László Á Gergely
email: gergely@physx.u-szeged.hu

Peter L Biermann
email: plbiermann@mpifr-bonn.mpg.de

Laurent ¸iu

I Caramete
email: lcaramete@gmail.com

Supermassive black hole spin-flip during the inspiral

HAL is

Introduction

During galaxy mergers, following a regime of slow approach due to dynamical friction, eventually the central supermassive black holes (SMBHs) approach each other to a separation of the order of 10 3 Schwarzschild radii, when gravitational radiation takes over as the leading order dissipative effect. The Laser Interferometer Space Antenna (LISA, see [1]) is expected to detect merging binary SMBHs with masses m 1 + m 2 ≤ 10 7 solar masses (M ) up to redshift z ≈ 30. A post-Newtonian approach is well suited to describe their forthcoming inspiral, a regime we define in terms of the post-Newtonian (PN) parameter ε = Gm/c 2 r ≈ v 2 /c 2 ∈ (ε in = 10 -3 , ε fin = 10 -1 ), where r and v characterize the orbital separation (from the center of mass) and speed of the reduced mass particle, G is the gravitational constant and c the speed of light. Various corrections to the conservative dynamics add up to 2 PN, while the gravitational radiation results in dissipation of energy, angular momentum and orbital angular momentum at 2.5 PN.

The leading order conservative correction to the Newtonian dynamics in a compact binary, which results in a change of the orbital plane (defined by the direction LN of the Newtonian orbital angular momentum L N = µr×v of the reduced mass particle µ) is the spin-orbit (SO) interaction [2], [START_REF] Apostolatos | [END_REF]. The precessional time-scale (the time during which the normal to the orbit LN undergoes a full rotation) is longer than the orbital period, however shorter than the characteristic time-scale of gravitational radiation (defined as L/ L, where L is the magnitude of the total orbital angular momentum). Combined with the leading order gravitational radiation backreaction averaged over one quasicircular orbit, the SO correction provides a fair approximation to orbital dynamics, explored in Refs. [START_REF] Apostolatos | [END_REF], [4].

X-shaped radio galaxies (XRGs) exhibit two pairs of radio lobes and jets [5], [6]. A recent review [START_REF] Gopal-Krishna | On the origin of X-shaped radio galaxies[END_REF] summarizes the four different models explaining XRGs: galaxy harbouring twin AGNs, back-flow diversion models, rapid jet reorientation models, finally a new jet-shell interaction model. A large subset of the observations (excepting cases, when the jets are aligned with the optical axes of the host ellipticals [START_REF] Battye | [END_REF]) are well-explained by the jet reorientation model, which in turn implies a spin-flip [5], [9] of the dominant black hole.

The details of how this would occur were worked out in Ref. [4]. A key element was the determination of the typical mass ratio at SMBH mergers by a series of estimates, which resulted in mass ratios ν = m 2 /m 1 = 1/30 to ν = 1/3. Because the spin scales with the mass squared, the second spin was neglected and only the dominant spin S 1 (with magnitude S 1 ) kept. We summarize the consequences of this model as follows.

a) For the typical mass ratio the dominance of L over S 1 is reversed as the separation in the binary decreases throughout the inspiral. In the last stages of the inspiral the spin dominates over the orbital angular momentum S 1 L. b) The angle α between the orbital angular momentum and total angular momentum J (with magnitude J), also the angle β between the dominant spin and total angular momentum evolve as:

α = - L J sin α > 0 , (1)
β = L J sin α < 0 . (2)
c) The approximate expression relating α to the post-Newtonian parameter ε, mass ratio ν and initial angle α + β span by the dominant spin with the orbital angular momentum (this angle being a constant during the inspiral) is:

tan α ≈ sin (α + β) ε -1/2 ν + cos (α + β) . (3)
(In Eq. ( 41) of Ref. [4] the left hand side was given as sin 2α/ (1 + cos 2α).)

In a criticism to the work presented in Ref. [4], Gopakumar recently argued that "it is unlikely that the spin-flip phenomenon will occur during the binary black hole inspiral phase" [START_REF] Gopakumar | invited and highlight talk presented at the Workshop on Massive Black Hole Binaries and Their Coalescence in Galactic Nuclei[END_REF]. This misconception comes from mixing up the instantaneous change in the direction of the total angular momentum, d Ĵ/dt = L/J L -L • Ĵ Ĵ = 0 with its averaged expression d Ĵ/dt = 0 over the precessional time-scale. The angles α and β are not constants during the post-Newtonian evolution, as claimed in Ref. [START_REF] Gopakumar | invited and highlight talk presented at the Workshop on Massive Black Hole Binaries and Their Coalescence in Galactic Nuclei[END_REF], they rather change as given in Eqs. ( 1)- (2). ‡

In the present paper we revisit some of the arguments of the spin-flip mechanism and also provide more details on it as compared to Ref [4]. In Section 2 we revisit the typical mass ratio argument, following a recent statistics of supermassive black hole candidates, resulting a newly established mass distribution. We comment on how these findings would affect the typical mass ratio range at SMBH encounters. In Section 3 we analyze how the spin-flip angle depends on the mass ratio and relative orientation of the spin and orbital angular momentum. We also derive a formula for the final spin during the inspiral. Finally we present our Concluding Remarks.

The sky in black holes: new statistics, consequences for the mass ratio at SMBH encounters and chances of the spin-flip during the inspiral

First we summarize the arguments of Ref. [4] on the mass ratios at SMBH encounters. The mass distribution Φ BH (m) of the galactic central SMBHs in the mass range 3 × 10 6 ÷ 3 × 10 9 M is well described by a power-law with an exponential cutoff, but for our purposes can be adequately approximated by a broken power-law [START_REF] Press | [END_REF]- [13] (confirmed by an observational survey [14]). The break is at about 10 8 M . In agreement with these arguments and observations we assume Φ BH (m) ∝ m -k , with k ∈ (1, 2) below, and Φ BH (m) ∝ m -h , with h ≥ 3 above the break. Then the probability for a specific mass ratio arose as an integral over the black hole mass distribution, folded with ‡ Only when the total and orbital angular momenta are aligned, become the angles α and β individually constant, as they identically vanish. Therefore in the aligned configuration no spin-flip could ever occur by the combined mechanism of SO precession and gravitational radiation.

the rate F to merge, and by adopting the lower values of the exponents. For the merger rate we assumed that it scales with the capture cross section S (the dependence on the relative velocity of the two galaxies was neglected, as the universe is not old enough for mass segregation). For the capture cross-section we assumed S ∝ ν -1/2 , motivated by the following arguments:

• for galaxies an increase with a factor of 10 in radius (10 2 in cross-section) accounts for an increase with a factor of 10 4 in mass (from the comparison of our Galaxy with dwarf spheroidals [15]- [16],

• there is a well established correlation between the SMBH mass and the mass of the host bulge [17],

• the mass of the central SMBH scales with both the spheroidal galaxy mass component and the total, dark matter dominated mass of a galaxy [18].

As a result of these considerations we have found that most likely the mass ratio is in the range ν ∈ (1/30 , 1/3). A typical value to consider would be ν = 1/10, thus one of the SMBHs being 10 times as massive as the other.

New work on the statistical analysis of 5,895 NED candidate sources [START_REF] Caramete | The mass function of nearby black hole candidates[END_REF] has been carried out in the mass range from 10 5 M to above 10 9 M . Below about 10 6 M all candidates are probably compact star clusters, however the rest are likely SMBHs. This work shows that the SMBH mass function is a broken power law with M -2 BH at low masses, and M -3

BH at high masses, with a break near 1.25 × 10 8 M ; this general behaviour has been long known, and has now been rederived with a very large sample.

The key difference with respect to previous work was the careful attention paid in order to have equal probability for detecting a SMBH in a galaxy, regardless to the Hubble type. The mass distribution of the SMBHs is represented on Fig 1 . This particular distribution can be interpreted in the context of the merger model [START_REF] Silk | [END_REF] with a merger rate scaling as (mass) +2 , very much stronger than what we favored in Ref [4]. The extreme mass dependence describes well a M -3 black hole mass distribution consistent with the high end of the mass distribution; on the other hand a mass dependence of the merger rate close to (mass) +4/3 , suggested by gravitational focusing arguments [START_REF] Silk | [END_REF] describes well the lower mass distribution nearer to M -2 . It remains to be seen, whether all details of the mass function can be understood using either of these mass ratio dependences. Of course these simple merger rate calculations assume an environment without cosmological expansion. However, for the densest part of the cosmos the local expansion is very weak [21], and that is where most of the mergers happen.

However, for the determination of the typical mass ratio the essential result is only slightly changed. A merger rate running with mass +2 analytically gives a M -3 BH mass function (see [START_REF] Silk | [END_REF]), as observed; we use this rate to estimate here the typical mass ratios for high BH masses. Redoing the integrals of Section 2 of Ref. [4] with k = 2, h = 3 (denoted there α, β) and ξ = 2 (as in [START_REF] Silk | [END_REF], so much more extreme than what was assumed in [4]), then all four integrals are still dominated by the lower bound; only the second of the integrals has q ≡ ν -1 in its lower bound, and so the four integrals have the q-dependencies of q 0 , q +1 , q -1 and again q -1 . We can ignore the second integral, since it all refers to lower masses merging with lower masses. The most important integrals are those combining a SMBH above the break with a SMBH either below or above the break. Then the distribution in q is found as q -1 , a logarithmically even distribution in (dq)/q over a range of q from 1 to 1000, so a logarithmic average of 30. Weighting the two parts of the distribution, the larger mass ratios are favored, which would skew the logarithmic average of the mass ratio to q > 30, thus ν < 0.03.

The logarithmically even distribution means that the mass ratio ranges ν from 1 to 1/3, from 1/3 to 1/10, from 1/10 to 1/30, from 1/30 to 1/100, from 1/100 to 1/300 and finally from 1/300 to 1/1000 are roughly equally likely. A glance at Table 1 of Ref. [4] shows, that concerning the behaviour of the ratio of the dominant spin and orbital angular momentum magnitudes, we have three regimes:

(1) ν ∈ (1/3, 1) when S 1 < L throughout the inspiral, (2) ν ∈ (1/30, 1/3) when the initial S 1 < L is reversed to S 1 > L during the inspiral and

(3) ν ∈ (1/1000, 1/30) when S 1 > L holds throughout the inspiral.

For the mass ratio ranges ( 1) and ( 3) no spin-flip can occur during the inspiral, while for (2) it should. For (1) there is chance for a spin-flip to occur during the plunge, as some numerical simulations have already found this for equal masses [22]. For (3) by contrast there is no possibility for a spin-flip by the combined mechanism of SO precession and gravitational radiation. These mass ratio ranges then occur with (1) 20%, (2) 40% and (3) again 40% probability. This means that the spin-flip still typically occurs during the inspiral.

Spin-flip angle distribution

In this section we will present an analysis of the spin-flip angle occurring during the inspiral phase in the mass ratio range ν ∈ (1/30, 1/3) as a function of the mass ratio.

The spin-flip model be understood as follows. Initially the galactic SMBH has conserved spin, along which the primary jet can form. When the two galaxies collide, the SO induced spin precession starts, while gravitational radiation is diminishing the orbital angular momentum. The direction of the total angular momentum stays unchanged. The constancy of Ĵ over the precessional time-scale is due to the fact, that the change in the total angular momentum J = LL is about the orbital angular momentum, which (disregarding gravitational radiation) undergoes a precessional motion about J. This shows that the averaged change in J is along J (simple precession, [START_REF] Apostolatos | [END_REF]). This conclusion, however, depends strongly on whether the precessional angular frequency Ω p is larger than α and β. Indeed, if these are comparable, the component perpendicular to J in the change J = LL will not average out during one precessional cycle, as due to the increase of α it can significantly differ at the beginning and at the end of the same precessional cycle. Such a situation would occur, when the spin and the orbital angular momentum are of comparable magnitude (S 1 ≈ L, a regime through which a binary with typical mass ratio will pass through during the inspiral) and also roughly aligned, a regime known as transitional precession. During simple precession Eqs. ( 1)-( 2) governing the evolution of the angles α and β also hold in an average sense over the precessional time-scale. In what follows, we assume simple precession.

The magnitude of the spin is unaffected by gravitational radiation, therefore by the simple rule of addition of vectors the spin has to align close to the Ĵ direction. The second jet then can start to form. In the intermediate phase when the spin precesses, instead of jet formation the precessing magnetic field creates a wind, sweeping away the base of the old jet, which in many cases can be observed.

Spin and orbital angular momentum orientations, final spin formula

The key equation to start with is Eq. ( 3). In order to see the validity of this equation, also to generalize it to the cases of non-extreme rotation, we need to evaluate

S 1 ≈ m 1 RV 1 ≈ m 1 Gm 1 c 2 c V 1 c ≈ G c m 2 1 χ 1 , L ≈ L N ≈ µrv = G c v c c 2 r Gm µm = G c ε -1/2 m 1 m 2 = G c m 2 1 ε -1/2 ν . ( 4
)
Here V is some characteristic rotational velocity, R the radius of the SMBH (of the order of its Schwarzschild radius) and χ 1 ∈ (0, 1) is the dimensionless (χ 1 = 1 for extreme rotation). Therefore

S 1 L ≈ χ 1 ε 1/2 ν -1 . ( 5
)
Next we express J/S 1 first from J = L cos α + S 1 cos β by rewriting β = (α + β) -α, and secondly from the equality of the projections perpendicular to L of the total and spin angular momenta J sin α = S 1 sin (α + β), so that we can equal them. By also employing Eq. ( 5) and basic trigonometry we obtain

tan α ≈ sin (α + β) χ -1 1 ε -1/2 ν + cos (α + β) . ( 6
)
This is the desired generalization of Eq. ( 3).

It is worth to note that when applied to the final configuration ε fin , Eq. ( 6) also stands as a formula for the final spin at the end of the inspiral, giving the polar angle of the final spin α fin with respect to the axis Ĵ in terms of the mass ratio, spin magnitude and angle span with the orbital angular momentum. Related formulae based on numerical runs were advanced in Refs. [23]. These results are not immediate to compare with ours, as Eq. ( 6) should be applied at the end of the inspiral; although in the mass ratio range where it is valid, one would intuitively expect that as not much orbital angular momentum is left at the end of the inspiral in comparison with the dominant spin, the direction of the latter will not be significantly changed during the plunge.

Particular cases.

There are three particular configurations worth to mention:

i) The spin is aligned with the orbital angular momentum: α + β = 0, thus from Eq. ( 6) α = 0 and there is no room for any spin-flip. This would be the situation for perfectly wet mergers, which align the spin with the orbital angular momentum.

ii) The spin is anti-aligned with the orbital angular momentum, α + β = π. Therefore depending on which of the S 1 and L are larger, the angle α is either 0 or π.

iii) For the parameter ranges when the denominator vanishes α + β = arccos -χ -1 1 ε -1/2 ν , from Eq. ( 6) we obtain α = π/2, therefore β is also determined.

Discussion as function of mass ratios.

Keeping in mind that due to Eqs. ( 1)-( 2) the angle α+β is a constant during the inspiral (a parameter), and the dimensionless spin χ 1 behaves similarly, also regarding the mass ratio as a third parameter characterizing the particular merger, the angle α in general remains a function of ε, thus it evolves together with the orbital separation r and velocity v.

For the mass ratio ν = 1/10 we have (

S 1 /L) in ≈ χ 1 ε 1/2 in ν -1 = 0.316χ 1 and (S 1 /L) fin ≈ χ 1 ε 1/2 fin ν -1 = 3.162χ 1 .
As tan α ≤ S 1 /L and tan β ≤ L/S 1 (the equalities arising when the spin and orbital angular momentum are perpendicular) we have tan α in ≤ 0.316χ 1 and tan β fin ≤ 0.316χ -1

1 . For extreme rotation (χ 1 = 1) we obtain α in , β fin ≤ 0.306 = 17.541 • .

For ν = 1/3 we obtain tan α in ≤ (S 1 /L) in ≈ χ 1 ε 1/2 in ν -1 = 0.095χ 1 and for extreme rotation α in ≤ 0.095 = 5.44 • . In fact at the beginning of the inspiral this latter condition (meaning that the orbital angular momentum is roughly the total angular momentum) holds in the whole mass ratio range ν ∈ [1/3, 1]. Under these conditions Eq. ( 6) can be approximated as

α in ≈ χ 1 ε 1/2 in ν -1 sin β in = 0.032χ 1 ν -1 sin β in . (7)
For ν = 1/30 we obtain tan

β fin ≤ (L/S 1 ) fin ≈ χ -1 1 ε -1/2 fin ν = 0.105χ -1 1
and for extreme rotation β fin ≤ 0.105 = 6.04 • . In fact at the beginning of the inspiral this latter condition (meaning that the dominant spin is roughly the total angular momentum) holds in the whole range ν ∈ [1/1000, 1/30]. Under these conditions Eq. ( 6) can be expanded (to first order in β fin , with χ -1 1 ε

-1/2
fin ν of the order of β) as

β fin ≈ χ -1 1 ε -1/2 fin ν sin α fin = 3.162χ -1 1 ν sin α fin . (8)
For slowly rotating SMBHs with χ 1 ≈ 0.1 the above formula would hold only in the range ν ∈ [1/1000, 1/300] .

The spin-flip angle during the inspiral

A minimal value for the spin-flip angle σ arises by forming the difference between the angles β, characterizing the orientation of the spin with respect to the inertial direction Ĵ. Thus

σ min = β in -β fin = α fin -α in . (9)
In the second equality we have used that α in + β in = α fin + β fin . However we have to take into account, that the above is only true in a 2-dimensional picture. In reality the 3-dimensional SO precession will complicate the situation, and the above angle emerges only if the number of precessions during the inspiral is an integer multiple of 2π. If instead is of the type (2k + 1) π the spin-flip angle will be maximal, to be calculated as

σ max = β in + β fin -lπ = 2 (α in + β in ) -lπ -(α in + α fin ) , (10)
where

l = 0 if β in + β fin ≤ π and l = 1 if π < β in + β fin < 2π.
The difference between σ max and σ min is due to the fact, that the realignment of the spin along Ĵ is not perfect. The closer S fin 1 is to Ĵ, the less their difference ought to be due a more perfect alignement, therefore σ max -σ min = β fin -β in should go to 0 with decreasing ν.

For generic mass ratios ν ∈ (1/30, 1/3), Eqs. ( 6), ( 9) and [START_REF] Gopakumar | invited and highlight talk presented at the Workshop on Massive Black Hole Binaries and Their Coalescence in Galactic Nuclei[END_REF] give the range of allowed spin-flip angle for each relative orientation α + β of the spin with respect to the plane of motion and each χ 1 . The generic numerical solution for σ min in the case χ 1 = 1 is represented on Fig 2 as function of the relative orientation of the spin and orbital angular momentum α + β and mass ratio ν. For a given mass ratio the spin-flip angle has a maximum shifted from π/2 towards the anti-aligned configurations. The figure confirms the prediction, that significant spin-flip will happen during the inspiral in the mass ratio range ν ∈ (1/30, 1/3). For mass ratios smaller than 1/100 the spin does not flip at all, as the infalling SMBH acts as a test particle. For a given mass ratio the spin-flip angle has a maximum shifted from π/2 towards the anti-aligned configurations. The mass ratios ν = 1; 1/3; 1/30 and 1/1000 are located on the log ν -1 axis at 0; 1.09; 3.40 and 6.91, respectively, confirming the prediction, that a significant spin-flip will happen in the mass ratio range ν ∈ (1/30, 1/3). For mass ratios smaller than 1/100 the spin does not flip at all, as the infalling SMBH acts as a test particle.

Concluding Remarks

In light of the new data on a large sample of SMBH candidates we have established that the mass ratios obey an even logarithmic distribution in ν. In the mass ratio range ν ∈ (1/30, 1/3) of SMBH mergers representing 40% of all possible cases, we have investigated the SO precession driven conservative and gravitational radiation driven dissipative contributions to the orbital evolution during the inspiral, averaged over the precession time-scale. In this mass range the ratio of the dominant spin magnitude and orbital angular momentum magnitude S/L changes from less than one to larger than one during the inspiral. As the direction of the total angular momentum is unchanged on all time-scales larger than the precession time-scale, while the magnitude of the the orbital angular momentum decreases due to gravitational radiation and the magnitude of the spin stays constant, the spin direction has to change. The spin-flip of the dominant spin therefore occurs during the inspiral. If jet activity is involved, X-shaped radio galaxies arise by this mechanism and a large set of observations on X-shaped radio galaxies could be explained.

In another 40% of the mergers with mass ratios ν ∈ (1/30, 1/1000) the spin-flip never happens by this mechanism, while in the remaining 20% of mergers with mass ratios ν ∈ (1/3, 1) it may occur during the plunge.

SMBH mergers of equal mass to ν = 1/3 are only half as likely as the mass ratios 1/30 to 1/3, therefore the occurrence of the spin-flip can be considered typical during the inspiral. We analyzed the magnitude of the spin-flip angle occurring during the inspiral as function of the mass ratio and original relative orientation of the spin and orbital angular momentum and supported by numerical analysis the theoretical prediction (Fig 2). We also derived a formula for the final spin at the end of the inspiral in this mass ratio range.

During the inspiral the following relations among the relevant time-scales hold: tilt / spin-flip time-scale ≥ inspiral time-scale precession time-scale orbital time-scale (for all mass ratios in the typical range). Interestingly enough, the spin-flip time-scale for a typical mass ratio of 1/10 is only about three years, while the precession time-scale is less then a day [4]. Thus rapidly rotating relativistic jets coming close to our line of sight could produce significant variability at all wavelengths years before the coalescence. Therefore electromagnetic counterparts / precursors to the strongest gravitational wave emission are also likely to occur.

Figure 1 .

 1 Figure 1. Aitoff projection in galactic coordinates of 5,895 NED SMBH candidate sources. The complete sample is complete in a sensitivity sense, in order to derive densities one needs a volume correction. The color code is Orange, Green, Blue, Red, Black corresponding to masses above 10 5 M , 10 6 M , 10 7 M , 10 8 M , 10 9 M , respectively. With the exception of the less numerous first range (Orange), representing compact star clusters, the rest are SMBHs.

Figure 2 .

 2 Figure 2. The spin-flip angle σ min as function of the relative orientation of the spin and orbital angular momentum α + β (a constant during inspiral), and mass ratio ν.For a given mass ratio the spin-flip angle has a maximum shifted from π/2 towards the anti-aligned configurations. The mass ratios ν = 1; 1/3; 1/30 and 1/1000 are located on the log ν -1 axis at 0; 1.09; 3.40 and 6.91, respectively, confirming the prediction, that a significant spin-flip will happen in the mass ratio range ν ∈ (1/30, 1/3). For mass ratios smaller than 1/100 the spin does not flip at all, as the infalling SMBH acts as a test particle.

Acknowledgements

This work was supported by the Polányi Program of the Hungarian National Office for Research and Technology (NKTH), the Hungarian Scientific Research Fund (OTKA) grant 69036 and the COST Action MP0905 (L ÁG); AUGER membership and theory grant 05 CU 5PD 1/2 via DESY/BMBF and VIHKOS (PLB); and contracts CNCSIS 539/2009 and CNMP 82077/2008 (LIC).


Piotr Bizoń

Arthur Wasserman

Remark on non-existence of self-similar solutions in 4 + 1 vacuum gravitational collapse

We give a very short proof that the vacuum Einstein equations in 4 + 1 dimensions have no cohomogeneity-two Bianchi IX continuously self-similar solutions.

Introduction. In a recent paper [1] it was shown that in five spacetime dimensions one can perform a consistent cohomogeneity-two symmetry reduction of the vacuum Einstein equations which -in contrast to the spherically symmetric reduction -admits time dependent asymptotically flat solutions. The key idea was to modify the standard spherically symmetric ansatz by replacing the round metric on the three-sphere with the homogeneously squashed metric, thereby breaking the SO(4) isometry to SO(3) × U (1). In this way the squashing parameter becomes a dynamical degree of freedom and Birkhoff's theorem is evaded. This model (which we shall refer to as the BCS model) provides a simple theoretical setting for studying the dynamics of gravitational collapse in vacuum. Numerical simulations indicate that the spherically symmetric solutions, Minkowski and Schwarzschild, play the role of attractors in the evolution of generic regular initial data (small and large ones, respectively) and the transition between these two outcomes of evolution exhibits a discretely self-similar critical behavior [1]. In this respect the BCS model is very similar to the Einstein-massless scalar field system [2][3][4]. However, there is one interesting difference between these two models which we want to point out here. The difference is concerned with the existence of continuously self-similar (CSS) solutions. In [5] Christodoulou proved that the Einstein-massless scalar field system possesses CSS solutions. These solutions, suitably truncated, provide examples of naked singularities developing from regular initial data (however, being unstable [6], they do not contradict the weak cosmic censorship conjecture). We will show below that the BCS model has no CSS solutions. This result indicates that the CSS naked singularities found by Christodoulou for the self-gravitating massless scalar field are, in a sense, matter generated (mathematically, they are due to the fact that only derivatives of the scalar field appear in the equations).

The BCS ansatz and self-similarity. After [1] we parametrize the metric as follows

ds 2 = -Ae -2δ dt 2 + A -1 dr 2 + 1 4 r 2 e 2B (σ 2 1 + σ 2 2 ) + e -4B σ 2 3 , (1)
where A, δ, and B are functions of (t, r), and σ i are left invariant one-forms on SU (2) which in terms of the Euler angles take the form σ 1 = cos ψ dθ + sin ψ sin θ dφ, σ 2 =sin ψ dθ + cos ψ sin θ dφ, σ 3 = dψ + cos θ dφ.

(

) 2
Substituting this ansatz into the vacuum Einstein equations we get the following system of PDEs

∂ r A = - 2A r + 2 3r 4e -2B -e -8B -2r e 2δ A -1 (∂ t B) 2 + A(∂ r B) 2 , (3)
∂ t A = -4rA (∂ t B) (∂ r B) , (4)
∂ r δ = -2r e 2δ A -2 (∂ t B) 2 + (∂ r B) 2 , (5)
∂ t e δ A -1 r 3 ∂ t B = ∂ r e -δ Ar 3 ∂ r B + 4 3 e -δ r e -2B -e -8B . (6)
These equations have the scaling symmetry (t, r) → (λt, λr) so it is natural to look for continuously self-similar (CSS) solutions, that is solutions which are scale invariant. Such solutions depend on a single variable ρ = r/t and then the system (3-6) reduces to ordinary differential equations (where prime is d/dρ and Z = e δ ρ/A)

ρA = -2A + 2 3 4e -2B -e -8B -2ρ 2 A(1 + Z 2 )B 2 , (7)
A = -4ρAB 2 , ( 8
) ρZ = Z + 2Z(1 -Z 2 )ρ 2 B 2 , (9) B = (2Z 2 -3)B + 2ρ 2 (1 -Z 4 )B 3 ρ(1 -Z 2 ) + 4 3 e -2B -e -8B ρ 2 A(1 -Z 2 ) . ( 10
)
Confidential: not for distribution. Submitted to IOP Publishing for peer review 17 March 2010

The combination of equations ( 7) and ( 8) yields the constraint

3A -3A(1 -Z 2 )ρ 2 B 2 -4e -2B + e -8B = 0. (11)
We are interested in regular solutions, where 'regular' means twice continuously differentiable. At the origin regular solutions must satisfy the following initial conditions

B(ρ) ∼ bρ 2 , A(ρ) ∼ 1 -4b 2 ρ 4 , Z(ρ) ∼ ρ , (12)
where we used the remaining scaling freedom to set Z (0) = 1 for convenience. It follows from (12) and Eq.( 9) that if Z < 1 then Z(ρ) ≥ ρ, hence there is an ρ 0 such that Z(ρ 0 ) = 1. Geometrically, ρ 0 corresponds to the similarity horizon (the light cone of the singularity).

Proof. We will show that solutions starting from initial conditions (12) cannot be regular at ρ 0 . Assume for contradiction that the solution (A(ρ), Z(ρ), B(ρ)) is regular on the closed interval I = {ρ : 0 ≤ ρ ≤ ρ 0 }. First, note that the function A is positive on I since from Eq.( 8) we have A(ρ) = exp -4 ρ 0 sB (s) 2 ds . Second, it follows from Eq.( 10) that if B (ρ 1 ) = 0 for some ρ 1 then B (ρ 1 ) has the same sign as B(ρ 1 ). Thus, the function B(ρ) is monotone on I and B (ρ) has the sign of b. Next, let us define the function

H = 8e -2B -5e -8B -3ρAB -3A . (13)
With the use of this function (which we found by an arduous trial and error), the rest of the proof amounts to a oneline exercise in elementary calculus The initial conditions (12) imply that H(ρ) ∼ 9bρ 2 near the origin. Differentating H and using the constraint (11), we obtain

H + 1 ρ(1 -Z 2 ) + 3B H = 27e -8B B , (14)
hence H(ρ) cannot have a zero for ρ < ρ 0 because if H(ρ) = 0 then H (ρ) has the same sign as B (ρ) and therefore b. Similarly, H(ρ 0 ) cannot vanish because L'Hopital's rule gives H (ρ 0 ) = 54e -2B(ρ0) B (ρ 0 ). However, H(ρ 0 ) must vanish for regular solutions, as follows immediately from ( 14). This contradiction ends the proof.


M A Lysaght
email: m.lysaght@qub.ac.uk

H W Van Der Hart

Influence of Excitation Pulse Length on a Correlated Two-Electron Wave-Packet

Keywords: numbers: 31.14.A-, 32.80.Qk, 32.80.Rm

We employ the time-dependent R-matrix approach to investigate how ultrafast dynamics in the excited 2s2p 2 configuration of C + is affected by the length of the excitation pulse. The excitation pulse is tuned closer to the 2 D state than to the 2 S state. We demonstrate that by varying the excitation pulse length, we can choose an excitation of the 2s2p 2 configuration which is dominated by the 2 D state or an excitation which is dominated by a breathing motion between the uncoupled | 2p 0 2p 0 state and the singlet | 2p -1 2p 1 states.

Introduction

One of the main challenges in physics is the development of understanding of the evolution of correlated dynamics within complex systems. The clearest way in which such correlated motion manifests itself within atomic physics is through the core excitation of atoms and molecules leading to collective electron dynamics that typically unfolds on the attosecond time scale [1,2]. Insight into the basic forms of correlated electron motion forms a prerequisite for the development of schemes to control the motion of electrons in more complex systems ranging from large biomolecules [3] to metallic nanostructures [4].

The recent emergence of attosecond light sources [5] now allows direct time-domain access to ultrafast electron dynamics in atoms [6], molecules [2] and condensed matter systems [7]. As such, attosecond technology not only provides a means of observing, or even controlling, the motion of single electrons, but also affords the promise of providing deeper insight into ultrafast correlated dynamics between electrons in complex matter. For example, a recent investigation [8] proposed using an attosecond light pulse to doubly ionize a superposition of low-lying doubly excited states in He. By measuring the momentum vectors of the two ejected electrons in coincidence, it would be possible to observe the time-varying collective behaviour of the two electrons.

Although some success has been achieved in understanding the correlated motion of electrons in the time-domain [8,9], to our knowledge, the possibility of selecting :;6:-,< 99;:=-,< >;>-,< 6?;@A-,< @>;AA-,< @B;>B-,< 616. @ ( 616. 9 ( correlated electron dynamics has received little attention. In this report we propose an ultrafast scheme in which we are able to choose the nature of the correlated dynamics occurring between two electrons. We explain the induced ultrafast dynamics and show that by choosing light field parameters carefully, it is possible to steer a multielectron wave-packet into a specific field-free evolution.

The pump-probe scheme

Our scheme is similar to a pump-probe scheme used in a previous study of collective electron dynamics in C + [9,10]. The basic features of the scheme are explained in figure 1. We consider a C + ion in its 2s 2 2p 2 P o ground state with total magnetic quantum number M=0. This ion is excited by an extreme ultraviolet (XUV) pulse, linearly polarized in the z direction, into a superposition of the 2s2p 2 2 S e and 2 D e excited states. The energy separation between these two states, caused by the Coulomb repulsion between the two equivalent 2p electrons, results in a temporal interference between the two states with a frequency determined by the energy separation. The fact that configurations in an ion are more separated in energy than in a neutral system provides the opportunity to investigate electron-electron dynamics within a single configuration. This is not the case for neutral systems where configurations lie close in energy with the consequence that inter-configuration interactions will lead to more complicated dielectronic repulsion dynamics [8]. In this naive single-configuration picture, the responsible dielectronic repulsion integral is the F 2 (2p,2p) integral [START_REF] Fischer | Computational Atomic Structure: An MCHF approach[END_REF]. Hence, this repulsion governs the dynamical behaviour of the multielectron wave-packet which continues to evolve after the end of the pump pulse. Subsequent irradiation of the C + ion with a time-delayed ultrashort XUV pulse results in ionization. It was found in a previous study that by varying the time-delay between the pump pulse and the ultrashort ionising probe pulse the ionisation probability of the coherent superposition of the 2s2p 2 2 D and 2 S states showed rapid modulation due to collective dynamics of the two equivalent 2p electrons [9]. Ejected electron momentum distributions demonstrated clearly that the oscillations could be ascribed to a breathing motion between two different angular distributions, explaining the modulation of the ionisation yield [10].

In the present study, we aim to investigate the role of electron-electron interactions in the transition from ultrafast excitation to long-pulse excitation. In long-pulse excitation (and the non-relativistic limit), transitions take place between LS-coupled states. In short-pulse excitation, the electrons have little time to adjust to the changes in, for example, angular-momentum couplings between the initial and the final state that occur in LS-coupling. For example, the M=0 level of an sp configuration can only be formed from electrons with m=0, whereas the M=0 levels of a p 2 configuration can be formed either by two m = 0 electron or by two | m |=1 electrons. As a consequence short-pulse excitations can lead to dynamics governed by electron-electron interactions [10]. In this study, we investigate how the transition from short to long excitation pulses affects the dynamics in the 2s2p 2 configuration of C + .

The present investigations are carried out using the recently developed timedependent R-matrix (TDRM) approach [START_REF] Burke | [END_REF]13]. This approach enables the response of multielectron atoms to ultrashort light fields to be determined from first principles. In it, the solution of the time-dependent Schrödinger equation is accomplished by the standard partitioning of space into an internal and an external region [START_REF] Burke | Atomic and Molecular Processes: an R-matrix Approach[END_REF]. In the internal region, exchange interactions involving the ejected electron and the residual electrons are important, while these interactions are negligible in the external region. Hence, in the latter region, the ejected electron moves in the local long-range potential of the residual ion together with the laser field. A detailed description of the theory can be found in Ref. [13].

As already mentioned, the pump-probe scheme studied is shown in figure 1, but the details differ slightly from the previous studies. The initial state remains the M=0 level of the C + ground state. The pump pulse now has a central photon energy ω 1 = 10.0 eV in order to be near-resonant with the 2s2p 2 2 S e and 2 D e states but with different detunings. It is defined by a three-cycle sin 2 ramp-on of the electric field followed by q cycles of constant intensity and finally by a three-cycle sin 2 ramp-off, giving a total of n = 6+q cycles of the electric field. After letting the pumped C + ion freely evolve for ∼ 1 fs, we probe the ion by irradiating it with a probe pulse. The probe pulse has a central photon energy ω 2 = 21.8 eV and is described by a three-cycle sin 2 ramp-on followed by a three-cycle sin 2 ramp-off of the electric field. This pulse duration remains fixed throughout the study. Both pulses have a maximum intensity of 5× 10 12 W/cm 2 . The probe pulse is sufficiently energetic to transfer the population from the 2s2p 2 2 S e and 2 D e excited states to continuum channels coupled to the first excited state of C 2+ , 2s2p 3 P o . After the probe pulse we allow the system to relax freely for ∼ 30 fs. We analyse the dynamics through the angular distributions of the emitted electrons [10]. We thus focus on the channels associated with the 2s2p 3 P o only, and not on those associated with the 2s 2 1 S state. This is because in the former case, the ejected electrons can have m= -1, 0 or 1, whereas in the latter they can only have m=0. A description of the structure used to describe C + and other TDRM parameters can be found in [9,10].

To describe the correlated electron dynamics in terms of the dynamics of individual electrons, we transform from the LS coupled basis to the uncoupled basis |2p m 1 2p m 2 . Since the light is linearly polarized, M is conserved. The 2s electron can be considered as a spectator electron for M=0, so that the LS coupled 2p 2 1 S and 1 D configurations can be decomposed as follows:

|2p 0 , 2p 0 = - 1 3 |2p 2 1 S + 2 3 |2p 2 1 D (1a) |2p 1 , 2p -1 S = 2 3 |2p 2 1 S + 1 3 |2p 2 1 D , (1b)
where the subscript S indicates singlet spin coupling between the m=1 and m = -1 electrons. The decomposition in eq.( 1) immediately suggests that angular dynamics in the 2s2p 2 configuration involves correlative dynamics of the two electrons: if the m-value of one electron changes, the other electron must also experience a change.

Results

Figure 2 shows the population of the | 2p m ,2p -m functions and the LS coupled states as a function of time during and after an n=18 cycle (7.5 fs) pump pulse. The figure shows clear evidence of Rabi oscillations between the ground and excited states during the pump pulse due to detuning. During the pump pulse, most population in the 2s2p 2 configuration can be found in the 2 D state. However, at the end of the pulse, its population has fallen back substantially. The final population in the 2s2p 2 configuration thus consists of populations in 2 S and 2 D at comparable magnitudes. Consequently, after the pulse, a striking oscillation in the population of |2p 0 ,2p 0 and |2p 1 ,2p -1 S can be seen. Due to the energy difference between 2 S and 2 D, this leads to a noticeable oscillation in the |2p 0 ,2p 0 and |2p 1 ,2p -1 S populations after the 18 cycle pump pulse with both populations reaching a minimum value close to 0. During the pulse, oscillations between |2p 0 ,2p 0 and |2p 1 ,2p -1 S can be seen as well, but they do not indicate as substantial a change in the nature of the wavepacket. Figure 3 shows the 2D momentum distributions of the ejected electron in the k x k z plane for several pump pulse lengths. For each of the momentum distributions, the probe pulse reaches its peak intensity at a time-delay which corresponds to a minimum in the |2p 0 ,2p 0 population and a maximum in the |2p 1 ,2p -1 S population. The values for these time delays are given in Table 1. The momentum distributions are obtained by transforming the continuum wave functions coupled to the 1s 2 2s2p 3 P o excited state of C 2+ at the end of the calculation into the momentum representation [13]. The figure shows that the ejected electrons are predominantly emitted along the laser polarization direction, except for pulse lengths of 6 and 18 cycles. Since |m|=1 electrons cannot be emitted along this direction, the figure shows a dominance for the emission of m=0 electrons. The exceptions are pulse lengths of 6 and 18 cycles, which are dominated by the emission of |m|=1 electrons. This is despite our choice of probing the configuration at a time when the population of |2p 0 ,2p 0 is at a minimum and |2p 1 ,2p -1 S at a maximum. The reason for this is that when the populations of |2p 0 ,2p 0 and |2p 1 ,2p -1 S are comparable, emission of an m = 0 electron is favoured. For most pulse lengths, population in the 2s2p 2 configuration is dominated by the 2 D state, and hence the |2p 0 ,2p 0 and |2p 1 ,2p -1 S have a comparable population. For pulse lengths of 6 and 18 cycles, however, population in the 2 D and 2 S states is comparable, and we can expect the dynamics to appear far more prominently.

Although the momentum distributions for the 6 and 18 cycle pulses in figure 3 show a similar angular dependence, there are clear differences between the structure in their respective momentum distributions. The most noticeable difference can be seen at small momenta. At low momentum, figure 3 shows little interference structure for the 6-cycle pulse. On the other hand, the 18-cycle pulse shows considerable interference structure. This interference structure is, furthermore, asymmetric with respect to k z = 0, which suggests interference between waves with even and odd parity. The most obvious explanation for this interference is thus interference between electrons emitted by the probe pulse, and electrons emitted during three-photon absorption from the pump pulse. The increased duration of the pump pulse should make three-photon ionization relatively more important for the 18-cycle pump pulse than for the 6-cycle pump pulse.

So far we have shown that emission of |m| = 1 electron occuring at a minimum in the population of |2p 0 ,2p 0 is dominant only after a 6 cycle and 18 cycle pump pulse. We can further demonstrate that this is a good measure of the correlated dynamics occuring between the two 2p electrons after the end of the pump pulse. Figure 4 shows the momentum distributions obtained for five separate lengths of the pump pulses of varying duration, and for three delays of the probe pulse. The first delay corresponds to a minimum of the |2p 0 ,2p 0 population, as in figure 3. The third delay corresponds to a maximum of the |2p 0 ,2p 0 population and the second delay correponds to the midpoint between these two delays. The values for these time delays are given in Table 1. Figure 4 clearly shows the dominant emission of |m| = 1 electrons for the 6-and 18-cycle pump pulses for delays at which the |2p 0 ,2p 0 population is at a minimum. At the third time-delay, the probe pulse interacts with the wave-packet when the |2p 0 ,2p 0 population is at a maximum. The dominance of the emission of m=0 electrons at this delay is clearly visible. This behaviour indicates that the wave-packet is undergoing a breathing motion in which it oscillates between two distinct angular distributions. The wave-packet will oscillate between these distributions after the end of the pulse with a period of 1.5 fs, determined by the energy separation between the 2 S e and 2 D e states. For a wave-packet prepared by the 9-, 12-or 15-cycle pump pulse, however, signs of electron dynamics are elusive. The dominance of population in the 2 D state means that the breathing motion between the two angular distributions only forms a minor part of the excitation into 2p 2 .

Figure 4 thus shows that by a careful choice of the pump pulse length, it is possible to choose the nature of the excitation of the 2p 2 configuration. For certain pulse lengths, the excitation will be dominated by a single LS-coupled state, but for other pulse lengths, the excitation will be dominated by a breathing motion between uncoupled states. Studies in this area may lead to further insight into the fundamental connection The distributions have been obtained for three different pump pulse durations: n = 6,9,12,15 and 18 cycles. For each pump pulse the probe pulse ionizes the pumped system at three increasing time delays, where 'intermediate' indicates a delay at which |2p 0 ,2p 0 is half way between its minimum and maximum population.

between atomic structure and correlated multi-electron dynamics.

The scheme presented here demonstrates the possibility of influencing the "dynamical" alignment of a multielectron wave-packet by varying the duration of single laser pulse. However, the use of a single pump pulse has significant drawbacks. For example, it is difficult to transfer significant population into the 2 D and 2 S states. However, this problem could be alleviated by the use of multiple pump lasers. For example, an XUV pulse could be used to resonantly populate the 2s2p 2 2 D state with a second long wavelength pulse used to couple the 2 D and 2 S states via a 2-photon transition. The long wavelength pulse could be tuned to this 2-photon transition, thereby reducing the effect of detuning seen in the present study. The amount of population pumped into the excited states could also be controlled by the long wavelength laser parameters.

Conclusion

In conclusion, we have applied time-dependent R-matrix theory to illustrate a technique that may provide the ability to steer multielectron wave-packets into a specific field-free evolution. In the present application to the 2s2p 2 configuration of C + , we have shown that the field-free angular dynamics of the two-electron wave-packet is dominated by a breathing motion between two 2p electrons with m=0 and two 2p electrons with |m|=1 for an excitation pulse of 10 eV with pulse lengths of 6 and 18 cycles, whereas for pulse lengths inbetween the two-electron wavepacket is dominated by the 2 D state. Although the present single pump pulse scheme allows a choice of dynamics for the twoelectron wave-packet, more elaborate pulse schemes are required to transfer significant population into such dynamics. It would also be interesting to extend this study to more complex configurations to obtain additional insight into the influence of electron-electron interaction on correlated multi-electron dynamics.

Figure 1 .

 1 Figure 1. (Color online). Diagram of the pump-probe scheme. The pump pulse coherently populates a superposition of the 2s2p 2 2 D and 2 S states. The time-delayed probe pulse has a central photon energy that allows the transfer of population above the C 2+ 2s2p 3 P ionization treshold.

3 Figure 2 .

 32 Figure 2. (Color online). Population of the excited 2s2p2 bound states as a function of time during and after an 18 cycle (7.5 fs) pump pulse. Populations are given for the LS-coupled 2 S state (thin blue dash-dot) and the 2 D state (red dash-dot), as well as for the uncoupled |2p 0 ,2p 0 states (solid blue) and |2p 1 ,2p -1 S states (red dash). No probe pulse is present here to show the free evolution of C + after the pump pulse. The inset shows the free evolution of the |2p 0 ,2p 0 and |2p 1 ,2p -1 S populations.

Figure 3 .

 3 Figure 3. (Color online). Momentum distributions |kφ(k)| 2 associated with the residual C 2+ ions left in the 2s2p 3 P o state in the k x k z plane for the ejection of an electron from the pumped C + ion for varying durations of the excitation pulse. In this figure, n indicates the total number of cycles of the excitation pulse. For each distribution the probe pulse interacts with the C + ion when the |2p 0 ,2p 0 population is at a minimum.

Figure 4 .

 4 Figure 4. (Color online). Momentum distributions similar to those shown in fig. 3.The distributions have been obtained for three different pump pulse durations: n = 6,9,12,15 and 18 cycles. For each pump pulse the probe pulse ionizes the pumped system at three increasing time delays, where 'intermediate' indicates a delay at which |2p 0 ,2p 0 is half way between its minimum and maximum population.

Table 1 .

 1 Time delays (fs) between end of n-cycle pump pulse and start of probe pulse.

	n	|2p 0 ,2p 0	intermediate |2p 0 ,2p 0
		min		max
	6	1.89	2.25	2.68
	7	1.70	-	-
	8	2.64	-	-
	9	2.06	2.42	2.76
	10 1.81	-	-
	11 1.64	-	-
	12 1.50	1.91	2.30
	14 1.72	-	-
	15 1.43	1.86	2.27
	18 1.69	2.03	2.35

Acknowledgments

ML acknowledges support through Grant No. EP/E000223/1 from the UK EPSRC.


T Grycuk
email: tgrycuk@fuw.edu.pl

M Kubkowska

Experimental studies of the 200 nm diffuse band in zinc vapours and its interpretation

Keywords: numbers: 32.70.-n, 32.70.Cs, 33.20.Lg, 33.70.-w, 34.20.-b

The absorption spectrum of zinc vapours is measured under various conditions in the 190-224 nm region and the absolute absorption coefficient per one pair of atoms is derived. A relatively narrow diffuse band at 200.22 nm (in a vacuum) is observed and its profile is carefully determined for several temperatures. It is found that the band maximum slightly decreases with a temperature increase. This indicates that the band is generated by transitions occurring within the range of negative energy values of the ground state potential.

The experimental results are utilized for deriving some information concerning the upper state potential and the oscillator strength function via semiclassical calculations of the band profile and its comparison with the experimental one. It is shown that theoretical potentials for the lowest Rydberg states available cannot be responsible for the band formation and some part of the previously unknown excited state potential near its minimum as well as the f(R) function reproducing the experimental band profile are found. The equilibrium position, R e , and the energy minimum, E(R e ), for this state are established to be 3.94 ± 0.15 and 49660 ± 50 cm -1 , respectively. It is also shown that the f(R), being equal to zero at large distances, for R lower than about 4 increases rapidly to the maximal value of 0.61 ± 0.03 at R = 3.65 ± 0.05 . Such a behaviour indicates that the upper state here can be coupled with a state characterized by strongly allowed transition to the ground state.

Discussion concerning this matter and identification of the upper state asymptote is presented.

Introduction

In our previous paper [1], devoted to the experimental and theoretical studies of the blue satellite of the Zn (4 1 P 1 -4 1 S 0 ) 213.8 nm line, the observation of the diffuse band appearing in the far UV region near 200 nm was reported. The present work being a continuation of those studies is aimed at deriving more experimental data concerning this band, hereafter referred to as the S2 band following [1]. The problem is important because, to our knowledge, the S2 band seems to be hitherto the only observed spectral feature which can provide information on the electronic transition in Zn 2 between its ground state and the poorly known excited states lying above the 1 u ( 1 u ) state of the (4 1 P 1 + 4 1 S 0 ) asymptote (see figure 5 in section 3).

It should be mentioned that although several UV diffuse bands in dense vapours of the zinc family elements have been discovered as early as in the 1920s [2 -4] the origin of the S2 band has not Confidential: not for distribution. Submitted to IOP Publishing for peer review 30 April 2010 been unambiguously established. Most frequently it has been suggested that the upper electronic state for this band is one of the lowest Rydberg states 1 u + or 3 u + correlated with, respectively, the ((n+1) 1 S 0 + n 1 S 0 ) [START_REF] Finkelnburg | Kontinuierliche Spectren[END_REF] or the ((n+1) 3 S 1 + n 1 S 0 ) [START_REF] Cram | [END_REF] atomic asymptote, where n = 6, 5, 4 for Hg, Cd, Zn. This hypothesis, unfortunately, has never been verified because potential energy curves of so highly excited states, in particular for Cd and Zn, were missing.

We have recently carried out the first quantitative studies of such a band in cadmium [6,7] where it appears at about 212.5 nm The spectroscopic measurements were performed in fluorescence and absorption and from the temperature variation of the band profile in the later case, some part of the upper state interaction potential was derived and the oscillator strength function for the involved transition was roughly estimated. On the other hand Czuchaj et al. have calculated interaction potentials for several electronic states of Cd 2 [8] and Zn 2 [9] including, for the first time, both above mentioned Rydberg states. It is noteworthy, that lower valence states predicted by those authors reproduce the known spectral features much better than other existing theoretical potentials as it was discussed in [1]. Contrary to this, a comparison of the experimental findings for Cd 2 [7] with the theoretical potentials for the Rydberg 1 u + (6 1 S 0 ) and 3 u + (6 3 S 1 ) states from [8] showed qualitative disagreement indicating that neither of these states can be a carrier for the S2 band.

The present studies were undertaken for examine this problem in zinc what would be an additional test for the S2 band interpretation. The experimental study of the S2 band for this element is, unfortunately, more difficult than in cadmium, not only in the case of conventional spectroscopy but with the new techniques as well. The main difficulties arise from the higher melting point for Zn, resulting in the necessarily of work at temperatures of about 1000 K or more to detect this rather weak band, as well as from the inconvenient spectral region where the band appears. Thus, although the great interest in the electronic structure of the group 12 diatomic systems have being observed since several years [10] the S2 band of Zn 2 has not been studied as yet.

Experimental details and results

The apparatus and procedure were generally the same as those described in [1]. Therefore, only those improvements that are important to the present investigation are described below.

Two quartz (suprasil) cells, 5 and 25 cm in length and 3 cm in diameter, containing the high purity (6n) zinc, distilled into relatively short side arms, were prepared. The shorter cell was used for the precise, quantitative determination of the wings profile of the 213.8 nm Zn line, utilized for an intensity calibration of the much weaker spectrum extended towards short wavelengths including the S2 band which was measured with the longer cell.

In order to eliminate the so-called hot Schumann-Runge (S-R) bands of the O 2 molecule [START_REF] Herzberg | Spectra of Diatomic Molecules 2nd Edition[END_REF] whose intensity strongly depends on temperature [START_REF] Cheung | [END_REF], the oven for heating the cell was flushed by a regulated argon flow as it was established in [1]. The gas was fed through two quartz narrow pipes in which it was heated to the oven temperature in order to minimize its influence on the thermal conditions inside the oven.

The high resolution Rowland spectrometer with a 6 m concave grating was used as a dispersive system. The great luminosity of this device in the UV region and a long optical path inside the spectrometer (~12 m) resulted in the detection of a very weak atmospheric O 2 absorption originating from the lowest vibrational level in the ground state potential of this molecule. This spectrum was then used to improve the wavelength calibration of the spectrometer in the far UV region where several rotational lines of the 2-0 and 1-0 ro-vibronic S-R band systems were observed as shown in figure 1. The J assignment here was adopted after [13] and very accurate vacuum wavelengths of the particular lines were taken from [14].

Although our concern in this work was with the S2 diffuse band near 200 nm, for an accurate intensity calibration the spectra were determined over a much broader spectral range, between about 199 -224 nm, including the wings of the Zn 213.8 nm resonance line. The measured quantity was the absorption coefficient of the Zn vapour as a function of the relative frequency

( ) ( ) ( ) [ ] d I I k ν ν ν ∆ ∆ - = ∆ 0 / ln (1)

Here

= -0 , with 0 being a frequency of the line centre assumed as 46745.37 cm -1 [START_REF] Moore | Atomic Energy Levels[END_REF] (here and hereafter the frequency, , is measured in cm -1 ), d is the cell length, and I( ) and I 0 ( ) are the signal amplitudes measured for Zn vapour and empty cell, respectively, normalized to their values at the same chosen in the region free of any absorption.

In view of the main goal of the present work, we have concentrated on possibly precise determination of the absolute (reduced) absorption coefficient per one pair of atoms

k (r) ( ) = k( )/ N 2 (2)
where N is the vapour number density. This value is usually calculated from the ideal gas law using an appropriate formula for the relation between the pressure (p) of the saturated vapour and the measured temperature (T s ). Such a method, however, proved to be insufficiently accurate in the case of the longer cell due to the considerable temperature gradient along its side arm and the cell body.

Therefore, the intensity calibration of the spectrum under study at the given mean temperature (T±5 K) of this cell was achieved by its comparison and adjusting to a suitable spectrum of the reduced absorption coefficient in the wings of the resonance 213.8 nm line, determined much more precisely for the short cell. In this case the cell together with its Zn reservoir occupied the central part of the 60 cm long oven where the temperature was well established and measured with accuracy of 0.2 K. The saturated vapour pressure in this case was calculated from p(T s ) curve received via a least square fit to sufficiently accurate vapour pressure measurements recommended by Nesmeyanov [START_REF] Nesmeyanov | Vapor pressure of the elements[END_REF] and Hultgren et al. [START_REF] Hultgren | Selected Values of the Thermodynamic Properties of the Elements[END_REF]. Fortunately, over the temperature range about 900 -1000 K a considerable number of such data exists and they do not show a great scattering of values, thus, the relative pressure error generated by that formula is estimated as 3 per cent yielding the uncertainty in the absolute absorption coefficient derived for the short cell not greater than 6 per cent.

Using the above method the absolute absorption coefficient spectrum, given by equation ( 2), for unsaturated vapour of the densities between about (2.5 -9.1) × 10 17 cm -3 was determined at several cell temperatures between about 940 -1030 K. These limitations were due to increasing influence of the atmospheric O 2 absorption for lower N and increasing instability of measurements at higher temperatures. In any case, the cell temperature T was appreciably higher than T s determining the vapour density.

The measurements showed that even for so high vapour density as of the order of 10 18 cm -3 the band under study is a relatively narrow continuum of nearly symmetric profile, well isolated from the blue satellite of the 213.8 nm line, S1, as it is seen in figure 2. Intensity of both bands presented somewhat decreases with a temperature increase, however in the case of the S2 band this effect seems to be slightly enhanced by influence of the O 2 (1-0) band system which gives a constant contribution to the Zn vapour absorption under study.

In order to determine accurately the temperature dependence of the S2 band, being very important for its interpretation in terms of the interaction potentials, it was necessary to remove the contribution from molecular oxygen. Because it was impossible to do this accurately for the total band profile we have decided to estimate the temperature effect only for the band centre. This was done by fitting the Gaussian profile, G( ), to the particular, experimentally derived spectrum as it is presented in figure 3. For the sake of comparison with figure 1, the spectrum is here plotted as a function of the vacuum wavelength ( v ). It was assumed that the optimal fitting is reached when the difference between the experimental spectrum and the fitted profile does not depend of the Zn vapour density and agrees reasonably well with that of the O 2 (1-0) band spectrum.

It is noteworthy, that all the experimental spectra could be fitted by the G( ) profile of the same width = 0.755(5) nm (FWHM) and the centre position 0 = 200.220(5) nm (or, alternatively, = 190(5) cm -1 and 0 = 49945(2) cm -1 ) while its maximum G( 0 ) depended on the square of the Zn vapour density with rather moderate scatter of values. In such a way k (r) ( 0 ) G( 0 )/N 2 for different cell temperatures, which represents the pure temperature effect for the band maximum, was determined and presented in figure 4. It is seen that, in spite of considerable errors originating mainly from the vapour density uncertainty mentioned above, the reduced absorption coefficient in the maximum of the S2 band, k (r) ( 0 ), clearly decreases with a temperature increase.

Since the temperature dependence of a thermally averaged absorption spectrum arises from a population of different scattering states (free states) in the ground state potential, V g (R), and is roughly described by the Boltzmann distribution as exp(-V g (R( ))/k B T), the effect observed for the S2 band maximum indicates that this band is formed by transitions originating mainly in the region of negative values of the ground state potential, i.e. within its well, similarly as it was observed for the analogical Cd 2 band in [6,7]. A slope of the straight line fitted to the data in figure 4 yields the ground state potential in the band maximum amounting to 360 ± 150 cm -1 which seems to be reasonable accounting for appreciable errors of the experimental data and rather narrow range of temperature explored.

The band interpretation

Taking into account the experimental findings presented in section 2, some information concerning the S2 band origin can be deduced.

In particular, the shape of the band is typical for thermally averaged spectral features associated with asymptotically forbidden transitions that have been observed in the past for different diatomic systems (see e.g. [START_REF] Veza | [END_REF]19] and interpreted in a frame of the quasimolecular theory of spectral line wings and satellites associated with optical collisions. This theory, comprehensively discussed in [20], is particularly useful in its classical quasistatic limit, which straightforwardly leads to the wellknown inversion technique developed for both the asymptotically allowed [21] and forbidden [22][23][24] electronic transitions. The method, based on analysis of the temperature dependence of the spectrum, has been successfully used by many authors for deriving the interaction potentials and the transition moment function involved. A review of these results may be found in [20].

In the case of the Zn 2 band studied in this work such a method could not be fully utilized because the sufficiently accurate temperature dependence of the total band profile was hardly attainable. However, it was possible to employ the experimental profile for verification of various potential energy curves assumed for the upper state responsible for the band formation. Looking for the suitable candidate we have considered both of the lowest Rydberg states from [9], referred to as R3( 3 u + ) and R1( 1 u + ) in figure 5, as well as the hypothetical highly excited Y( u ) state of the equilibrium distance close to one of the ground state potential well from [9].

The calculation of the band profile were, in general, performed using the well known semiclassical Szudy and Baylis formula [25,20]. According to this approach the adiabatic line wing profile associated with an electronic transition between the lower (l) and the upper (u) state, characterized by the respective potential energy curves V l (R) and V u (R) of the homonuclear diatomic system, in the case of absorption is given by

( ) ( ) ( ) ( ) ( ) ( ) c c c B c c c c J z L z R V T k R V R R f g g mc e N k 2 1 36 / exp 4 1 2 2 2 2 2 π π ν ′ ∆ - ⋅ = ∆ Ω (3) with ( ) ( ) ( ) 3 4 3 1 2 2 2 1 - ′ ′ ∆ ′ ∆ = c c B c R V R V T k c z µ (3a)
where and V are expressed in cm -1 .

In the above equations e, m, c and k B have their usual meanings, g and g J are the statistical weights of the excited molecular and parent atomic state, f(R c ) is the oscillator strength at the internuclear separation R c , µ is the reduced mass and L(z c ) is the universal line shape function defined in [25].

The subscript c denotes the so-called Condon points at which the separation R c satisfies the equation:

h = V(R c ) V u (R c ) -V l (R c ).
Only the real Condon points were accounted for and all calculations of the band profile were performed for the temperature of 1031 K at which the experimental spectrum, being derived for high vapour density, was practically unaffected by the oxygen molecule v = 0 bands.

The Zn 2 potential energy curves of higher excited states available from [9] are presented in figure 5. This figure also shows that the potential differences between the ground state (X) and each excited state (R1, R3) have a monotonic character, so that in both cases there is only one Condon point R c for each . In such a situation the semiclassical profile given by Eqs. ( 3) simplifies to the classical quasistatic one and for the reduced absorption coefficient we have

( ) ( ) ( ) ( ) ( ) ( ) c B c c c J r R V T k R V R R f g g mc e k ′ ∆ - ⋅ = ∆ Ω / exp 4 1 2 2 2 2 π ν (4)
It is noteworthy that this simple classical wing profile was verified via quantum-mechanical calculation and found a quite adequate in the case of the monotonic difference potentials i.e. far from any extremum of V(R) (see e.g. [26]).

For the R1 excited state both statistical weights are equal to 1 and f(R c ) can be deduced from an electric dipole moment function for the R1 X transition obtained also in [9]. Although due to the character of the V(R) curve any band for this transition is expected, the shape of the f(R) function (see inset of figure 5) leads to an appearance of the weak diffuse band in the far wing of the 5 1 S 1 -4 1 S 0 atomic line, as shown in figure 6. The predicted position of the calculated band agrees surprisingly well with experimental data, however, the intensity is about a hundred times weaker and the width of band is much more larger. So big disagreement proves clearly that the interaction data used here can not be responsible for the S2 band generation.

For the R3 X transition, a transition moment function is not available. However, from Eq. ( 4) through its inversion one can provide the oscillator strength which together with responsible potentials could reproduce the experimental band profile k (r) ( ). It is give by

( ) ( ) ( ) ( ) ( ) ( ) ( ) R V T k R R R k g g C R f g B r J / exp 2 ν ν ′ ∆ ∆ = Ω (5)
where C includes all the constant factors and for k (r) in cm 5 , in cm -1 and R in it amounts to 8.988937×10 -2 cm -1 .

Unfortunately, this formula, for theoretical potentials V 3 ( 3 u + ) and V g ( u + ) from [9] and the statistical weights of both states amounting 3, for related to the frequency 0 = 53672.24 cm -1 of the 5 3 S 1 -4 1 S 0 atomic transition [START_REF] Moore | Atomic Energy Levels[END_REF] gives unreasonably high f-values of about 5 at the band maximum.

Therefore, we conclude that the R3 molecular state from [9], also can not be a carrier of the S2 band.

Finally, we have considered the excited state of the ungerade symmetry, Y( u ), for which the potential difference, V(R), is not a monotonic function but it has a minimum for the energy roughly determined by the S2 band spectral position at separation close to one for the ground state potential from [9]. In such a situation it can be expected that the band observed is the so-called rainbow satellite appearing in the red wing of any forbidden line of the frequency determined by an atomic asymptote of the Y( u ) state. Because the shape of the band calculated depends only on derivatives of the difference potential (see Eqs. 3) it is not necessary to assume any real state for this asymptote.

For some simplicity of calculations, the part of this V(R) curve which could be responsible for the formation of the S2 band was approximated with the quadratic function and f(R) was assumed as a Gaussian of two different slopes. Several combinations of such interaction data were tested receiving finally quite good agreement between the calculated quasiclassical band profile (the antistatic part was omitted) and the experimental one as it is presented in figure 7

  . The profile calculated for f(R) = const, normalized to the maximum of the experimental band, is also shown in this figure. Although in this case the spectrum calculated shows a distinct band the experimental profile can be reproduced only for the f(R) function which is different from zero in a relatively narrow range of the internuclear separation. The part of the difference potential curve, V, and f(R) for the X-Y transition used for the calculation of spectra from figure7are presented in figure8. Numerical values for the difference potential over limited range of R (as shown) are given by parabola aR 2 + bR +C with a=1299.282 , b=-10126.55, c=69608.7. The upper state potential curve Y( u ) is found as a sum of this V(R) and V g (R). It is seen that the minimum of the upper state appears just above the maximum of the barrier in the 1 u ( 1 P + 1 S) state producing the diffuse band S1 and that the X-Y transition being forbidden at large separations becomes allowed at distances close to the minimum of the upper state potential. These results indicate that the Y( u ) state can be here coupled with a state characterized by strongly allowed transition to the ground state. Systematic tests were made for possible variations in the calculated band profile due to variations of the position and value of the V(R) minimum and the f(R) maximum. Accounting for the results of these tests the main parameters for the Y( u ) state i.e the equilibrium position R e and the V(R e ) energy are derived to be 3.94 ± 0.15 and 49660 ± 50 cm -1 respectively, with f(R) having a quite reasonable maximal value of 0.61± 0.03 at R = 3.65± 0.05 . The error bounds for the upper state are estimated taking into account also the ground state potential uncertainty.

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. The reduced absorption coefficient in the far blue wing of the self-broadened Zn 213.8 nm line including its satellite band, S1, and the S2 band measured during the same scan. The spectra were derived at different temperatures: ___ , T = 983 K; ••••• T = 1031 K.

Figure 3 .Figure 4 .

 34 Figure 3. The absorption coefficient of the S2 band observed at N = 8.2246×10 17 cm -3 and T =1023 K (upper plot) in comparison with the one derived for an empty cell at room temperature scanned in smaller step (lower plot). The upper smoothed contour represents a Gaussian profile of the parameters: G( 0 ) = 0.087 cm -1 , 0 = 200.22 nm, = 0.75 nm. (For particular description see the text)

Figure 5 .

 5 Figure 5.Theoretical potential energy curves of Zn 2 for the ground state, X, and ungerade excited states correlated with higher atomic asymptotes from Ref.[9]. Dashed curves represent respective difference potentials. The oscillator strength function for the transition R1-X is presented in the inset.

Figure 6 .

 6 Figure 6. The reduced absorption coefficient as a function of difference frequency for the R1-X transition. The comparison of the experimental spectrum, (•••), with this calculated multiplied by 100, (-), using the suitable potentials and f(R) function presented in figure 5. The dashes curve represents the spectrum calculated for f(R) = 1.

Figure 7 .

 7 Figure 7. The comparison of the experimental ( ) S2 band profile with the calculated ones (-) using the difference potential V(R) and the f(R) function presented in figure 8. The dashes curve represents the spectrum calculated for f(R) = const resulting from normalization of the band maximum to the experimental intensity.

Figure 8 .

 8 Figure 8. The established part of the difference potential energy curve, V, responsible for the S2 band formation and the respective excited state potential, Y( u ). The oscillator strength function for the transition X-Y is presented in the inset. The potential energy curve of the lower close-lying 1 u( 1 u ) state dissociating to (4 1 P 1 + 4 1 S 0 ) is also presented.

Figure 1 (Figure 2 Figure 8 (

 128 Figure 1 (gryc0430/figure1.tif)

Discussion

The existence of highly excited states with a minimum immediately above the barrier in the 1 u ( 1 P + 1 S) state have been suggested in the earlier less sophisticated ab initio calculations of the electronic structure of the Group II (A and B) metal diatomic molecules [27][28][29][30][31]. For Zn 2 such a state was first predicted by Hay et al. [27,28] who have shown the important effects in the excited valence potential energy curves arising from the Zn + ( 2 S) + Zn --( 2 P) ion pair states although they are asymptotically unstable. In particular, the 1 u diabatic ion pair state which is strongly attractive at relatively long distances dramatically changes the originally repulsive lower states of the same symmetry due to an avoided crossing. It deals, in particular, with the 1 u state of the asymptote 3 P + 3 P and the lower 1 u state correlated with the 1 P + 1 S which, finally, become attractive at smaller R as a result of mixing with that ion-pair state. In this way the resulting 1 u ( 3 P + 3 P) adiabatic state has a potential well close above the barrier in the 1 u ( 1 P + 1 S) adiabatic state. More recently, Ellingsen et al. [32] have performed all electron calculations of the ground state and the eight lowest-lying excited states of Zn 2 at several levels of theory including spin-orbit coupling. Those calculations, among other results, also showed that the 1 u ( 1u ) ( 1 P+ 1 S) state is perturbed by the interaction with the 1 u ( 1 u ) state dissociating to the 3 P + 3 P. The resulting avoided crossing yielded the barrier of about 5600 cm -1 in the 1 u ( 1u ) ( 1 P + 1 S) state and the well depth of about 9600 cm -1 in the 1 u ( 1 u ) ( 3 P + 3 P) state. Assuming that the 1 u ( 1 u ) ( 3 P + 3 P) state of the asymptotic energy of 2×32696.34 cm -1 is the upper state for the S2 band under study, its spectral position should be expected at about 56000 cm -1 whereas it is observed at 49945 cm -1 . Moreover, the transition between the 1 u ( 1 u ) ( 3 P + 3 P) and ground states at long distances is fully allowed that disagrees with the present situation.

Unfortunately, the above mentioned early calculations according to their authors opinion, could not provide any data concerning the Rydberg states of interest because of limitations of the basis set being inadequate for describing such states. However, the van der Waals coefficients, describing long-range interactions for the R1 and R3 states were calculated by Ellingsen et al. [33] and proved to be very large. This indicates that these states can be more attractive than those received by Czuchaj so, if the spin-orbit interaction (not considered in [9]) is taken into account, the potential energy curve for the 1 u fine structure components of the adiabatic R3 state can undergo the avoided crossing with the 1 u ( 1 u ) ( 1 P + 1 S) excited state. Hence, the here postulated Y state would be attributed to the 1 u ( 3 u + )

Rydberg state of the Zn(5 3 S 1 + 4 1 S 0 ) asymptote. The well depth of this state would be then estimated as about 4000 cm -1 .

The above conclusion is consistent with that inferred in [7] for the S2 band interpretation in Cd 2 . Moreover, basing on the combining results of that paper with findings derived from the extensive studies of fluorescence spectra of the Cd vapour following strong laser excitation [6,34], in [7] we have argued that the 1 u ( 3 u + ) state could be the state of the double minimum and the interaction data received from the S2 band analysis deal with its outer well. This suggestion need further theoretical studies of the electronic structure of the zinc family diatomic systems, particularly in highly excited valence and Rydberg states among which several avoided crossings one can expect. The adequate description of these states over wide range of separations requires the knowledge of states arising from higher-lying asymptotes which are not yet available.