L Safari
email: laleh.safari@oulu.fi

P Amaro

J P Santos

F Fratini

Spin effects probed by Rayleigh X-ray scattering off hydrogenic ions

Keywords: numbers: 32.80.Wr, 32.90.+a, 31.30.jc

We study the polarization characteristics of x-ray photons scattered by hydrogenic atoms, based on the Dirac equation and second-order perturbation theory. The relativistic states used in calculations are obtained using the finite basis set method and expressed in terms of B-splines and B-polynomials. We derive general analytical expressions for the polarization-dependent total cross sections, which are applicable to any atom and ion, and evaluate them separately for linear and circular polarization of photons. In particular, detailed calculations are performed for the integrated Stokes parameters of the scattered light for hydrogen as well as hydrogenlike neon and argon. Analyzing such integrated Stokes parameters, special attention is given to the electron-photon spin-spin interaction, which mostly stems from the magnetic-dipole contribution of the electron-photon interaction. Subsequently, we find an energy window for the selected targets in which such spin-spin interactions can be probed.

I. INTRODUCTION

Photonic spin, which is defined by its polarization, is one of the fundamental characteristics of light. Understanding the effects that spin induces to physical phenomena is of great importance in many branches of applied science and technology like quantum communications, astronomy, chemistry, electronics and so on. Owing to the recent advances in x-ray polarization sensitive detectors [1][2][3], tunable polarization and orbital angular momentum (OAM) free-electron lasers (FELs) [4][5][6][7] as well as synchrotron radiation sources [8], an increasing demand for accurate theoretical prediction on polarization-dependent atomic phenomena is arising. On the other hand, due to the recent progresses in storage rings and trapping techniques the scattering of light by ionic targets is of special interest nowadays since it allows us to obtain information about electron-photon interactions as well as atomic structures in the presence of strong Coulomb field. This also calls for accurate theoretical predictions in order to complement the experimental data.

The mentioned theoretical predictions address fundamental level issues, which are, consequently, not only essential for the new generations of experiments but also are of high importance for other fields. For instance, very recently, a measurement of the angle-differential cross section for Rayleigh scattering off some selected neutral atoms with fully linearly polarized incident x-rays has been performed in the PETRA III synchrotron at DESY [9]. Soon after, there has been a theoretical work focused on analyzing angle-differential cross sections for Rayleigh scattering of linearly polarized x-rays by some hydrogenlike ions (see. [10] and references therein). On the other hand, polarization-dependent total cross section (PDTCS) for scattered light can be measured via 4π detectors. Furthermore, highly energetic photons mostly scatter forwardly. Therefore, PDTCSs may be measured up to a very good approximation by current techniques of measuring angle-differential cross sections. Polarization analysis in scattering of highly energetic photons is also of great interest in astrophysics and cosmology. Related to PDTCS, the polarization power spectra in Rayleigh scattering has been regarded as a promising tool for investigating Cosmic Microwave Background (CMB) anisotropies from recombination processes [11]. Moreover, the PRISM (Polarized Radiation Imaging and Spectroscopy Mission) collaboration aims to measure temperature signal and polarization from Rayleigh scattering with high accuracy in order to set constraints in the early recombination history [12]. Polarized light in the interstellar medium can be originated by the scattering of light by inhomogeneously populated hydrogen and helium atoms. The processes that generate inhomogeneous atomic population are often the presence of a stellar magnetic field (Zeeman and Paschen-Back effects) or anisotropic radiation pumping. Specifically, a magnetic field that is inclined with respect to the symmetry axis of the pumping radiation can produce significant linear polarization signals in spectral lines (the so-called Hanle effect) [START_REF] Bueno | Gentle Introduction to the Physics of Spectral Line Polarization[END_REF][START_REF] Landi Degl'innocenti | Polarization in Spectral Lines[END_REF]. Polarization analysis of starlight is therefore essential for providing reliable information on stellar magnetic fields [START_REF] Stenflo | [END_REF]16].

When light travels through the matter it partially loses its polarization due to atom-photon interactions. In this article we address the investigation of such interactions. Specifically, in Sec. II, we derive analytical expressions for PDTCS, which are valid for scattering by hydrogenic as well as by many-electron systems. The main advantage of such analytical expressions is that allows us to perform faster and more accurate calculations. We then evaluate those expressions for two experimental scenarios that uses light circularly (Sec. II C) and linearly (Sec. II D) polarized. By making use of Wigner-Racah algebra, we write those expressions in terms of angular parts and reduced matrix elements. In Sec. III, the fully relativistic numerical evaluation of the reduced matrix elements is carried out through the use of finite basis sets for the scattering by hydrogen as well as hydrogenlike neon (Ne 9+ ) and argon (Ar 17+ ). In Sec. IV, particular attention is paid to the analysis of integrated Stokes parameters (ISPs) of the scattered light as a powerful tool for probing effects of spin-spin interactions during the scattering events. Such effects are found to be within 10 -3 % to 2 % from hydrogen to hydrogenlike argon. The investigated energy range for all targets lies above their 1s ionization thresholds. It has been recently showed by us that, within that energy range, the finite basis set approach to Rayleigh scattering gives good agreement with NIST (National Institute of Standards and Technology) data values and other calculations [17,18]. Finally, a summary of our findings and links to future prospects are given in Sec. V.

II. THEORY A. Geometry of scattering event

The geometry under which our studies are conducted is displayed in Fig. 1. Hydrogenic atom irradiated by energetic light is considered in the ground state. Incident (scattered) photons have energies E γ 1(2) = ω 1(2) , propagation vectors k 1(2) and polarization unit vectors ǫ1 (2) , where is the reduced Planck constant. For both, the incident and scattered photons, ǫ is assumed to be either linear (ǫ l χ ) or circular (ǫ c λ ). The polarization angle 0 ≤ χ ≤ π and the helicity λ = ±1 are the variables used to parametrize the linear and circular polarization states, respectively. In such a notation, λ = +1 describes right-handed and λ = -1 left-handed polarized photons, respectively. The (definition of the) polarization angles are also displayed in Fig. 1 if incident and scattered photons polarizations are taken to be linear. The direction of the incident photon (k 1 ) defines the quatization (z) axis. This selection of the quantization axis simplifies the multipole expansion of the electron-photon interaction operator. The scattered photon propagates along the direction k 2 at angle θ with respect to the z axis. The scattering plane (xz) is defined by the incident and scattered photon directions.

B. Second-order scattering transition amplitude

The second-order transition amplitude for Rayleigh (and Raman) scattering is given by [17][18][START_REF] Fratini | One-and Two-photon decays in Atoms and Ions[END_REF][START_REF] Akhiezer | Quantum Electrodynamics[END_REF]

M if (ǫ 1 , ǫ2 ) = ν f | R † (k 2 , ǫ2 ) |ν ν| R(k 1 , ǫ1 ) |i ω νi -ω 1 + ν f | R(k 1 , ǫ1 ) |ν ν| R † (k 2 , ǫ2 ) |i ω νi + ω 2 , (1)
where ω νi = (E ν -E i )/ is the transition frequency between states |ν and |i . Here, the transition operator R(k 1(2) , ǫ1(2) ) describes the relativistic interaction between the bound electron and the incident (scattered) photon. In the Coulomb gauge, the explicit expression of this transition operator is

R(k, ǫ) = α • ǫ e ik•r , (2)
where α is the vector of Dirac matrices. The initial |i and final |f atomic states have well-defined angular momentum j, angular momentum projection m j and parity (-1) l , where l is the orbital angular momentum of the larger component of the Dirac spinor. In the following, such states will be labeled by |i ≡ |β i , j i , m ji and |f ≡ |β f , j f , m j f , where β is a collective label used to denote all the additional quantum numbers needed to specify the atomic states but for j and m j . For hydrogenic ions, specifically, β refers to the principal quantum number n and the orbital angular momentum quantum number l. Since Rayleigh scattering is an elastic process, the initial and final states coincide, |i = |f and consequently the energy of incident and scattered photons becomes equal (E γ1 = E γ2 ≡ E γ ).

In this work, we shall separately consider the two experimental scenarios corresponding to circularly and linearly polarized light. In the following subsections, we will show how PDTCs and ISPs are defined in terms of the transition amplitude (1), and evaluate them separately for each scenario. Moreover, in both scenarios, is assumed that the initial and final atomic states remain unobserved as typical in most experiments.

C. Circular polarization scenario

We here consider an experimental scenario in which the incident light is circularly polarized and the polarization of the scattered light is measured in the circular base (i.e., the helicity of the scattered photon is measured). In this scenario, the angular distribution function (i.e., the angle-differential cross section) can be written in SI units as [18]

dσ cc dΩ (λ 1 , λ 2 ) = α 2 c 2 (2j i + 1) mj i mj f M cc (λ 1 , λ 2 ) 2 , (3)
where α is the electromagnetic coupling constant, c the speed of light in vacuum, and we considered

M cc (λ 1 , λ 2 ) ≡ M if (ǫ c λ1 , ǫc λ2
). To further facilitate the analysis of the angular and polarization properties of the transition amplitude M cc (λ 1 , λ 2 ), we expand ǫc λ e ik•r in terms of spherical tensors with well-defined angular momentum properties [START_REF] Rose | Elementary Theory of Angular Momentum[END_REF] ǫc

λ e ik•r = √ 2π +∞ L=1 L M=-L p=0,1 i L [L] 1/2 (iλ) p a p LM (k, r) × D L Mλ (ϕ k , θ k , 0) , (4)
where

[L 1 , L 2 , ..., L n ] = (2L 1 + 1)(2L 2 + 1)...(2L n + 1),
while the spherical tensor a p LM (k, r) refers to the magnetic (p = 0) and electric (p = 1) multipoles. On the other hand, k refers to |k|. Each term a p LM (k, r) has momentum L, angular momentum projection M and parity (-1) L+1+p . Expressions for a p LM (k, r) can be found in [18]. Combining Eqs. ( 1) and ( 4), and making use of the Wigner-Eckart theorem [START_REF] Sakurai | Modern Quantum Mechanics[END_REF], the transition amplitude can be written as

M cc (λ 1 , λ 2 ) = 2π L1L2 M2 p1p2 (+i) L1-L2+p1+p2 [L 1 , L 2 ] 1/2 (λ 1 ) p1 (λ 2 ) p2 d L2 M2-λ2 (θ) × jν (-1) -jν 1 (2j ν + 1) 1/2 Θ jν (1, 2)S jν (1, 2) + Θ jν (2, 1)S jν (2, 1) , (5)
where the reduced (second-order) matrix element is given by

S jν (1, 2) = βν β i , j i ||α • a p1 L1 (k 1 , r)||β ν , j ν β ν , j ν ||α • a p2 L2 (k 2 , r)||β i , j i ω νi + ω 2 , (6)
and S jν (2, 1) is obtained from (6) by i) interchanging the label 1 with 2 and ii) replacing the positive sign in the denominator with a negative sign. The reduced amplitude β ν , j ν ||α • a p L (k, r)||β i , j i does not depend on the angle θ and is zero unless Π i (-1) L+p+1 = Π ν , where Π i(ν) is the parity of initial (intermediate) hydrogenic state and has the well-known form of (-1) l i(ν) . Following the notation used in [18,[START_REF] Goldman | [END_REF], in Eq. ( 5) we have furthermore defined

Θ jν (1, 2) = mj ν (-1) mj f +mj ν (2j ν + 1) 1/2 (7) × j f L 1 j ν -m j f λ 1 m jν j ν L 2 j i -m jν M 2 m ji ,
where Θ jν (2, 1) is obtained from Eq. ( 7) by replacing L 1 ↔ L 2 and λ 1 ↔ M 2 . From Eqs. ( 3)-( 6), and by integrating the angle-differential cross section (3) over the angle θ, the PDTCS can be written in SI units as

σ cc (λ 1 , λ 2 ) = α 2 c 2 (2j i + 1) mj i mj f L2M2 Mcc (λ 1 , λ 2 ) 2 , (8)
where

Mcc (λ 1 , λ 2 ) = 4π √ π L1 p1p2 (+i) L1+p1+p2 [L 1 ] 1/2 (λ 1 ) p1 (λ 2 ) p2 × jν (-1) -jν 1 (2j ν + 1) 1/2 Θ jν (1, 2)S jν (1, 2) + Θ jν (2, 1)S jν (2, 1) . (9)
In order to obtain Eq. ( 8), the integral result

2π 0 dα π 0 dβ sin β D J1 * m1 b (α, β, 0) D J2 m2 b (α, β, 0) = 4π 2J 1 + 1 δ m1,m2 δ J1,J2 (10)
has been used. Although, from Eq. ( 8), the quantity Mcc may seem a quantum-mechanical amplitude, we underline that, rigorously, it is not. The quantity Mcc has been obtained after having integrated over the scattered photon direction, and, therefore, does not fulfill the superposition rule that applies to any quantum-mechanical amplitude.

The only Stokes parameter that is relevant for circularly polarized light is the third one and is defined as

P cc C ≡ P cc 3 (λ 1 , E γ ) = σ cc (λ 1 , +1) -σ cc (λ 1 , -1) σ cc (λ 1 , +1) + σ cc (λ 1 , -1) . (11)
The quantity P cc 3 (λ 1 , E γ ) does not depend on the scattering angle θ but, rather, depends only on the energy (E γ ) and on the circular polarization (λ 1 ) of the incident photon. It measures the circular polarization of the scattered light for events in which the incident light is circularly polarized. It ranges from -1 to +1 and the value -1 (+1) corresponds to events in which scattered light is fully left(right)-handed polarized.

For the unpolarized incident light case, we can obtain the total cross sections by averaging Eq. ( 8) over the polarizations of the incident light [24]. Upon doing this, we obtain

σ uc (λ 2 ) = α 2 c 2 2(2j i + 1) mj i mj f λ1 L2M2 Mcc (λ 1 , λ 2 ) 2 . (12)
Correspondingly, the third ISP for this case takes the form

P uc 3 (E γ ) = σ uc (+1) -σ uc (-1) σ uc (+1) + σ uc (-1) , (13)
where P uc 3 measures the circular polarization of the scattered light for events in which the incident light is unpolarized.

D. Linear polarization scenario

We will address in this section an experimental scenario in which the incident light is linearly polarized, and the polarization of the scattered light is measured in the linear base. The angular distribution is given by [17]

dσ ll dΩ (χ 1 , χ 2 ) = α 2 c 2 (2j i + 1) mj i mj f M ll (χ 1 , χ 2 ) 2 , (14)
where we considered M ll (χ 1 , χ 2 ) ≡ M if (ǫ l χ1 , ǫl χ2 ). The amplitude for linear polarization scenario is linked to the amplitude for circular polarization scenario as [17] M

ll (χ 1 , χ 2 ) = 1 2 λ1λ2 e -iλ1χ1 e iλ2χ2 M cc (λ 1 , λ 2 ) . ( 15
)
Upon integrating dσ ll /dΩ over the scattered photon direction, we get the total cross section as

σ ll (χ 1 , χ 2 ) = dΩ dσ ll dΩ = α 2 c 2 (2j i + 1) mj i mj f L ll (χ 1 , χ 2 ) , (16)
where

L ll (χ 1 , χ 2 ) = (2π) L1L2 L ′ 1 L ′ 2 p1p2 p ′ 1 p ′ 2 λ1λ2 λ ′ 1 λ ′ 2 mm ′ M2 V L1L2p1p2 λ1λ2M2 e i π 2 λ2 h L2 M2mλ2 V L ′ 1 L ′ 2 p ′ 1 p ′ 2 λ ′ 1 λ ′ 2 M2 e i π 2 λ ′ 2 h L ′ 2 M2m ′ λ ′ 2 * f (m, m ′ ) , (17)
V L1L2p1p2 λ1λ2M2 = πe i(λ2χ2-λ1χ1) (i) L1-L2+p1+p2 [L 1 , L 2 ] 1/2 (λ 1 ) p1 (λ 2 ) p2 × jν (-1) -jν 1 (2j ν + 1) 1/2 Θ jν (1, 2)S jν (1, 2) + Θ jν (2, 1)S jν (2, 1) , f (n, m) = 2δ n,m + i π 2 δ n,m+1 -δ n,m-1 , and h J nmk = d J n m (-π/2) d J m k (π/2) . (18)
In order to obtain Eq. ( 16), the integral

2π 0 dα π 0 dβ sin β D J1 * m1 b1 (α, β, 0) D J2 m2 b2 (α, β, 0) = (2π) δ m1,m2 e -i π 2 (m1-m2) e +i π 2 (b1-b2) × J1 m3=-J1 J2 m5=-J2 f (m 3 , m 5 ) h J1 m1m3b1 h J2 m2m5b2 (19)
has been used, where J 1 and J 2 are integers. Equation ( 19) can be in turn obtained by decomposing a rotation of a certain angle β around the y axis into a rotation of the same angle around the z axis, followed and preceded by (non-commutative) rotations of angles ±90 • around y and z axes [START_REF] Edmond | Angular momentum in quantum mechanics[END_REF]. If b 1 = b 2 ≡ b, the integral ( 19) simplifies into Eq. ( 10). As mentioned before, Eq. ( 16) allows us to calculate the total cross section faster and more accurately in comparison with numerical integrations. The integrated degree of linear polarization is given by

PL = ( P ll 1 ) 2 + ( P ll 2 ) 2 , ( 20
)
where the first and the second ISPs are defined as

P ll 1 (χ 1 , E γ ) = σ ll (χ 1 , 0 • ) -σ ll (χ 1 , 90 • ) σ ll (χ 1 , 0 • ) + σ ll (χ 1 , 90 • ) , P ll 2 (χ 1 , E γ ) = σ ll (χ 1 , 45 • ) -σ ll (χ 1 , 135 • ) σ ll (χ 1 , 45 • ) + σ ll (χ 1 , 135 • ) , (21)
respectively. P ll 1(2) ranges from -1 to +1 and measures the linear polarization of the scattered light for events in which the incident light is linearly polarized. More specifically, P ll 1(2) measures the linear polarization along the chosen pair of axes defined by the angles

χ 1 = 0 • (45 • ), χ 2 = 90 • (135 • ).
In the case the incident light is unpolarized, while the scattered light is measured in the linear base, we can obtain the total cross sections by averaging Eq. ( 16) over the polarizations of the incident light as [24]

σ ul (χ 2 ) = α 2 c 2 2(2j i + 1) mj i mj f χ1 L ll (χ 1 , χ 2 ) . ( 22
)
The first and the second ISPs are thus defined as

P ul 1 (E γ ) = σ ul (0 • ) -σ ul (90 • ) σ ul (0 • ) + σ ul (90 • ) , P ul 2 (E γ ) = σ ul (45 • ) -σ ul (135 • ) σ ul (45 • ) + σ ul (135 • ) , (23)
where P ul 1(2) measures the linear polarization of the scattered light for events in which the incident light in unpolarized.

III. NUMERICAL CALCULATIONS

The most difficult task of obtaining the transition amplitude (1) is the calculation of S-function in Eq. ( 6). In the present work, we calculate the S-function by adopting the finite basis set method, using B splines and B polynomials as finite basis sets [START_REF] Sapirstein | [END_REF][27][28][29][30][START_REF] Amaro | Study of Forbidden transitions in Atomic Systems[END_REF]. The parameters of the B splines basis set are the radius of the cavity (R bs ), the number of B splines (n bs ) and their degree (k). As for the B polynomials, the parameters are the radius of the cavity (R bp ) and the number of B polynomials (n bp ) (the degree of the B polynomials is n bp -1). The parameters used in both basis sets were optimized in order to obtain stability and an agreement of six digits between the results produced by the two approaches. The optimal used parameters are: R bs = 60 a.u., n bs = 60, k = 9, R bp = 50 a.u. and n bp = 40. There has been also alternative numerical methods such as the Coulomb-Green function for calculating second-order transition amplitudes in the literature (see Refs. [START_REF] Fratini | [END_REF][33][34][35][36][37]).

IV. RESULTS AND DISCUSSION

In this section, we present the results for the integrated degrees of circular and linear polarization of the scattered light by hydrogen as well as hydrogenlike neon (Ne 9+ ) and argon (Ar 17+ ). Results have been obtained by using the relations presented in Secs. (II C) and (II D) and by implementing the computation technique presented in Sec. III.

A. Circular polarization scenario

Here we consider the circular polarization scenario as described in Sec. II C. Fig. 2 displays the third ISP, P cc 3 , for scattering by hydrogen, as a function of the photon energy E γ within the energy range 0.5 to 10 keV and for the two initial photon helicities λ 1 = ±1. Results are shown by taking into account only the contribution of electric-dipole (E1E1) multipole (dashed-red-line) and the contribution of all multipoles of the electron-photon interaction (solid-black-line). As seen in the figure, for the E1E1 curve, P cc 3 is zero for all energy values. This can be understood in view of the symmetric shape of angular distribution for E1E1 term [17,18]: The photon is scattered at small angles (in which case helicity is conserved) or large angles (in which case helicity is flipped) with the same probability. This naturally determines a vanishing third ISP. A similar discussion also applies to the higher multipoles: For low photon energies, the E1E1 multipole in the photonic interaction operator has the most contribution and, consequently, the third ISP turns out to be around zero in either case the photon initial helicity being positive or negative; For high photon energies, forward scattering is strongly preferred to backward scattering [18], which results in the conservation of the photon helicity during the scattering process. Although scaled at higher photon energies, plots of P cc 3 for scattering by targets Ne 9+ and Ar 17+ , which we do not show here for the sake of simplicity, follow the same behavior as for hydrogen. Therefore, the foregoing discussion also applies to them.

If incident light is unpolarized, P uc 3 vanishes for all energies and for all targets. This result can be traced back to conservation of parity [18].

B. Linear polarization scenario

Here, we consider the linear polarization scenario as described in Sec. II D. Fig. 3 displays the first ISP, P ll 1 , for the two incident linear polarizations χ 1 = 0 • (left column) and 90 • (right column). Results are shown for hydrogen (top row), Ne 9+ (middle row) and Ar 17+ (bottom row), as a function of the incident photon energy. We can see that P ll 1 turns out to be ≈ 1 (-1) for χ 1 = 0 • (90 • ) polarization direction, which indicates that polarizationflip (spin-flip) transitions are suppressed (see Eq. ( 21)).

This indicates that electron-photon spin-spin interaction is weak, since such interaction plays a role in spin-flip transitions. However, the deviations that | P ll 1 | has from unity are meaningful for understanding the magnitude of the spin-spin interactions, and at which energy they achieve their maximum. In addition, we readily see: i) spin-spin interaction effects are of the order ∼ 10 -3 %, ∼ 0.5 % and ∼ 2 % for H, Ne 9+ and Ar 17+ , respectively; ii) P ll 1 has a peak (or a dip) at photon energy value (called E max γ (Z) hereinafter) ≈ 5 keV, ≈ 50 keV and ≈ 90 keV, for H, Ne 9+ and Ar 17+ , respectively. Based on our calculations, we find that the magneticdipole term (M1) of the electron-photon interaction operator is mainly responsible for the deviations that | P ll 1 | has from unity. Magnetic-dipole M1 can be rigorously shown to be responsible for spin-spin and spin-orbit interactions between the photon and the atomic electron [START_REF] Fratini | One-and Two-photon decays in Atoms and Ions[END_REF]: M1∝ (k × ǫ l χ ) • (g s S + L), where S, L are the spin and orbital angular momentum of the atomic electron, while g s is the gyromagnetic factor. Since the orbital angular momentum of 1s orbital in hydrogenic atoms is zero, M1 here reflects the electron-photon spin-spin interaction. Therefore, we may label the quantity 1 -| P ll 1 | as indicating the magnitude of electron-photon spin-spin interaction. Generally, we find that spin-spin interaction effects are ∝ Z 2 , where Z is the atomic number. In order to have more quantitative idea regarding the energies at which spin-spin interaction is maximized, we recall the multipole expansion in Eq. ( 4) where spherical tensor a p LM (k, r) can be expressed in terms of spherical Bessel functions j L (kr) [18]. The first magnetic contribution, M1, depends on the spherical Bessel function of first order: M1∝ j 1 (kr). The first term responsible for the spinspin interaction in the squared amplitude will thus be ∝ (E1) * M 1 ∝ j 0 (kr)j 1 (kr). The condition kr ≈ 1.3 maximizes such term and thus it maximizes the contribution of M1. It is easily seen that such condition, with r ≈ r A , where r A is hydrogenic Bohr radius, is satisfied by energy values E max γ (Z) in Fig. 3. Moreover, since r A ∝ Z -1 , the condition kr A ≈ 1.3 implies that E max γ (Z) ∝ Z. This fact is evident from Fig. 3. We further remark that, since the electron binding energy (E b ) in the hydrogenic atoms grows with atomic number as E b ∝ Z 2 , the ratio E max γ (Z)/E b decreases as ∝ Z -1 . This implies that, the higher the atomic number is, the closer E max γ (Z) is to the binding energies, which results in enhancing spin-spin interaction effects. In other words, photons which maximize spin-spin interaction in hydrogen atom have energy E max γ (1), which is far away from the atomic binding energy and therefore electronphoton interactions are very weak. As the atomic number grows, E max γ (Z) approaches the atomic binding energy from above and this allows the ion to interact more effectively with photon of such energy resulting in larger spin-spin interaction effects.

We remark that P ll 1 is here equivalent to the degree of linear polarization of the scattered light since, because of symmetry reasons, P ll 2 vanishes for all photon energies. Let us consider the case in which incident light is unpolarized and the PDTCS is measured in linear basis. For such a case, the first ISP, P ul 1 , is displayed as a function of the photon energy in Fig. 4 for hydrogen. As we can see, the first ISP ( P ul 1 ) is -0.5 at low photon energies. This means that half of the scattered light is polarized along y direction. On the other hand, such (partial) polarization of the scattered light is almost completely lost at high energies. Similar plots for other targets (Ne 9+ and Ar 17+ ), which we do not show here for the sake of simplicity, follow the same behavior as for the hydrogen. Similar to the previous case, P ul 2 vanishes for all photon energies and all targets.

V. SUMMARY AND FUTURE PROSPECTS

We studied the polarization characteristics of Rayleigh scattered x-ray photons, based on second-order perturbation theory and Dirac relativistic equation. Experimental scenarios were considered for which light has circular or linear polarization. We derived general analytical expressions for the polarization-dependent total cross sections and integrated Stokes parameters by decomposing the transition amplitude in terms of spherical tensors (angular part) and reduced amplitudes (radial part). Results have been obtained for scattering by hydrogen as well as hydrogenlike neon and argon by means of the finite basis-set method based on the Dirac equation. The Stokes parameters of the scattered light were plotted for the photon energies higher than 1s ionization threshold of the targets. Special attention has been paid to electronphoton spin-spin interaction. We found that, circularly polarization of the incident photon, while is totally lost for Rayleigh scattering at low photon energies, is preserved and conveyed to the scattered photon at high photon energies for all targets. We also found that, linear polarization of the incident photon is in general conserved and conveyed to the scattered photon in all energy values. However, there is an energy window at which the linear polarization of the scattered photon is slightly jeopardized by the effect of electron-photon spin-spin interactions, being the latter brought by the magnetic-dipole term in the expansion of the electron-photon interaction operator. Such effect can be explored by analyzing the Stokes parameters and it has a contribution from 10 -3 % to 2 %, from hydrogen to hydrogenlike argon. Highly energetic photons mostly scatter forwardly. Therefore, the PDTCSs and ISP here theoretically studied may be measured up to a very good approximation by current techniques of measuring angle-differential cross sections.

1 FIG. 1 :

 11 FIG.1:(Color online) The adopted geometry for the scattering event. The polar angle θ uniquely defines the direction of the scattered photon in the scattering plane. The angle χ1 (χ2) parametrizes the linear polarization of the incident (scattered) photon. The hydrogenic atom is located at the origin of the coordinate axes xyz.

FIG. 2 :

 2 FIG.2:(Color online) Third ISP of the scattered light by hydrogen when incident light is circularly polarized ( P cc 3 ), as a function of the photon energy. The curves corresponding to positive (λ1 = +1) and negative (λ1 = -1) helicity of the incident photon are displayed. Results are calculated with the account of all photon multipoles (solid-black line) and within the electric dipole approximation (dashed-red line).

9 FIG. 3 :

 93 FIG.3: First ISP of the scattered light when incident light is linearly polarized ( P ll 1 ), as a function of the photon energy. Results are displyed for the hydrogen (top row), hydrogenlike neon (middle row) and hydrogenlike argon (bottom row) and for the x linearly polarized incident light (left column) and ŷ linearly polarized incident light (right column).

FIG. 4 :

 4 FIG.4: (Color online) First ISP of scattered light when incident light is unpolarized ( P ul 1 ), as a function of the photon energy. Results are calculated with the account of all photon multipoles (solid-black line) and within the electric dipole approximation (dashed-red line).

  . and F. F. acknowledge financial support by the Research Council for Natural Sciences and Engineering of the Academy of Finland. P. A. acknowledges the support of German Research Foundation (DFG) within the Emmy Noether program under Contract No. TA 740 1-1. J. P. S. and P. A. acknowledge the support by FCT -Fundação para a Ciência e a Tecnologia (Portugal), through the Projects No. PEstOE/FIS/UI0303/2011 and PTDC/FIS/117606/2010, financed by the European Community Fund FEDER through the COMPETE -Competitiveness Factors Operational Programme. F.F. acknowledges support by Fundação de Amparo à Pesquisa do estado de Minas Gerais (FAPEMIG) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).


Douglas Kell

Professor of Bio-analytical sciences) of University of Manchester considers data intensive research as -a new paradigm beyond experimental and theoretical research and computer simulations of natural phenomena-one that requires new tools, techniques, and ways of working‖. Different areas of physics research such as high-energy particle physics or research on nuclear fusion use large data sets. Data intensive research based on huge volumes of data is the current trend and is considered as fourth paradigm in science research. RRI as an Institute specializing in physics has witnessed data intensive research in the past four decades. This paper is a case study aiming to understand the challenges faced by graduate students and their perspectives in data intensive research at RRI. -Big Data‖ -being around in the corner, our objective is to find whether research at RRI is heading towards this buzz phenomenon.

Research Methodology:

3.1 To understand the perspectives of the past students who have already graduated from RRI, we plan to elicit information by consulting their theses (ETDs) archived in RRI digital repository.

3.2 Our research methodology includes a questionnaire survey of the current graduate students with an aim to understand their perspective regarding the following: -Data types -Data collection methods -Data storage and access -Data processing challenges -Legal issues -Ethical issues/ plagiarism and many more.

4.

5.2.3

What is the mode of data collection was the next question. We had provided four options and they are a) Instrument based data; b) Observational data; c) Experimental data and d) Simulation data. Respondents were given the freedom to choose all those options that were relevant to them.

Mode of Data collection

Responses As a result, we observed different combinations of data collection methods. However, more than 50% are resorting to single method of data collection.

The responses are graphically represented in figure -7 .

From the figure-7 it is very clear that Instrument based research is the most popular category at RRI followed by experimentation. This is an expected result as Astronomy & Astrophysics; Soft Condensed Matter are the two core areas of research at RRI producing maximum number of doctorates.

5.2.4

We wanted to ascertain the data format collected by researchers at RRI, digital being the most popular in recent times. Their response is given in the adjacent pie chart Figure -8 .

Our assumption was to get a response of 100% digital data, and it is surprising to note that 13% are of the view that their data is in non-digital format. Also, 6% have not answered this question. We have identified 9 categories of digital content. For each one of them, we have offered more than two options. They were allowed to choose all the relevant options. Figure -9 lists them all along with options. Responses received for these questions are graphically represented in the figure -10

a) Structured text i) HTML ii) JSON iii) TEX iv) XML b) Spreadsheets i) XLS ii) ODS iii ) CSV iv) SAS v) Stata vi) SPSS c) Databases i) MS Access ii) MySql iii) Oracle d) Graphics/Images i) JPEG ii) SVG iii) PNG iv) GIF v) TIFF vi) PS e) Audio i) MP3 ii) WAV iii) AIFF iv) OGG f) Video/Film i) MPEG ii) AVI iii) WMV iv) MP4 g) Software applications Source code a)CSS ii) JavaScript iii) Java h) Configuration data i)INI ii)CONF i)
Software applications Any other, Please specify………………………

It is clear from above figure-10 that highest hit is for JPEG followed by PNG in graphics/images category. This seems like JPEG and PNG are the most favored output for image analysis and this is an indication that research happening at RRI is centered on image processing. Another highest peak is observed in structured text category for TeX having 36 hits. It has been an observed norm in all the scientific publications to employ Latex for typesetting system. Our hit results at RRI also confirms to this norm. XLS for spreadsheets is another important digital category, which is generally used for data analysis. This survey endorses this fact by having 26 hits. Rest of the hits are distributed and insignificant for analysis. In addition to these digital content, graduate students at RRI have used many other specific software's such as Mathematica, Labview, Python, Origin Lab, ImageJ, MatLab and many more. 5.2.12 -Research data loss-is a cause of concern. Data can be lost inadvertently, and measures need to be taken to handle such situations. There are 69% of researchers who have not experienced data loss, indicating conducive research atmosphere at RRI. There are 60 % of them who have protected data by using password protected, limited access computers.

The other options such as data encryption, working in limited access secured data rooms and working on machines without internet connectivity are the measures taken by very few respondents which does not gives scope for analysis.

5.2.13

5.1.2

The mode of data collections was ascertained by consulting each thesis during the study. It was observed that most of the thesis had employed more than one method of data collection. As a result, there are 211 observed hits for 139 theses. Data obtained is graphically represented in below figure -4.

Theses and dissertations is an important branch of scholarly communication. They are one of the least tapped information resources unlike journal papers or conference papers or for that matter any other primary information resource. Theses and dissertations can be attributed as grey literature which generally does not undergo publishing process. However, it is to be noted here that the content value of scholarship in them cannot be under estimated.

Considering the importance of Theses and dissertations, policy decision was taken to include them in RRI digital repository (RRIDR) initiative thus aiming to bring global visibility to them. Theses submitted before 2004 were in print format only. So, they had to be converted into e-format and then uploaded on the RRIDR.

Employing survey questionnaire to elicit perspectives from current graduate students

Survey questionnaire comprising of sixteen questions was administered to 95 current research scholars to get their perspective with respect to research data at RRI. Data collected from both the methods mentioned above are analyzed in following section.

Data Analysis:

Through ETD of RRIDR

The first method of data collection using ETD repository for the past research from 1972 to 2015 gave us the following results. We had a question to assess the volume of data that will be collected by our scholars in the process of doctoral research. To our surprise, as per figure -11, maximum hits is for small data disproving our hypotheses. As per this, we are forced to come to a conclusion that data intensive research is not happening at RRI. It was hypothesized that research at RRI would be heading towards big data for the very reason that astronomy & astrophysics and soft condensed matter being two core areas of research, which generally tend to be data intensive. In the recent past, RRI had a directive from Jawaharlal Nehru University, New Delhi, India to which RRI is affiliated for the award of doctoral degree. As per the directive, it has become mandatory for all the thesis to pass through plagiarism check before they are submitted to the award of degree. As a result majority of our graduate students (80%) are aware of plagiarism. However, there are 13% of respondents who are not aware of plagiarism and balance 7% who have not answered the question. We presume 13% are fresh graduate students.

5.2.7

5.1.16

Our respondents were questioned about the kind of support options that they expect from the institute while handling research data. They were asked to choose all the relevant options, so that we get a clear idea about their requirements. Their response is given in figure -19. The majority of graduate students have felt the need for better technical infrastructure followed by training courses and data processing facilities. The fourth position is taken by data management followed by ethical issues, legal advice and creating data. 5.1.3 Theses were also consulted to find if the researchers had used computer for data analysis. It was apparent that more than 97% have used computer for data analysis followed by 3% of manual method. While doing so, many software's such as DYNALS, CAD, DAQ card, LabView, Mathematica and MBR Software were used by graduate students. They have also used AxioVision software for capturing videos and images and SPIP for image processing. Finally, we wanted to know, if the current graduate students wish to share any additional information about data and research. There has been a request for an online data storage system for the entire institute, quite similar to cloud facility. There are few respondents who have suggested data acquisition software and training associated with that. However not many research scholar have taken the trouble of expressing their wish , which otherwise would have thrown some light for further analysis.

Inferences

1. Our hypothesis was that research at RRI is increasingly becoming data intense. But, the present study disproves our hypothesis and it indicates that research centers around -Small Data‖ at RRI. 2. The most favored data collection methods are instrument based and experimental methods, thus proving our hypothesis. 3. It was also hypothesized that the data intensive research critically hinges more upon ease of access to data which in turn depends on facilities like storage capacity, download speed, data processing and computational facility available, and not so much on legal or ethical issues, as the later would have been addressed already by the graduate students and if not, by their experienced mentors.

Our data positively respond to this, as our respondents are satisfied with all the infrastructural facility at the institute. They are also knowledgeable about variety of skills required for the data collection and analysis.

As hypothesized, the ethical issues are not of much concern amongst researchers as they are well built into the system. 4. Our next hypothesis was about tools and techniques and their major role in data related research. This hypothesis is proved as our graduate students have responded that they use many software's, programming languages and statistical tools and techniques for data analysis. 5. As regards data sharing and collaboration network in basic science research, our data indicates 52% of respondents being favorable. So logistically our hypothesis is proved.

Conclusion

These inferences are drawn based on the analysis of the responses received in the questionnaire survey and the information captured from the ETD repository of RRI. Since our first hypothesis is disproved, we are forced to conclude that research data at RRI is not heading towards --Big Data".

A study like this will help in understanding the challenges faced by graduate students in a research institute of medium size. Majority of the graduate students have expressed their desire for a better technical infrastructure and they wish for training courses on different aspects of data related research. Legal aspects, ethical issues, Plagiarism and data citation attributions are some of the social issues associated with data and research. Our observations in this study indicate that researchers are not well informed about these issues. However trivial they may be, research scholars needs to be informed as they come a long way in their scientific pursuit.

5.2.9 Data processing is an important aspect of research. It could be done either manually or the process can be automated. A question to find the choice at RRI fetched us the following result represented through pie chart in figure-14.

  Next question was to find different types of digital content that research scholars create, while generating the research data.

  Next question was to get the opinion of researchers regarding sharing of research data. There are 52% of the respondents who have said ‗yes 'to share the research data and they share data with their research guides, Lab/ Group members and collaborators. Data sharing pattern is represented in the adjacent figure -17 . While sharing / using external data, legal issues are of major concern. We wanted to know the perception of our graduate students about this aspect which is very important. To our surprise, majority of them are unaware of the legal clauses that are associated in data sharing process. Out of 80 respondents only 14% have heard about legal aspects, without having any clarity. So we suggest an awareness program about legal aspects of sharing research data. Today, the thrust areas of research at the Institute are Astronomy & Astrophysics; Light & Matter Physics; Soft Condensed Matter and Theoretical Physics. The research activities include work in Chemistry, Liquid Crystals, Physics in Biology, , a medium sized research institute has graduate program leading to Doctoral degree in these areas of research. Since 1972, there are 159 theses submitted for the award of Doctoral degree from RRI.

  Figure -1 is the screen shot of ETD on RRI Digital Repository.

Figure - 1 :

 1 Figure -1 : Screen shot of ETD on RRIDR

  Below figure -2 represents total number of theses at RRI Digital Repository, followed by classified representation at the second level giving number of theses submitted by the four different groups. At the third level there is a representation bifurcating -theses with data‖ and -theses without data‖. The same information is graphically represented in Figure-3 .

  Figure -4

  Data storage facility is a major concern while handling big data. 85 % of the respondents have expressed their satisfaction with respect to the storage facility available at RRI. They use multiple devices to store research data. Their response with respect to data storage is represented graphically in the adjacent figure -12. Storing research data on personal/workplace computer is the obvious choice which is preferred by 46% of the respondents followed by External hard drive such as USB/CD/DVDs. The other three choices are marginally used by scholars.5.2.8Skills such as a) programming language b) knowledge about different software's and c) statistical tools are prerequisites for research while handling data. They play a major role both during data collection and data processing in all disciplines of research. Research Scholars were asked to choose all the relevant options. Therefore, we have 8 options and they are represented below in the pie chart Figure -13. 50% of the respondents have expressed their opinion that all the three skills are important. Programming language and software are considered to be important skills by 18% of the people. Hits for the other combinations of skills or independent skills are almost identical and hence has less scope for analysis. 70% of the respondents are knowledgeable about the skills required. 22% of them have expressed their desire for training in programming languages. responsibilities of researchers is an important attribute, be it science or social science. This has been of great concern during the last 3-4 decades where data surge has grown tremendously with the advent and usage of internet and many other ICT tools. Since the objective of our present study is to understand the perspectives of researchers regarding data and dissertation, we felt the necessity to learn their understanding of moral responsibilities such as a) Ethical Issues, b) Plagiarism and c) Data citation attribution. Our question as to know whether they are aware of their moral responsibility fetched us the following result represented in Figure -18

  Figure -2 : ETD of RRIDR

  Figure -15

Volume of Digital data

				5.2.10 Archiving research data for posterity is widely accepted especially in disciplines like Astronomy & Astrophysics and	Archiving Research Data
				nuclear physics. During this survey when we sought their opinion, 47% percent were not in favor of archiving, 40% supported archiving and the rest 13% left it unanswered. All those who are in favor, have voiced their choice as -self	3% 6%	10%	Self External agency
				archiving‖.	Computer
					group of the
					81%	institute Library
				5
				Small (< 50 GB)
	13%			Medium (50-100 GB)
	1% 0% 8%	41%	Large (101 GB -1 TB)
				Very large (1TB -
				1PB)
	20%			Massive
	17%		Am not sure
				Unanswerd
	Skills Required
				Programming Language
	5%	7%		Software
		6%	1%	Statistical analysis
				Programming Language &
		18%	Software
				Programming Language &
	50%			Statistical analysis
		5%		Software & Statistical analysis
		8%		Programming Language, Software
				& Statistical analysis
				unanswerd

.2.11 Using External data from

  repositories and published data sources are well accepted phenomenon in Science research. Majority of the graduate students (77%) in this survey are not using external data of any kind. The balance 23% (16 Respondents ) use external data. 14 out of 16 are from A& A group of research and rest 2 are from SCM group.

Raman Research Institute (RRI), a pioneering institute of research in physics was started by Noble Laureate Sir C V Raman in 1948 to carry forward his research soon after his retirement from Indian Institute of Science. This self-funded Institute became an autonomous research institute receiving grants from Department of Science and Technology, Government of India, in 1972, after the demise of its founder.

Data Intensive Research at Raman Research Institute:

A case study of challenges and perspectives employing questionnaire survey and ETD repository 2.


Dina Irofti
email: dina.irofti@l2s.centralesupelec.fr

Fatihcan M Atay
email: atay@member.ams.org

On the Delay Margin for Consensus in Directed Networks of Anticipatory Agents

Keywords: Consensus, prediction, anticipation, delay, network, graph theory, stability analysis

We consider a linear consensus problem involving a time delay that arises from predicting the future states of agents based on their past history. In case the agents are coupled in a connected and undirected network, the exact condition for consensus is that the delay be less than a constant threshold that is independent of the network topology or size. In directed networks, however, the situation is quite different. We show that the allowable maximum delay for consensus depends on the network topology in a nontrivial way. We study this delay margin in several network constellations, including various circulant networks with directed links. We show that the delay margin depends not only on the number of neighbors, but also on the directionality of connections with those neighbors. Furthermore, the delay margin improves as the circulant networks are rewired en route to a small-world configuration.

INTRODUCTION

Consensus and coordination problems arise in a wide range of applications where multi-agent systems interact to agree on a common value of a certain quantity of interest. We can cite here, among others, [START_REF] Lynch | Distributed Algorithms. CA: Morgan Kaufmann[END_REF] in distributed computing, [START_REF] Degroot | Reaching a consensus[END_REF] in management science, [START_REF] Vicsek | Novel type of phase transition in a system of self-driven particles[END_REF] in flocking and swarming theory, [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF] in distributed control, and Olfati-Saber and Shamma (2005) in sensor networks. The classical linear consensus problem can be formulated in the form ẋi = n j=1 a ij (x j (t) -x i (t)), i = 1, . . . , n

where n is the number of agents in the network, x i ∈ R is the state of the agent i at time t, which changes under the interaction with other agents, and a ij are nonnegative numbers describing the interaction strength between agents i and j. Consensus can then be formally defined as follows.

Definition 1. The system (1) is said to reach consensus if for any set of initial conditions {x i (0)} there exists c ∈ R such that lim t→∞ x i (t) = c for all i, in which case the number c is called the consensus value.

Under mild conditions related to the connectivity of the network, it can be shown that the system (1) reaches consensus from arbitrary initial conditions, and the consensus value equals the average of initial conditions of the agents. The problem becomes more interesting when the system involves a time delay τ , for example an information processing delay modeled by ẋi = n j=1

a ij (x j (t -τ ) -x i (t -τ )) (2)
which has been studied in Olfati-Saber and Murray (2004).

In this case, it is known that there exists an upper limit τ max such that the system (2) reaches consensus from arbitrary initial conditions if and only if τ < τ max (see, for instance, Olfati-Saber and Murray ( 2004)). Another model, which involves an information transmission delay, is given by (Moreau, 2004;[START_REF] Seuret | Consensus under communication delays[END_REF][START_REF] Atay | Consensus in networks under transmission delays and the normalized Laplacian[END_REF][START_REF] Atay | The consensus problem in networks with transmission delays[END_REF] ẋi

= n j=1 a ij (x j (t -τ ) -x i (t)) . (3)
It has been shown that such a system reaches consensus from arbitrary initial conditions regardless of the value of the delay τ as long as the network contains a spanning tree; however, the consensus value depends on the system's history in a nontrivial way [START_REF] Atay | Consensus in networks under transmission delays and the normalized Laplacian[END_REF][START_REF] Atay | The consensus problem in networks with transmission delays[END_REF].

In this paper we are concerned with a rather different source of delay, arising from the anticipatory nature of the agents. More precisely, we consider a network of intelligent agents who try to predict the future states of their neighbors in their interaction. Formulating in the context of system (1), agent i uses, instead of the current state x j (t) of its neighbor, a predicted value xj (t + τ ) of its future state, yielding

ẋi = n j=1 a ij (x j (t + τ ) -x i (t)), i = 1, . . . , n (4)
Using a first order estimation [START_REF] Atay | A delayed consensus algorithm in networks of anticipatory agents[END_REF], the prediction of the future can be done by a linear time

x j t -τ t t + τ
Fig. 1. Linear prediction of the future state x j (t + τ ) of an agent j using its present and past states.

extrapolation from past values, namely

xj (t + τ ) = x j (t) + x j (t) -x j (t -τ ) τ τ, = 2x j (t) -x j (t -τ ).
(5) (See Figure 1 for a graphical depiction.) Using ( 5) in (4), we arrive at the main model that we will study in this paper:

ẋi (t) = 1 d i n j=1 a ij (2x j (t) -x j (t -τ ) -x i (t)), (6)
Note that we have additionally normalized the summation term via dividing by the (generalized) degree d i of node i, d i = n j=1 a ij . This normalization gives rise to a normalized Laplace operator, which is a natural choice in several applications and has some advantageous properties, as will be briefly reviewed in Section 2. In particular, the normalization bounds the spectrum of Laplacian regardless of the network size, thus allowing comparison of networks of very different sizes.

When the network is undirected (i.e. a ij = a ji ∀i, j) and connected, it has been shown (Atay andIrofti, 2015, 2016) that system (6) reaches consensus from arbitrary initial conditions if and only if τ < 1. (7) In other words, in the undirected case, the maximum allowable delay for consensus in (6) (the delay margin) equals 1 regardless of the network topology. The situation for directed networks is different, however, as we show in this paper. In particular, the network topology turns out to play an important role in affecting the delay margin.

In the following, we first prove that (7) is a necessary condition for consensus, but it is not sufficient. Moreover, as already mentioned above, the undelayed network (τ = 0) always reaches consensus (as long as it contains a directed spanning tree). It follows by continuity that, sufficiently small delays will not destroy stability of the consensus. Hence, the delay margin for (6) is some positive number less than one. We calculate the locus of (complex) Laplacian eigenvalues that are detrimental for consensus. Just like undirected networks, many directed networks also turn out to enjoy (7) as the exact condition for consensus. However, we show that some specific networks that are actually frequently used in the literature do have much lower delay margins. We study these circulant networks in detail with respect to their Laplacian eigenvalues and determine their delay margins. We also consider random rewiring of circulant networks en route to small-world configuration and show that a few rewirings improve the delay margin already, although the improvement is not monotone with further rewirings.

DIRECTED NETWORKS AND CHARACTERISTIC ROOTS

A directed graph (or digraph) G = (V, E) consists of a finite set V of vertices and a set of directed edges E ⊂ V × V consisting of ordered pairs of vertices. We consider only simple, non-trivial graphs without self-loops or multiple edges. We denote by A = [a ij ] the (weighted) adjacency matrix of the graph, where a ij > 0 if there is a directed link from node j to node i, and a ij = 0 otherwise. The in-degree d i of node i is defined as

d i = n j=1 a ij , i.e.
, the sum of the elements of the i th row of A, and D = diag(d 1 , . . . , d n ) denotes the diagonal matrix of in-degrees.

Assuming that d i = 0 ∀i, the normalized Laplacian matrix is defined as

L = I n -D -1 A, ( 8
)
where n is the number of nodes in the network and I n is the identity matrix of size n. The normalized Laplacian L naturally arises in a class of important problems, in particular in random walks on networks, as D -1 A is the transition matrix for probability distributions arising from such walks [START_REF] Chung | Spectral graph theory[END_REF]).

An application of Gershgorin's theorem to the definition of L shows that the Laplacian eigenvalues λ k are complex numbers satisfying

|1 -λ k | ≤ 1, k = 1, 2, . . . , n.
(9) Furthermore, the first eigenvalue λ 1 is always zero and corresponds to the eigenvector (1, 1, . . . , 1) . In the special case of undirected networks the eigenvalues are all real, because D -1 A is similar to a symmetric matrix,

D -1 A = D -1/2 (D -1/2 AD -1/2 )D 1/2 , as A is symmetric and D is diagonal. In matrix form, (6) becomes ẋ(t) = D -1 A(2x(t) -x(t -τ )) -x(t), (10)
with x = (x 1 , x 2 , . . . , x n ) . Suppose that L has a complete set of eigenvectors {v k } corresponding to the eigenvalues {λ k }. Then one can write x(t) = n k=1 u k (t)v k for some scalar coefficients u k , which transforms (10) into a system of n decoupled scalar equations uk

(t) = (1 -2λ k ) u k (t) -(1 -λ k )u k (t -τ ), ( 11
) for k = 1, . . . , n. The characteristic equation correspond- ing to the eigenmode (11) is ψ k (s) := s -2(1 -λ k ) + 1 + (1 -λ k )e -sτ = 0, ( 12
) and the characteristic equation for the whole system (10) can be written as

Ψ(s) := n k=1 ψ k (s) = 0. ( 13
)
Note that s = 0 is always a characteristic root for the first factor ψ 1 (s) = s -1 + e -sτ corresponding to the first eigenmode, λ 1 = 0. Thus, points on the synchronization subspace spanned by v 1 = (1, 1, . . . , 1) can be at best neutrally stable. If that is the case, and in addition lim t→∞ u k (t) = 0 for all k ≥ 2, then the system converges to a point on v 1 , i.e. it reaches consensus, from arbitrary initial conditions. This clearly happens if and only if zero is a simple root of Ψ and all other roots of Ψ have negative real parts.

UPPER BOUND FOR THE DELAY MARGIN

Based on the considerations of the previous section, we can state a necessary condition for consensus, which serves as an upper bound for the delay margin. Theorem 1. The inequality τ < 1 ( 14) is a necessary condition for (6) to reach consensus from arbitrary initial conditions. In other words, the delay margin for the stability of consensus in ( 6) is at most 1.

Proof. Consider the first factor ψ 1 (s) = s -1 + e -sτ in (13). Since ψ 1 (0) = 0 and ψ 1 (0) = 1 -τ , zero is always a root of ψ 1 and is a simple root unless τ = 1. For τ > 1, on the other hand, ψ 1 is unstable; in fact, it has a real and positive root, which can easily be seen by plotting the real functions 1-s and e -sτ and observing that they must intersect at a positive value of s if τ > 1. Therefore, ( 14) is a necessary condition that ψ 1 , and hence the characteristic equation ( 13), has a simple zero root and all the remaining roots have strictly negative real parts.

As remarked in the introduction, unlike the case of undirected networks, the condition ( 14) is not sufficient for consensus in directed networks. The distinction clearly lies in the fact that the Laplacian eigenvalues are real for undirected networks but not necessarily for directed ones. It is therefore of interest to have some understanding of the network structures and the corresponding Laplacian eigenvalues that give rise to reduced delay margins. We take up this task in the next section.

LAPLACIAN EIGENVALUES RESPONSIBLE FOR INSTABILITY

To understand the role of Laplacian eigenvalues on the stability of consensus, we consider a generic factor of the characteristic equation ( 12)-( 13), namely ψ(s) := s -2(1 -λ) + 1 + (1 -λ)e -sτ = 0 (15) for an arbitrary λ ∈ C that might have come from the spectrum of the Laplacian of some network. The latter condition requires |1 -λ| ≤ 1 (16) in view of (9). The stability of (15) can be conveniently established using the Lambert W function, as follows.

Recall that the Lambert W function is defined as the inverse function of the mapping z → ze z for z ∈ C (see [START_REF] Corless | On the Lambert W function[END_REF]). In other words, the (multi-valued) function W (z) satisfies Fig. 2. Fraction of complex numbers λ that make the characteristic equation ( 15) unstable, from among 10 6 randomly selected complex numbers inside the shifted unit circle ( 16).

W (z)e W (z) = z. ( 17
Letting a = 2(1 -λ) -1 and b = -(1 -λ), the roots of ψ satisfy s = a + be -sτ , which can be re-written as s -a = be -(s-a)τ e -aτ . A change of variables s → (s -a)τ gives s = τ be -aτ e -s , from which, by the definition ( 17), one has s = W (τ be -aτ ). In the original variable s, the roots of ψ satisfy s = 1 τ W (τ be -aτ ) + a.

Furthermore, by a result of [START_REF] Shinozaki | Robust stability analysis of linear time-delay systems by Lambert W function: Some extreme point results[END_REF], the root having the largest real part is given by the principal branch W 0 of the Lambert function. Hence, for the stability of (15) it suffices to calculate

s = 1 τ W 0 (τ be -aτ ) + a, (18)
and check whether its real part is negative.

As a first step in numerical investigations, we randomly generate complex numbers λ satisfying ( 16) and check the stability of ( 15) using ( 18) for the range of delay values τ ∈ (0, 1). The results of the experiment are shown in Figure 2 for one million randomly generated λ. It can be seen that only a small fraction of λ actually yield an unstable ψ.

To gain further insight into the nature of instability, we plot in Figure 3 the location of the stable and unstable Laplacian eigenvalues λ in the complex plane. As expected, for sufficiently small delays consensus is stable, and as the delay increases, two regions of unstable eigenvalues (shown with red color) grow from the circle boundary, eventually meeting at λ = 0 as τ ↑ 1 (Figure 3 (a) through (d)).

The value τ = 1 is the upper value of allowable delay for any network, since at this value the characteristic equation ( 15) has a double zero root (see Theorem 1 and its proof).

At first it might appear from Figures 2-3 that most networks would have a delay margin equal to 1. While this is true in a certain sense, a class of networks commonly used in the literature does turn out to belong to the exceptional unstable category. In the next section we investigate in detail these networks that have a poor delay tolerance.

DIRECTED CIRCULANT NETWORKS AS PROTOTYPES FOR WORST DELAY TOLERANCE

An important observation from Figure 3 is that the red regions for unstable Laplacian eigenvalues border on the shifted unit circle (16), and in fact disappear by merging into the circle border near the origin as τ is decreased. Therefore, networks having the worst delay tolerance must be those that have a Laplacian eigenvalue λ on the circle near the origin. Therefore, as the next step in our investigation, we consider a network configuration that has all its eigenvalues on the circle, namely the directed cycle shown in Figure 4(a). We refer to this configuration as Network A. Indeed, it is easy to calculate that the characteristic polynomial for the Laplacian for Network A is

(1 -λ) n = 1; thus the eigenvalues λ k are roots of unity shifted by one: 4(e). It follows that for very small network sizes n, the eigenvalues are outside the red region of Figure 3 (although they are still all on the circle); so Network A will have a delay margin equal to 1. As the network size n becomes larger, the eigenvalues move into the red region and the delay margin decreases, as shown in Figure 5.

λ k = 1 -exp(2πi(k -1)/n), k = 1, . . . , n; see Figure
We next consider circulant networks with in-degree equal to two. Here we have two natural variations as extensions of Network A: either all connections follow the same direction, or connections of distance one and two have opposite directions. We refer to these networks as Network B and Network C, respectively; see Figure 4 (b)-(c).

For directed circulant networks with three connections per node, there are more possibilities for the choice of connection directions. We only show one possibility in Figure 4(d), where all connections point in the same direction. We refer to this connection topology as Network D.

The Laplacian eigenvalues of network types B, C and D no longer fall on the shifted unit circle ( 16), as those of Network A did. Nevertheless, the locus of eigenvalues still comes close to the circle near the origin, as shown in Figure 4(f)-(h). Hence, we have a similar conclusion: up to a certain network size, these networks have a delay margin of 1, and for larger sizes the delay tolerance of the networks decreases. This critical network size, for which the delay margin equals 1, is smallest for Network A and largest for Network C, as confirmed by the horizontal segments of the curves in Figure 5. Interestingly, Network C has a much higher delay tolerance than B, although all nodes in both networks have the same in-degrees.

We also test the accuracy of the curves in Figure 5 by direct simulation of the system (6). We pick τ = 0.9 and choose network sizes close to the curves in Figure 5. Starting from random initial conditions, the time evolution of ( 6) is depicted in Figure 6. The simulations confirm that there exists a critical network size, whose value agrees with the information in Figure 5, below which the systems reaches consensus and above which it diverges.

As a final step in our investigation, we consider random re-wiring of directed circulant networks, in a well-known process that produces small-world networks starting from regular ones [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF]. In a directed version of this re-wiring algorithm, we start with a network of type B or C, and at each step we randomly pick one node and reconnect one of existing outgoing links to a different, randomly-selected node. Figure 7 shows that the delay margin improves with re-wiring (although the effect is not always monotone, due to the randomness of the process and the relative sparsity of the network). This shows from yet another viewpoint that directed circulant networks, especially at large sizes, have poor delay tolerance, which can be improved significantly by a few re-wiring of links.

  Fig. 3. Values of random complex numbers λ inside the shifted unit circle, colored according to the stability of the characteristic equation (15). Red points represent unstable values and blue points represent stable values of λ, shown for four different values of the delay.

  (a) Circulant network A: every node is connected to the next one counterclockwise. (b) Circulant network B: every node is connected to the next two nodes counterclockwise. (c) Circulant network C: similar to B, except the outer connection direction is counterclockwise and the inner direction is clockwise. (d) Circular network D: every node is connected to the next three nodes counterclockwise.

Fig. 4 .

 4 Fig. 4. Example of directed circulant networks (upper row) and their Laplacian eigenvalue patterns (lower row). The eigenvalues are plotted in the complex plane for the network sizes n = 20 (black dots), and n = 100 (blue crosses); the red circle depicts the shifted unit circle (16) in the complex plane.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Delay margin as a function of the network size n for various types of circulant networks: type A (blue circles), type B (green points), type C (red crosses), and type D (purple triangles).

Fig. 7 .

 7 Fig. 7. Improving the delay margin of circulant networks by random rewiring of the links.

  )

		0.09
		0.08
		0.07
	Fraction unstable	0.03 0.04 0.05 0.06
		0.02
		0.01
		0.6 0	0.7	0.8 τ	0.9	1

ACKNOWLEDGEMENTS

The authors thank the ZiF (Center for Interdisciplinary Research) of Bielefeld University for their support within the cooperation program Discrete and Continuous Models in the Theory of Networks, where this problem was initiated. Part of the research was carried out when DI visited the Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, in Summer 2015 under financial support from DAAD Short-Term Grants 2015 (57130097) and from L2S Laboratory, France. FMA acknowledges support from the European Union's Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 318723 (MATHEMACS).