where Q χT = 6.4 • 10 -5 is in fact slightly smaller than the present value, however Q M = 5.3 • 10 -3 differs by three orders of magnitude from the present result. The relative errors listed in Table 1 reveal that the set of the best fit parameters .24] reported in [START_REF] Stefańczyk | Irradiation temperature dependence of the photomagnetic mechanisms in a cyanidobridged Cu(II) 2 Mo(IV) trinuclear molecule[END_REF]. Finally, the fractions f ω00 and f ω10 (ω ∈ Ω) are all positive and attain consistently values below 1.

f D10 = f E10 =
parameter J 1 [cm -1 ] J 2 [cm -1 ] ∆ 1 [K] ρ ∆ 2 [K] T 0 [
The curves corresponding to the parameter values in Table 1 are depicted with solid lines in Fig. 2 (susceptibility) and Fig. 3 (magnetization). The agreement of the calculated values (solid lines) with the experimental ones (symbols) is satisfactory although not perfect. The largest discrepancies in susceptibility (see Fig. 2)can be observed in the low temperature limit, which may be due to the fact that for the sake of simplicity we neglected the intermolecular interactions between the clusters in their excited states. The introduction of two further parameters would certainly have an adverse effect on the convergence of the optimizing procedure. The agreement of the calculated values (solid lines) with the experimental ones (symbols) of the molar magnetization (see Fig. 3) is apparently better at 5 K than at 1.

Conclusions

Let us note that the approach presented here represents our second attempt to quantitatively describe a magnetic photoinduced effect. A similar approach has been taken

for the first time in [START_REF] Korzeniak | Photoswitchable Cu II 4 Mo IV and Cu II 2 Mo IV cyanidobridged molecules[END_REF], where two cluster compounds based on Cu(II) and Mo(IV) revealed a slight enhancement of their magnetic properties due to irradiation. Although the modeling of the photoinduced signals involved as many as eighteen parameters, the best fit parameter values seem physically acceptable. Moreover, the fact that both the susceptibility and the isothermal magnetization were reproduced to a satisfactory extent with these parameters bears out the correctness and reliability of the whole procedure. The initial fractions of excited trimer units diminish from run to run, which can be understood by remembering that the excited states are metastable. The successful analysis corroborates the assumption that the irradiation of the present sample triggers two independent processes: the metal to metal charge transfer (MMCT) leading to a state with the Arrhenius-type relaxation and the spin crossover (SC) transition ending in a state whose relaxation displays a threshold behavior. The extended procedure gave a unique insight into the values of the spectroscopic factors of both excited states of the Mo ion. They turned out to be comparable and exceed the free electron value, which indicates crucial anisotropy of both photo-generated states.

Figure 1 :

 1 Figure 1: Solid state UV-vis-NIR absorption spectrum of the optically diluted 1 in BaSO 4 measured at room temperature (black curve) and the Gaussian deconvolution of the experimental spectrum (red curve). Purple vertical line represents the wavelength used for photomagnetic studies.

Figure 2 :

 2 Figure 2: Temperature dependence of χT of 1 before irradiation (BFI), upon irradiation in subsequent runs A -E (A -irradiated at 10 K; B -irradiated at 50 K; C -irradiated at 100 K; D -irradiated at 150 K; Eirradiated at 200 K), and after irradiation runs on heating up to 300 K (AFI&H).

Figure 3 :

 3 Figure 3: Field dependence of the isothermal magnetization of 1 before irradiation (BFI), after irradiation runs on heating up to 300 K (AFI&H), and upon irradiation in subsequent runs A -E (A -irradiated at 10 K; B -irradiated at 50 K; C -irradiated at 100 K; D -irradiated at 150 K; E -irradiated at 200 K) at 1.8 and 5 K.

  all the available experimental points, which was optimized for a single common value of g Cu . The corresponding procedure was implemented in the Mathematica8.0 environment and yielded g Cu = 2.02[START_REF] Muñoz | Thermo-, piezo-, photo-and chemo-switchable spin crossover iron(II)-metallocyanate based coordination polymers[END_REF] and R M = 2.2 • 10 -4 . The sizeable value and the ferromagnetic character of the intermolecular coupling may point to the presence of intramolecular interaction between the Cu(II) ions mediated through the diamagnetic -N-C-Mo(IV)-C-N-linkage. The set of parameters zJ ′ = 0.13(1) cm -1 and g Cu = 2.02[START_REF] Muñoz | Thermo-, piezo-, photo-and chemo-switchable spin crossover iron(II)-metallocyanate based coordination polymers[END_REF] will serve in what follows as an input for the analysis of the susceptibility signal upon irradiation.The irradiation of the sample in each run (from A to E) resulted in an apparent increase of the χT values (see Fig.2). At the lowest temperatures all the curves display consistently a downturn, which to a large extent may be due to the substantial dc field (10 kOe) present during the measurement, but the antiferromagnetic intermolecular coupling is not excluded. For runs A to C three regimes are apparent in the flow of the χT curves on increasing the temperature. The first regime is characterized by a rather steep fall of the χT values followed by a more gentle decrease (second regime) and finally an abrupt drop reaching the level detected before irradiation (third regime). Runs D and E lack the first regime, while maintaining the other two ones. The overall increase of the χT signal upon irradiation may originate from two independent mechanisms. The first is the metal to metal charge transfer (MMCT), where an electron from the spinless Mo(IV) configuration is transferred to one of the Cu(II) ions transforming the trimer into the state Cu(II)-N-C-Mo(V)-C-N-Cu(I), with spin 1/2 on the Mo(V) ion and the spinless Cu(I) ion. The spin carriers in thus excited state may be exchange-coupled through the diamagnetic cyanido bridge. The other is a spin crossover (SC) process centered on the Mo(IV) ion due to irradiation in the LF region of the [Mo IV (CN) 8 ] 4-spectra, giving rise to an excited paramagnetic Mo(IV) * linking two paramagnetic Cu(II) centers with a possible superexchange interaction. The spin of the excited Mo(IV) * species is equal to 1 and associated to a disruption of the 5s-electronic pair. It is important to note two facts. Firstly, the Cu(II)-N-C-Mo(V)-C-N-Cu(I) state as well as the Cu(II)-N-C-Mo(IV) * -C-N-Cu(II) state are excited and thus metastable, and with the course of time they will relax back to the ground state of the trimer. Secondly, only a certain part of the trimer molecules in the sample shall undergo either an MMCT process or a SC process leaving the remaining molecules in 135 the unperturbed state described by model "0". Pertinent to the new photoinduced configurations of the trimer molecule we assume model I and II depicted in Fig.4, where J 1 and J 2 denote superexchange constants of the coupling between, respectively, the spin 1/2 state of the Mo(V) ion and the spin 1/2 state of the remaining Cu(II) ion and the spin 1 state of the Mo(IV) * ion and the spin 1/2 states of each of the Cu(II) ions. One comment is necessary here. Although the unit cell of 1 contains two symmetry inequivalent trimer entities, the bond lengths and angles in the cyanido bridges mediating the superexchange coupling differ only little between both entities and between either side of each entity. Therefore, we resort to a natural simplification consisting in assuming that all the trimers in compound 1 are structurally identical. Hence our models involve a single exchange coupling constant each. This assumption reduces considerably the number of free parameters to be determined. The

Figure 4 :

 4 Figure 4: Schematics of models I and II of MMCT (top) and SC (bottom) excited states.

  0 the optimizing procedure converged returning an acceptable solution, it did converge in its full yielding reasonable parameter values. The minimum thus obtained amounts to Q = 8.6 • 10 -5 (Q χT = 8.1 • 10 -5 and Q M = 5.5 • 10 -6

  8 K. At 1.8 K the calculated values overestimate the experimental signal especially for the intermediate field values.This again may be due to the lack of accounting for the intermolecular interactions of the timer units in their excited states, which most probably are antiferromagnetic in character. Another source of discrepancy may be the fact that the magnetization measurement preceded the susceptibility measurement in each run while the values of the initial fractions f ω00 and f ω10 (ω ∈ Ω) correspond precisely to the starting moment of the susceptibility measurement. However, this should rather result in a slight underes-timation of the fractions and therefore lead to a further enhancement of the calculated values. For the sake of transparency of presentation the initial abundances (fractions) of the ground state and the MMCT and SC excited states are shown in Figure5in a form of a stack column chart. It can be seen that the level of population of both the last two runs sufficiently long for the MMCT excited trimer units to relax back to the ground state.

Figure 5 :

 5 Figure 5: Initial abundances of the ground state (GS) and the MMCT and SC excited states for all the measurement runs A-E (A -irradiation at 10 K; B -irradiation at 50 K; C -irradiation at 100 K; Dirradiation at 150 K; E -irradiation at 200 K).

Table 1 :

 1 Parameter values corresponding to the minimum of Q breaks up into two groups: the group of six parameters (J 1 , ∆ 1 , ρ, ∆ 2 , f D10 , f E10 ) to which function Q is only weakly sensitive (considerable relative errors) and the remaining twelve parameters (T 0 , g Mo 1 , g Mo 2 and the remaining fractions f ω00 and f ω10 ) with fairly strong influence on Q (small relative errors). Let us at the same time stress that the relative errors within the first group do not exceed 100 % but for the fraction f E10 which has the smallest absolute value. The exchange couplings in both the MMCT excited state and the SC excited state turn out to be of ferromagnetic character, which is consistent with the general increase of the susceptibility signal detected upon irradiation in all the measurement runs. Parameter ρ implies that τ 0 ≈ 1.77 h, which is a reasonable result comparing well with the time scale of the photomagnetic experiment.Moreover, the height of the energy barrier for the relaxation of the MMCT excited state ∆ 1 ≈ 80 K is also plausible. Parameter ∆ 2 ≈ 4300 K constitutes a fraction (12%) of the energy of the light quantum (≈ 35430 K for the purple light with λ = 405 nm) used in the irradiation of the sample, which should be expected. The values of the Landé factors g Mo 1 = 2.22 and g Mo 2 = 2.25 are comparable and consistent with the upper bound of the interval [2.13,2

	K]

MMCT and SC states diminishes with increasing temperature of irradiation (along the series of the runs). The SC state has a nonvanishing contribution in all the runs, while the contribution of the MMCT state is negligibly small for the two final runs. Apparently, the time that elapses between the moment of irradiation finish and the moment of susceptibility measurement start, which gradually extends from run to run, is for the


Wei Liu
email: wei.liu@etu.univ-amu.fr

L U Xiao-Gang

Boulet Pascal
email: pascal.boulet@univ-amu.fr

H U Qing-Miao

Influence of atomic order on the enthalpy of formation and bulk modulus of the sigma phase

Keywords: sigma phase, atomic order, enthalpy of formation, bulk modulus, EMTO-CPA

We have calculated the enthalpy of formation and bulk modulus for 19 typical binary sigma compounds with different atomic order (i.e. atomic constituent distribution or site occupancy preference on inequivalent sites of a crystal structure) based on the experimental site occupancy as well as completely ordered and hypothetically disordered states by using the EMTO-CPA (Exact Muffin-Tin Orbitals -Coherent Potential Approximation) method. The calculation results show that at 0 K the sigma phase in ordered state bears a lower enthalpy of formation and a larger bulk modulus than the ones in less ordered state.

Introduction

The topologically close-packed (TCP) phases are one of the largest groups of intermetallic compounds, which are composed of metallic atoms with different atomic size. These atoms adapt each other forming a very complex crystal structure with high space utilization and high coordination number (CN). The characteristic of the structure is that it is stacked by coordination polyhedron of CN 12,[START_REF] Vitos | Anisotropic Lattice Distortions in Random Alloys from First-Principles Theory[END_REF][START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] and 16 [START_REF] Hu | Fundamentals of materials science[END_REF]. TCP phase is common precipitation in Fe-, Ni-, Co-based superalloys and austenitic and duplex stainless steels [START_REF] Seiser | TCP phase predictions in Ni-based superalloys: Structure maps revisited[END_REF][START_REF] Ladines | Structural stability of Fe-based topologically close-packed phases[END_REF][START_REF] Rettig | Numerical modelling of precipitation of topologically close-packed phases in nickel-base superalloys[END_REF][START_REF] Palumbo | First-principles-based phase diagrams and thermodynamic properties of TCP phases in Re-X systems (X=Ta, V, W)[END_REF]. The common members of TCP phases are sigma, chi, mu, laves, A15, R, P, delta and M phases [START_REF] Hu | Fundamentals of materials science[END_REF][START_REF] Seiser | TCP phase predictions in Ni-based superalloys: Structure maps revisited[END_REF]. A small amount of TCP precipitations can cause the effect of precipitation hardening and grain boundary strengthening [START_REF] Ladines | Structural stability of Fe-based topologically close-packed phases[END_REF][START_REF] Zhao | Assessments of molar volume of the binary C14 Laves phase[END_REF]. However, in most cases, TCP phase is a detrimental phase in alloys. It is hard and brittle, which make alloys crack under the service conditions in tensile stress. Moreover, the TCP precipitation will lead to poor alloying elements in matrix, and thus decreases the mechanical properties and corrosion resistance of materials [START_REF] Seiser | TCP phase predictions in Ni-based superalloys: Structure maps revisited[END_REF][START_REF] Ladines | Structural stability of Fe-based topologically close-packed phases[END_REF]. Therefore, accurate prediction and reasonable control of the precipitation of TCP phase is obviously the key to alloy design of iron-based, nickel-based, cobalt-based superalloys and stainless steels.

The sigma phase is a typical example of TCP phases, which is with large homogeneity range and designated as tetragonal structure (space group 𝐷 !"# $! -P42/mnm) with 30 atoms distributed on five inequivalent sites [START_REF] Hall | The sigma phase[END_REF][START_REF] Joubert | Crystal chemistry and Calphad modeling of the σ phase[END_REF] as presented in Table 1. It can serve as a prototype of TCP phases. In the present work, the EMTO_CPA method was conducted to reveal the influence of atomic order (i.e. atomic constituent distribution or site occupancy preference on inequivalent sites of a crystal structure) on the enthalpy of formation and the bulk modulus of the sigma phase. To facilitate explanation, the ordered and disordered state is designated with atomic occupancy as presented in Table 1. All the binary sigma phase systems studied were designated as A-B where atom A holds a larger atomic size than atom B.

Table 1. Crystal structure of the sigma phase with atomic occupancy corresponding to the ordered and disordered compounds A2B and AxB(1-x) (VA>VB, where VA and VB are the molar volumes of the constitutive elements in their hypothetic sigma phase structure from the CALPHAD assessments [START_REF] Liu | Modeling of molar volume of the sigma phase involving transition elements[END_REF]).

-x)) x•A x•A x•A x•A x•A

Methodology and calculation details

The first-principles calculations were performed using the exact muffin-tin orbitals (EMTO) method [START_REF] Vitos | Computational quantum mechanics for materials engineers: the EMTO method and applications[END_REF][START_REF] Vitos | Total-energy method based on the exact muffin-tin orbitals theory[END_REF], with which, the Green's function technique is used to solve the one electron Kohn-Sham equation. The optimized overlapping muffintin approximation is used when dealing with the effective potential in the one-electron equation. Besides, the total energy is corrected with the full charge density (FCD) method [START_REF] Vitos | Computational quantum mechanics for materials engineers: the EMTO method and applications[END_REF]. The wave function is expanded by using the basis sets of the exact muffin-tin orbitals [START_REF] Vitos | Computational quantum mechanics for materials engineers: the EMTO method and applications[END_REF][START_REF] Vitos | Total-energy method based on the exact muffin-tin orbitals theory[END_REF]. The coherent potential approximation (CPA) [START_REF] Soven | Coherent-potential model of substitutional disordered alloys[END_REF][START_REF] Gyorffy | Coherent-potential approximation for a nonoverlapping-muffin-tin-potential model of random substitutional alloys[END_REF][START_REF] Vitos | Anisotropic Lattice Distortions in Random Alloys from First-Principles Theory[END_REF] is incorporated within the EMTO code.

Green's function has been calculated for 16 complex energy points distributed exponentially on a semicircular contour.

We adopted the scalar-relativistic and soft-core approximations. The electronic exchange-correlation functional within the generalized-gradient approximation (GGA) as parametrized by Perdew et al. [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] has been used. The Brillouin zone is sampled by a uniform k-point mesh (3×3×6) without any smearing technique. Fig. 1 presents the enthalpy of formation of the sigma phase calculated by using the EMTO_CPA method. EMTO-CPA-ord, EMTO-CPA-exp and EMTO-CPA-x% represent results for enthalpy calculated with site occupancies corresponding to that of completely ordered state, experimental measurements [START_REF] Joubert | Crystal chemistry and Calphad modeling of the σ phase[END_REF][START_REF] Cieslak | Site occupancy and lattice parameters in sigma-phase Co-Cr alloys[END_REF][START_REF] Cieślak | Influence of composition and annealing conditions on the siteoccupation in the σ-phase of Fe-Cr and Fe-V systems[END_REF][START_REF] Spooner | Ordering in binary σ phases[END_REF][START_REF] Farzadfar | Thermodynamic assessment of the Molybdenum-Rhenium system[END_REF][START_REF] Boulineau | Structural characterization of the Ta-rich part of the Ta-Al system[END_REF] and completely disordered state, respectively. The binary systems of interest are Cr-Co, Cr-Fe, Cr-Mn, Mo-Co, Mo-Mn, Mo-Os, Mo-Re, Nb-Al, Nb-Ir, Os-Cr, Re-Cr, Re-Mn, Re-V, Ru-Cr, Ta-Al, V-Co, V-Fe, V-Mn and V-Ni binary systems.

Results and discussion

Influence of atomic order on enthalpy of formation

Based on Liu et al.'s work [START_REF] Liu | Influencing factors of atomic order in the binary sigma phase[END_REF], these systems have been labelled SMe, SMs-SMe, LAs-SMe, LAs, LAs-LAe or X-Al (X=Nb, Ta) according to the difference in total number of electron shells and total number of valence electrons between A and B atoms (see Fig. 1). LAs and LAe mean that atom A has larger total number of electron shells and larger number of valence electrons than atom B, respectively. SMs and SMe mean that atom A has smaller total number of electron shells and smaller number of valence electrons than atom B, respectively. The calculation results indicate that the sigma compound in a more ordered state holds a lower enthalpy of formation (more negative), i.e. the ordered structure is more stable than the less ordered ones at 0 K.

Besides, it has been found out that the formation enthalpy of the sigma phase depends on both the volume difference and the electron configuration of its constitutive elements. Table 2 presents the volume difference between the two constitutive elements (A and B) and the enthalpy of formation of the A66.7B33.3 sigma phase in ordered and disordered states calculated by EMTO-CPA method at 0 K.

EMTO_CPA-x%

EMTO_CPA-ord

It indicates that for SMe and SMs-SMe categories, generally, the enthalpy of formation for compounds in both ordered and disordered states decreases with increasing the size difference between the two constitutive elements (except for V-Fe and V-Ni systems between which inversion is observed). For LAs and LAs-LAe categories, the enthalpy of formation in disordered state for systems with large size difference (Re-Mn and Re-Cr systems), is positive, which indicates a less stable state. This behavior is related to the fact that the effects of the size and electrons factors (LAs or LAe factors) on atomic order for these categories are contradictory see Ref. [START_REF] Liu | Influencing factors of atomic order in the binary sigma phase[END_REF]. On the one hand, by considering the size factor, atom A tends to occupy large CN sites, namely 4f, 8i1, 8j; on the other hand, by considering LAs (i.e. Atom A is with the larger total number of electron shells than atom B) or LAe (i.e. Atom A is with the larger number of valence electrons than atom B) factors, atom A tends to occupy small CN sites, namely 2a, 8i2. For LAs-SMe category, due to the influence of LAs factor and the large size difference, for both Mo-Co and Mo-Mn systems, the enthalpy of formation in disordered state is also positive.

Moreover, the enthalpy of formation for Mo-Co system is smaller than that for Mo-Mn system, due to the large effect of SMe factor (i.e. Atom A is with the smaller number of valence electrons than atom B, which makes atom A tend to occupy large CN sites [START_REF] Liu | Influencing factors of atomic order in the binary sigma phase[END_REF]) for Mo-Co system reduces the effect of LAs factor on atomic order. By considering electron loss and gain, X-Al (X= Nb, Ta) category behave similarly as LAs-SMe category [START_REF] Liu | Influencing factors of atomic order in the binary sigma phase[END_REF]. The strong effect of LAs factor and a relatively large size difference for Ta-Al system cause the enthalpy of formation in disordered state to be positive.

In addition, calculations considering magnetism were conducted for Cr-Fe and V-Fe systems as presented in Fig. 1 (b,e).

The calculation results show that at 0 K when magnetism is accounted for compounds are more stable than those for which magnetism is not accounted for, especially for compounds with high content in Fe, which is agree with the experimental measurements [START_REF] Cieślak | On the magnetism of the σ-phase Fe-Cr alloys[END_REF][START_REF] Cieślak | Magnetic ordering above room temperature in the sigma-phase of Fe66V34[END_REF].

Table 2. Size difference between the constitutive elements (A and B) and enthalpy of formation (Hf) of the A66.7B33.3 sigma phase in both ordered and disordered states calculated by EMTO-CPA method at 0 K. VA and VB are the molar volumes of element A and B in their hypothetic sigma phase structure from the CALPHAD assessments [START_REF] Liu | Modeling of molar volume of the sigma phase involving transition elements[END_REF].

Influence of atomic order on bulk modulus

The bulk modulus of the sigma phase in the Cr-Co, Cr-Fe, Cr-Mn, Mo-Co, Mo-Fe, Mo-Mn, Mo-Os, Mo-Re, Nb-Al, Nb-Ir, Re-Cr, Re-Mn, Ta-Al, V-Co, V-Fe, V-Mn and V-Ni systems has also been evaluated by fitting the Birch-Murnaghan equation of state [START_REF] Murnaghan | The Compressibility of Media under Extreme Pressures[END_REF][START_REF] Birch | Finite elastic strain of cubic crystals[END_REF]. The corresponding results are listed in Table 3. For most systems, sigma phase in ordered state has a large bulk modulus and a small molar volume, which indicates a strong binding between atoms and a compact structure, respectively. Only for Mo-Re system, sigma phase in ordered state has a large bulk modulus but a large molar volume, which indicates a strong binding between atoms but a loose structure, respectively.

Conclusion

The influence of atomic order on both the enthalpy of formation and bulk modulus of the sigma phase has been studied by using the EMTO_CPA calculations. At 0 K, the sigma phase in the ordered state bears a low enthalpy of formation, which indicates that the ordered structure is more stable than the less ordered ones. Besides, the sigma phase in the ordered state bears a large bulk modulus, which indicates a strong binding between atoms. On the other hand, we found that the formation enthalpy of the sigma phase depends on the size difference and the electron configuration between the two constitutive elements.

1 EnthalpyFig. 1 .

 11 Fig.1. Enthalpy of formation of A-B binary sigma phase systems referred to the enthalpy of the pure elements in a hypothetic sigma phase structure calculated by EMTO-CPA method at 0 K.

Table 3 .

 3 EMTO_CPA calculated bulk modulus (B0) and corresponding molar volume (Vm) of the A66.7B33.3 sigma phase in both ordered and disordered states at 0 K.

		B0, Gpa	Vm, 10 -6 m 3 /mol
	AxBy
	ordered	disordered	ordered	disordered

Acknowledgements

This work was supported by the Chinese National Key Project of Science and Technology (Grant number: 2012ZX06004-012). W. Liu gratefully acknowledges the financial support from the China Scholarship Council. X.-G. Lu acknowledges the High Performance Computing Center of Shanghai University for providing the calculation facility.


Wei Liu

Xiao-Gang Lu

Pascal Boulet

Marie-Christine Record

Influence of nearest neighbor atoms and coordination polyhedron on atomic volume of the sigma phase

Keywords: sigma phase, atomic volume, nearest neighbor (NN) atoms, coordination polyhedron (CP), atomic charge

We have calculated the atomic volume with respect to the atomic charge of the constituent elements in the complete set of ordered configurations of several binary sigma systems by using VASP and Bader's AIM approach. The calculation results indicate that both the nearest neighbor (NN) atoms and coordination polyhedron (CP) affect the atomic volume of the sigma phase. For a specific element, on a specific site, atoms with more negative atomic charge bear a larger volume; the more the NN large atoms and the higher the symmetry of CP, the larger the space to accommodate the specific atom.

Introduction

The sigma phase, known as Frank Kasper phase, crystallizes in a tetragonal structure with space group P42/mnm. It is a non-stoichiometric intermetallic compound with large homogeneity range.

There are 30 atoms in a unit cell, which distribute on five inequivalent sites, namely 2a, 4f, 8i1, 8i2 and 8j. Fig. 1 presents the crystal structure of the sigma phase. As a typical topologically closepacked (TCP) phase, it is stacked by coordination polyhedrons (CP) of coordination number (CN) 12, 14 and 15 around sites 2a, 8i2; 8i1, 8j and 4f, respectively, which is clearly displayed in Fig. 2.

Table 1 lists the nearest neighbor (NN) sites for each crystal sites of the sigma phase. More details about the crystal chemistry of the sigma phase is recommended to Ref. [START_REF] Hall | The sigma phase[END_REF][START_REF] Joubert | Crystal chemistry and Calphad modeling of the σ phase[END_REF]. The sigma phase is a detrimental precipitation in a lot of technologically important materials, such as duplex stainless steels and Ni-based supperalloys. The occurrence of the sigma phase should be carefully avoided or controlled. Thus thermodynamic [START_REF] Dupin | A thermodynamic database for Ni-base superalloys[END_REF][START_REF] Mathieu | CALPHAD description of the Mo-Re system focused on the sigma phase modeling[END_REF] and molar volume [START_REF] Liu | Modeling of molar volume of the sigma phase involving transition elements[END_REF] databases of the sigma phase have been successively built by using the CALPHAD method [START_REF] Kaufman | Computer Calculations of Phase Diagrams[END_REF][START_REF] Saunders | CALPHAD (calculation of phase diagrams): a comprehensive guide[END_REF][START_REF] Lukas | Computational thermodynamics: the CALPHAD method[END_REF].

Besides, other properties of the sigma phase also attracts lots of interests, like site occupancy preference [START_REF] Joubert | Crystal chemistry and Calphad modeling of the σ phase[END_REF][START_REF] Liu | Influencing factors of atomic order in the binary sigma phase[END_REF][START_REF] Sluiter | Prediction of Site Preference and Phase Stability of Transition Metal Based Frank-Kasper Phases (PDF Download Available)[END_REF], enthalpy of formation [START_REF] Crivello | χ and σ Phases in Binary Rhenium-Transition Metal Systems: a Systematic First-Principles Investigation[END_REF][START_REF] Liu | Influence of atomic order on the enthalpy of formation and bulk modulus of the sigma phase[END_REF] and magnetism [START_REF] Cieślak | Magnetic moments and Curie temperatures of σ-FeCr alloys[END_REF][START_REF] Cieślak | Magnetic ordering above room temperature in the sigma-phase of Fe66V34[END_REF].

In the present work, VASP and Bader's AIM approach were conducted to elucidate the influencing factors of the atomic volume in the binary sigma phase. To facilitate explanation, for all the A-B binary sigma systems investigated, atom A always holds a larger size than atom B. The size of atom A or B is based on the molar volume assessments of the sigma phase [START_REF] Liu | Influencing factors of atomic order in the binary sigma phase[END_REF][START_REF] Liu | Modeling of molar volume of the sigma phase involving transition elements[END_REF]. Besides, we follow the divided categories of the A-B sigma phases by Liu et al. [START_REF] Liu | Influencing factors of atomic order in the binary sigma phase[END_REF], which is categorized according to the electron configuration of the two constituent elements, i.e. SMe, SMs-SMe, LAs, LAs-SMe, LAs-LAe, A-Al (A-Nb, Ta), where SMe and LAe indicates that atom A bears smaller or larger number of valence electrons, respectively, than atom B; SMs and LAs indicates that atom A bears smaller or larger total number of electron shells, respectively, than atom B.

Methodology and calculation details

First-principles calculations

The first-principles calculations were performed by using the plane wave method with projector augmented wave (PAW) pseudo-potentials [START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF] based on density-functional theory (DFT) [START_REF] Dreizler | Density Functional Theory[END_REF].

The method is implemented in the Vienna ab initio simulation package (VASP) [START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF]. The exchange-correlation functional within the generalized gradient approximation (GGA) as parameterized by Perdew and Wang (i.e. GGA-PW91) [START_REF] Perdew | Accurate and simple analytic representation of the electron-gas correlation energy[END_REF] was employed. A large plane-wave cutoff energy of 400 eV was used. The k-point meshes for Brillouin zone sampling were constructed using the Monkhorst-Pack scheme [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF], which were set as 8×8×15. To facilitate the calculations, ZenGen script-tool [START_REF] Crivello | a tool to generate ordered configurations for systematic first-principles calculations: The Cr-Mo-Ni-Re system as a case study[END_REF] was used to automatically generate input files.

Bader Charge Analysis

In the present work, we adopted the Bader charge analysis code [21] developed by the Henkelman Group. It is based on Richard Bader's theory of Atoms in Molecules (AIM) [START_REF] Bader | Atoms in Molecules: A Quantum Theory[END_REF],

which proposes an intuitive way to separate molecules into atoms based purely on the electronic charge density. The code provides an efficient and robust algorithm for the Bader decomposition of charge density [START_REF] Henkelman | A fast and robust algorithm for Bader decomposition of charge density[END_REF][START_REF] Sanville | Improved grid-based algorithm for Bader charge allocation[END_REF][START_REF] Tang | A grid-based Bader analysis algorithm without lattice bias[END_REF][START_REF] Yu | Accurate and efficient algorithm for Bader charge integration[END_REF].

During the calculation, the total electron charge was achieved by summing the core charge and the valence charge from the VASP. The Bader analysis was performed on the total charge density and then output the atomic volume and the total charge associated with each atom [21]. To get the atomic charge for each atom, the valence electrons (the valence states set in VASP) minus the obtained electronic charge were conducted. For the atomic charge, the negative value indicates that an atom tends to gain electrons and the positive one indicates that an atom tends to lose electrons.

Results and discussion

In the sigma phase structure, for a specific element, the atomic size varies with the change of occupying sites. Normally, the atomic size sequence follows the occupied site size sequence (i.e. 4f(CN15)>8i1(CN14)>8j(CN14)>8i2(CN12)>2a(CN12) [START_REF] Liu | Influencing factors of atomic order in the binary sigma phase[END_REF]) when atomic charges are close, e.g. Fig. 3 (ac). However, it may not be always true. For some binary sigma compounds, the atomic volumes sometimes may not follow the size sequence of the correspondingly occupied site owing to the influences of NN atoms and CP on atomic volume, e.g. Fig. 3 (d-g).

To figure this out, the influencing factors on atomic volume are specially discussed. For a specific atom in the sigma phase structure, two factors determine its atomic volume. The first one is its occupying crystal site (i.e. one among 2a, 4f, 8i1, 8i2 and 8j) and the corresponding CP symmetry. The second concerns the NN atoms around the atom. The effect of the next NN atoms is assumed small that can be neglected.

- stoichiometric sigma end-member compounds (e.g. W2W4W8W8W8, Re2W4W8W8W8 etc.) for W-Re, Mo-Re, Mo-Mn, Ta-Pd, Re-Mn. Ru-Cr and Ta-Al binary systems calculated using VASP and Bader's AIM approach. (Color online).

Influence of nearest neighbor (NN) atoms on atomic volume

The NN atoms mainly affect the atomic charge and the space to accommodate the specific atom, and thus affect its atomic volume. In the following section 3.1.1 and 3.1.2, we respectively discussed how the atomic charge and the accommodation space affect the atomic volume of the sigma phase. For Re-Mn, Ru-Cr, Mo-Mn (which belong to LAs, LAs-LAe and LAs-SMe categories respectively) and Ta-Al systems, for the same or a small variation of the atomic charge, the atomic volume difference for a specific element on a specific site can be very large. This is due to the fact that for these systems, the atomic volume is more sensitive to the accommodation space. For LAs, LAs-LAe categories and some system in LAs-SMe category, large atom A tend to gain electrons and the atomic size further increases; small atom B tend to lose electrons and the atomic size further decreases [START_REF] Liu | Influencing factors of atomic order in the binary sigma phase[END_REF]. Thus the size difference between the two constituent elements becomes very large, which makes the factor of accommodation space crucial. Properties of A-Al (A-Nb, Ta) category behaves similarly as LAs-SMe category [START_REF] Liu | Influencing factors of atomic order in the binary sigma phase[END_REF], such as the tendency of electron loss or gain, and the site occupancy preference.

Atomic charge

Accommodation space

As mentioned in the last section, for LAs, LAs-LAe categories, the atomic volume is more sensitive to accommodation space. Thus we selected Re-Mn and Ru-Cr systems, and analyzed the atomic volume among the 32 stoichiometric end-member compounds, compound-by-compound.

Fig. 4 and Fig. 5 show the atomic volume with respect to atomic charge of the constituent elements in several stoichiometric sigma end-member compounds for Re-Mn and Ru-Cr system, respectively. Table 2 shows the number of NN atom A around the 8i1, 8j and 8i2 sites of the corresponding compounds in Fig. 4 and Fig. 5, which can be conveniently achieved from Fig. 1 and Table 1. It was found that when large atom A (V(A)>V(B)) occupy 8i1 and 8j sites, and in the meanwhile the number of the NN atom A surrounding 8j site is large enough as indicated in Table 2, the atomic sizes of B on 8i1 and 8j follows 8j>8i1 (normally the atomic size sequence for a specific element follows the occupied site size sequence, i.e. 8i1(CN14)>8j(CN14) [START_REF] Liu | Influencing factors of atomic order in the binary sigma phase[END_REF]) as seen in Fig.

4 (e-h) and Fig. 5 (e-h). Hence we can conclude that the more the NN large atoms, the larger the space to accommodate the specific atom. There are more evidences to support this, but to avoid redundance, we will not state.

-

Fig. 1 .Fig. 2 .

 12 Fig. 1. Crystal structure of the sigma phase with different fillings indicating atoms occupying on different crystal sites, namely 2a, 4f, 8i 1 , 8i 2 and 8j. (Color online)

Fig. 3

 3 Fig.3shows the calculated atomic volume with respect to atomic charge of the constituent elements in the complete set (2 5 =32) of ordered configurations of the A-B (V(A) >V(B)) binary sigma phase. It indicates that for W-Re, Mo-Re and Ta-Pd systems (which belong to SMe, SMs-SMe and LAs-SMe categories respectively), the atomic size and site size follow the same sequence. It can be obviously seen that for a specific element, on a specific site, atoms with more negative atomic charge bear a larger volume.

Table 1

 1 Number of nearest neighbor (NN) sites for the five inequivalent sites of the sigma phase

					NN sites
	Site
		2a 4f 8i1 8i2 8j	Total
	2a	-	4	-	4	4	12
	4f	2	1	2	4	6	15
	8i1	-	1	5	4	4	14
	8i2	1	2	4	1	4	12
	8j	1	3	4	4	2	14

[START_REF] Liu | Influencing factors of atomic order in the binary sigma phase[END_REF]

.

Atomic Charge / eRu-Cr Cr Ru

Acknowledgements

This work was supported by the Chinese National Key Project of Science and Technology (Grant number: 2012ZX06004-012). W. Liu gratefully acknowledges the financial support from the China Scholarship Council. X.-G. Lu acknowledges the High Performance Computing Center of Shanghai University for providing the calculation facility.

Table 2 Number of nearest neighbor (NN) atom A for the 8i1, 8j and 8i2 sites of the corresponding A-B (V(A)>V(B)) binary sigma compounds in Fig. 4 and Fig. 5.

NN atom A around the site

Influence of coordination polyhedron (CP) on atomic volume

The CP mainly affects the space to accommodate the specific atom, and thus affect its atomic volume. Fig. 4 (i-l) and Fig. 5 (i-l) show that when larger atom A(V(A)>V(B)) occupy 8i2 and 8j sites, and smaller atom B occupy 8i1 site, 4f and 2a sites occupied by either A or B, the atomic sizes of A on 8i2 and 8j sites become close (normally it follows 8j(CN14)>8i2(CN12) [START_REF] Liu | Influencing factors of atomic order in the binary sigma phase[END_REF]). Since atom B occupy 8i1 site, it makes the NN atom A around 8j and 8i2 sites less. Thus the space of 8j and 8i2 sites cannot loosely accommodate the large atom A, and the CP symmetry becomes a primary factor determining the atomic size on 8j and 8i2 sites. The symmetry of CN12 sites is higher than that of CN14 sites [START_REF] Berne | Ab initio study of metastability in refractory metal based systems[END_REF], which can be clearly observed in Fig. 2. Thus even though, 8j site is with larger CN and more NN atom A as compared to 8i2 site as indicated in Table 2, for compounds (il) in Table 2 , the atomic sizes of A on 8j and 8i2 sites are close.

Conclusion

In the present work, the influencing factors of the atomic volume in the A-B (V(A)>V(B)) binary sigma systems were investigated. The main conclusion can be summarized as follows:

1. Both the nearest neighbor (NN) atoms and coordination polyhedron (CP) affect the atomic volume of the sigma phase and cause the atomic volumes sometimes unfollow the size sequence of the correspondingly occupied site.

2.

The CP mainly affects the accommodate space, the NN atoms mainly affect the atomic charge and the accommodate space, and thus affect the atomic volume. For a specific element, on a specific site, the greater the ability of an atom to gain electrons, the larger the volume of the atom;

the more the NN large atoms (atom A), the larger the space to accommodate the specific atom.

3. For SMe and SMs-SMe categories, the atomic volume of the sigma phase is mainly affected by the atomic charge; For LAs, LAs-LAe categories, the atomic volume is more sensitive to the accommodation space. For LAs-SMe category, it is ultimately determined by the compromised effect of LAs and SMe factors on the tendency of electron loss or gain of the two constituent elements.