Christopher L Alexander

Bernard Tribollet

Mark E Orazem

Contribution of Surface Distributions to Constant-Phase-Element (CPE) Behavior: 2. Capacitance

Keywords: Frequency Dispersion, capacitance, impedance spectroscopy

published or not. The documents may come

Introduction

Impedance measurements on solid electrodes often yield constant-phaseelement(CPE) behavior over a large range of frequencies. A CPE describes a distribution of time constants which can be physically categorized as oc-curring either normal to or along the electrode surface [START_REF] Jorcin | CPE analysis by local electrochemical impedance spectroscopy[END_REF].

Hirschorn et al. [START_REF] Hirschorn | Constant-phase-element behavior caused by resistivity distributions in films: 1. theory[END_REF][START_REF] Hirschorn | Constant-phase-element behavior caused by resistivity distributions in films: 2. applications[END_REF] used a measurement model incorporating a series of elements to show that a power-law distribution of resistivity through an oxide film gives rise to CPE behavior. This model has been used to determine the film thickness of oxides on stainless steel and aluminum electrodes [START_REF] Orazem | Dielectric properties of materials showing constant-phase-element (CPE) impedance response[END_REF]. It has also been used to determine physical properties of human skin [START_REF] Orazem | Dielectric properties of materials showing constant-phase-element (CPE) impedance response[END_REF][START_REF] White | Characterization of damaged skin by impedance spectroscopy: Chemical damage by dimethyl sulfoxide[END_REF] and polymer coatings [START_REF] Musiani | Determination of resistivity profiles in anti-corrosion coatings from constantphase-element parameters[END_REF].

Distributions of solution resistance are observed in planar disk electrode experiments. Huang et al. showed that the geometry of a disk electrode within an insulating plane leads to frequency dispersion on blocking electrodes at frequencies greater than dimensionless frequency ωC 0 r 0 /κ=1 [START_REF] Huang | The apparent CPE behavior of an ideally polarized disk electrode: A global and local impedance analysis[END_REF].

Jorcin et al. [START_REF] Jorcin | CPE analysis by local electrochemical impedance spectroscopy[END_REF] confirmed these results experimentally with the use of local electrochemical impedance spectroscopy.

Surface roughness of the electrode, once believed to contribute to the cause for frequency dispersion [START_REF] Borisova | Determination of the zero voltage points of solid metals from measurements of the capacity of the double layer[END_REF], will also yield a distribution of ohmic resistance. Alexander et al. [START_REF] Alexander | Contribution of surface distributions to constant-phase-element (cpe) behavior: 1. influence of roughness[END_REF] showed that surface roughness at the micron scale leads to frequency dispersion at frequencies larger than those due to the disk geometry if the radius of the disk is greater than the characteristic length, f r P , where f r is the roughness factor and P is the period of the roughness. The period of the roughness may represent the average width of the rough grooves. The disk radius will in most cases be greater than this characteristic dimension except in cases of very large roughness factors such as those that yield porous electrode behavior. This result contradicts the theory that surface roughness may be a source of CPE behavior over a large range of frequencies.

Another proposed cause of frequency dispersion is the surface distribution of capacitance [START_REF] Leek | The dispersion of double-layer capacitance with frequency i. smooth solid electrodes[END_REF]. Brug et al. developed a formula which relates the parameters of the CPE to an effective double layer capacitance [START_REF] Brug | The analysis of electrode impedances complicated by the pres-ence of a constant phase element[END_REF]. They postulated that the distribution was due to a variation in capacitance across the electrode surface.

In 1978, Leek and Hampson [START_REF] Leek | The dispersion of double-layer capacitance with frequency i. smooth solid electrodes[END_REF] calculated the frequency dispersion using a circuit element ladder with different capacitance values per surface area of electrode. They concluded that surface heterogeneity was an important factor contributing to frequency dispersion. In 1992, Pajkossy and Nyikos [START_REF] Pajkossy | Impedance of planar electrodes with scaleinvariant capacitance distribution[END_REF] used a self-similar capacitance distribution to show that frequency dispersion in the form of constant-phase-element behavior cannot be explained by a capacitance distribution since the CPE behavior is only observed for physically impossible variations in capacitance. Also in 1992, Kurtkya and de Levie [START_REF] Kurtyka | Frequency dispersion associated with a non-homogeneous interfacial capacitance[END_REF] conducted numerical simulations to explain that the frequency dispersion due to nonuniform capacitance is caused by the shift of the current lines with frequency from locations of high impedance to locations of low impedance.

The objective of this work is to explore, by use of finite-element models, whether a surface distribution of capacitance could provide a valid physical explanation for CPE behavior over a broad range of frequencies and to determine the characteristic frequency and dimension associated with this form of surface heterogeneity.

Impedance Calculations

Two disk electrode configurations were simulated, including a disk electrode embedded within an insulating plane, and a recessed disk electrode.

The calculations were performed in COMSOL Multiphysics 4.3 using cylindrical coordinates for a quarter of a circle domain, provided in Figure 1 which represented the electrolyte. Calculations were performed for electrode radii of 0.12 cm, 0.24 cm, and 0.48 cm. The counterelectrode was located at r 2 + y 2 =500 cm, making the domain size at least 1000 times larger than the radius of the disk (see Alexander et al. [START_REF] Alexander | Contribution of surface distributions to constant-phase-element (cpe) behavior: 1. influence of roughness[END_REF]). This geometry was chosen to ensure that the counterelectrode could be considered to be infinitely far from the working electrode. A free triangular mesh was used with a greater density of elements near the working electrode.

A capacitance distribution was simulated by a Fourier series that represented a square wave distribution, as shown in Figure 2 as a function of radial position. The Fourier series [START_REF] Fong | Advanced Mathematics for Applied and Pure Sciences[END_REF] was expressed as

C 0 (r) = C + ∞ n=1 (C max -C min ) cos (2nπr/P ) (1)
where the constants C max and C min represented the maximum and minimum capacitance. The average capacitance of the electrode surface C was calculated as

C 0 = 2 r 2 0 r 0 0 C 0 rdr (2)
where r 0 is the radius of the disk and C 0 is given by equation [START_REF] Jorcin | CPE analysis by local electrochemical impedance spectroscopy[END_REF]. The capacitance as a function of the radial position was integrated over the surface of the electrode and then divided by the area of the electrode. The period P of the square wave may be representative of an elemental grain size in which the capacitance is assumed to be relatively uniform across a grain and then jump to another value at an adjacent grain. Simulations were performed with 5, 10, 20, and 40 terms of the Fourier series. Numerical artifacts were observed at high frequencies when the series was truncated at 5 and 10 terms. As more terms were added to the series the artifacts shifted to higher frequencies and with 20 terms in the series this behavior was eliminated from the simulated frequency range. The Fourier series was therefore truncated after 20 terms.

The electrode was assumed to be purely capacitive, i.e., the contributions of slow reaction kinetics and mass-transfer were neglected. The potential distribution within the electrolyte domain was solved using Laplace's equation

∇ 2 Φ = 0. (3)
The potential comprises steady-state and oscillating components and may be expressed as

Φ = Φ + Re Φexp(jωt) (4)
The potential of the electrode surface may also be expressed in the same manner as

V = V + Re Ṽ exp(jωt) (5)
The normal oscillating current density at the surface of the electrode was expressed as

ĩ = jωC 0 (r)( Ṽ -Φ) ( 6
)
where Ṽ is the potential perturbation at the electrode and Φ is the complex oscillating potential within the electrolyte. The value of the potential perturbation will not affect the impedance response, but, for all of the stimulations presented in this work, a value of 10 mV was used. The oscillating potential was constrained to be equal to zero at the counterelectrode. The impedance was calculated as

Z(ω) = Ṽ ĩ (7)
for a specified range of frequencies.

Frequency dispersion in impedance measurements may occur for blocking electrodes that yield a nonuniform current distribution across the working electrode surface. A dimensionless frequency K may be expressed as the ratio of ohmic resistance and interfacial impedance as

K = ωC 0 R e (8)
such that frequency dispersion is induced at K ≥ 1. In the case of a blocking disk electrode within an insulating plane, the ohmic resistance in units of Ωcm 2 is given by [START_REF] Newman | Resistance for flow of current to a disk[END_REF] R e = πr 0 4κ [START_REF] Alexander | Contribution of surface distributions to constant-phase-element (cpe) behavior: 1. influence of roughness[END_REF] such that the dimensionless frequency is

K = π 4 ωC 0 r 0 κ (10)
The frequency at which dispersion occurs is found at K = 1 as

f c = 2κ π 2 C 0 r 0 (11)
Equation 11 may be used to estimate the frequency at which the disk geometry will begin to influence the impedance response.

Results and Discussion

Impedance simulations are presented to show the influence of a heterogeneous surface capacitance on disk electrodes. Results are shown first for a recessed electrode configuration and then a disk electrode within an insulating plane. Dimensionless results are used to determine the characteristic lengths and frequencies associated with a radially-periodic distribution of capacitance.

Capacitance Distribution on Recessed Electrodes

A recessed electrode model was used to isolate the effect of surface heterogeneity on the impedance response of blocking electrodes. The simulated impedance response of a recessed disk electrode with the square wave capacitance distribution shown in Figure 2 is presented in Figure 3 as a function of frequency with the period of distribution as a parameter. The minimum and maximum capacitance values expressed in Equation 1 were set to 1µF/cm 2 and 10µF/cm 2 respectively to represent a surface with an oxide film and a bare metal surface. The conductivity of the solution was 10 -5 S/cm. The impedance was representative of an ideal capacitor at frequencies below 100 kHz, shown by the vertical lines that are perpendicular to the real axis.

The figure inset shows a magnified view of the impedance at high frequencies. As the period of the capacitance distribution increases, the frequency at which dispersion begins decreases.

The simulated real impedance as a function of frequency is presented in Figure 4(a) with the period of the capacitance distribution as a parameter.

The low-frequency limit of the real impedance increased as the period of the distribution increased. The high-frequency limit, which represents the ohmic resistance, was unaffected by the distribution of capacitance. A phase angle dependent only on the imaginary part of the impedance was defined by Alexander et al. [START_REF] Alexander | Contribution of surface distributions to constant-phase-element (cpe) behavior: 1. influence of roughness[END_REF] as

P = 6 0 µm Z r / k Ωc m 2 f / H z (a) 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5 1 0 -1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5
C m i n = 1 µF / c m 2 C m a x = 1 0 µF / c m 2 -Z j / Ωc m 2 f / H z (b)
ϕ dZj = dlog(Z j ) dlog(f ) × 90 • . (14)
As compared to other definitions of phase angle, the imaginary-impedancederived phase angle is more sensitive to the onset of frequency dispersion.

The imaginary-impedance-derived phase angle is presented in Figure 7(a)

as a function of frequency with the period of distribution as a parameter.

The effect of a varying capacitance along the surface of the electrode did not influence the imaginary impedance at low frequencies as indicated by a phase angle, φ dZj = -90 • . However, frequency dispersion did occur in all cases at high frequencies with a minimum phase angle of approximately -50 • . The deviation from the expected capacitive behavior occurred at lower frequencies as the period of the distribution increased. If the period of the distribution is taken as the grain size within a polycrystalline surface, grain sizes on the order of 1µm and less should not lead to frequency dispersion at frequencies less than 1 kHz.

A dimensionless frequency can be expressed for the distribution of capacitance by amending Equation 8to include the surface-averaged capacitance expressed as

K = ω C 0 R e,C (15)
where R e,C represents the ohmic resistance associated with the nonuniform capacitance distribution. The ohmic resistance associated with the capacitance distribution is proportional to the difference between the asymptotic values of the real impedance at high and low frequency. The ohmic resistance per unit area associated with a radially distributed capacitance may be approximated as

R e,C ≈ P κ ( 16
)
where P represents the period of the distribution. According to Equation 10, the dimensionless frequency may now be expressed as

K4P/πr 0 = ω C 0 P κ (17)
The impedance was scaled by the approximated ohmic resistance as Zκ/P .

The imaginary-impedance-derived phase angle is presented in Figure 7(b) as a function of dimensionless frequency. The results were superposed, confirming that the period is the appropriate characteristic length to use to describe surface heterogeneity of capacitance. The characteristic frequency at which a non-uniform capacitance began to influence the impedance was determined to be K4P/πr 0 = 1. Since impedance is often used to measure interfacial capacitance, it is important to ensure that a distribution of capacitance along the surface does not complicate the use of this technique. The effective capacitance of the electrode/electrolyte interface may be determined from the imaginary part of the impedance as

1 0 -8 1 0 -6 1 0 -4 1 0 -2 1 0 0 1 0 2 1
C eff = -1 ωZ j (18)
The ratio of the calculated capacitance and the surface-averaged capacitance is shown in Figure 8 as a function of dimensionless frequency with the period of the distribution as a parameter. In all cases, the simulated capacitance closely matched the surface-averaged capacitance of the electrode at low frequencies.

Capacitance Distribution on Disk Electrodes

The impedance of a disk electrode within an insulating plane containing a frequency in which the period of the distribution was used as the characteristic length. The frequency dispersion due to the surface heterogeneity was superposed, indicating that the period of the capacitance distribution is the appropriate characteristic length. The results obtained for a recessed electrode provided in Figure 7 show only the frequency dispersion due to the surface distribution of capacitance.

0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 -3 -4 -5 -6 -7 -8 -9 S
0 -4 1 0 -3 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 -3 -4 -5 -6 -7 -8 -9 P
The imaginary-impedance-derived phase angles are presented in Figure 11(a) as a function of frequency with the radius of the disk as a parameter.

The period of the distribution was fixed at 60 µm. Changes in the radius of the electrode influenced only the frequency dispersion associated with the geometry of the disk. The imaginary-impedance-derived phase angle is shown in Figure 11(b) as a function of dimensionless frequency, Equation 10. The frequency dispersion associated with disk geometry was superposed and the frequency dispersion associated with the capacitance distribution was not.

Simulations were performed to explore whether the amplitude of the capacitance distribution may influence the frequency at which the impedance is influenced. The imaginary-impedance-derived-phase angle was calculated from simulations on planar disk electrodes with different amplitudes of capacitance distribution and are presented in Figure 12 as a function of frequency. As the amplitude of the distribution increased, the deviation from -90 • shifted to lower frequencies. However, when the frequency is made dimensionless with the use of the period as the characteristic length and the average value of the capacitance, shown in Figure 12(b), the characteristic frequency at which dispersion occurred did not change with amplitude and only the magnitude of the dispersion increased.

The ratio of the calculated capacitance and the average capacitance is presented as a function of dimensionless frequency in Figure 13 with the period of the distribution as a parameter. In all cases, the simulated capacitance closely matched the averaged-surface capacitance of the electrode at low frequencies, indicated by a capacitance ratio value equal to 1. At high frequencies, an initial decrease of the capacitance ratio was caused by the nonuniform current distribution on the disk surface. At a dimensionless frequency equal to one, a further decrease was observed due to the capacitance heterogeneity.

1 0 -3 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 1 0 3 -0 -0 -0 -0 -0 -0 -0 -0 -0 -1 0 S u
1 0 -5 1 0 -4 1 0 -3 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 -2 -3 -4 -5 -6 -7 -8 -9 -1 0
The characteristic frequency at which frequency dispersion occurs for a planar disk electrode exhibiting a distribution of capacitance may be expressed as where C 0 represents the surface-averaged capacitance. The characteristic frequency at which frequency dispersion begins due to the capacitance distribution may be expressed as

f c,r 0 = 2κ π 2 C 0 r 0 (19) 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5 1 0 - 1 0 - 1 0 - 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 r 0 = 0 . 5 c m κ/ < C 0 > = 1 0 -1 c m / s κ/ < C 0 > = 1 0 c m / s κ/ < C 0 > = 1 c m / s f c / H z P o r r 0 / µm P = 1 µm
f c,P = κ 2π C 0 P (20)
in which the period of the distribution is the characteristic length associated with the distribution. The frequency at which dispersion begins is presented in Figure 14 as a function of both the disk radius as well as the period of the capacitance distribution with the ratio, κ/ C 0 , as a parameter. The period of the distribution may be associated with the average width of grain sizes within a noncrystalline surface. For a 0.5 cm radius electrode in a system with κ/ C 0 = 1 cm/s (corresponding for example, to C 0 = 20 µF/cm 2 and a 0.16 mM sodium chloride concentration) and an average grain size of 1 µm, the frequency at which dispersion occurs due to the disk geometry is 400 mHz while the frequency at which dispersion occurs due to the nonuniform capacitance is 1.6 kHz. The frequency dispersion due to the capacitance distribution will always occur at greater frequencies than the frequency dispersion due to the disk geometry.

Conclusions

The role of surface capacitance heterogeneity on solid electrodes was studied using numerical finite element simulations under the assumption of blocking electrode behavior. Multiple cell geometries including a recessed electrode, and an electrode within an insulating plane were used to assess the effect of a radial distribution of capacitance. It was shown that surface heterogeneity due to non-uniform capacitance only caused frequency dispersion at frequencies greater than 1 kHz for a period of 1 µm.

Two distinct deviations from ideality were observed for a disk electrode in an insulating plane containing a periodic distribution of capacitance. The first is indicative of the effect of the disk geometry, and the second represents the influence of the surface heterogeneity. The characteristic length associated with a periodic distribution of capacitance is the period itself. Physically the period may represent the average width of grain sizes, which is usually on the order of microns. The characteristic frequency determined by the second deviation associated with capacitance distribution is f c = κ/2π C 0 P in which the period of the distribution is used as the characteristic length.

Surface heterogeneity due to a nonuniform distribution of capacitance does not provide a physical explanation for CPE behavior over a broad range of frequencies. A non-uniform capacitance only causes frequency dispersion at frequencies greater than those due to the geometry of a disk electrode contained within an insulating plane. The frequency dispersion due to the surface heterogeneity would only influence the impedance at frequencies lower than those influenced by the disk geometry if the period of the surface distribution were greater than the radius of the disk, which is physically impossible.

Figure 1 :Figure 2 :

 12 Figure 1: Finite-element mesh for the disk electrode simulations. a) Axi-symmetric domain used to represent disk electrode experiments; b) mesh showing a detailed view of a recessed electrode. The outer domain is shown in Figure 1(a).

  Ωc m 2 P e r i o d = 6 0 µm P e r i o d = 3 0 µm P e r i o d = 1 5 µm U n i f o r m C a p a c i t a n c e 1 k H z

Figure 3 :

 3 Figure 3: The impedance in Nyquist format of a recessed disk electrode with the square wave capacitance distribution shown in Figure 2 and the period of distribution as a parameter.

1 0 6 P

 16 e r i o d = 6 0 µm P e r i o d = 3 0 µm P e r i o d = 1 5 µm C a v g = 5 . 5 µF / c m 2

Figure 4 :+ ĩj 2 ( 13 )Figure 5 :

 42135 Figure 4: The simulated impedance as a function of frequency of a recessed disk electrode with a square wave capacitance distribution and the period of distribution as a parameter: a) the real part of the impedance; b) the imaginary part of the impedance.

Figure 6 :Figure 7 :

 67 Figure 6: Normal current distribution at the electrode surface due to a nonuniform capacitance distribution with a period of 60µm of a recessed electrode as a function of radial position: a) current distribution at 10mHz; b) current distribution at the high frequency limit of the simulations f = 100 kHz.

Figure 8 :

 8 Figure 8: The ratio of the calculated effective capacitance and the surface-averaged input capacitance as a function of dimensionless frequency K4P/πr0 with the period of the distribution as a parameter for recessed disk electrodes.

Figure 9 :

 9 Figure 9: Normal current distribution at a disk electrode surface as a function of radial position: a) current distribution at 10 mHz; b) current distribution at 100 kHz.

  u r f a c e D i s t r i b u t i o n P e r i o d = 6 0 µm P e r i o d = 3 0 µm P e r i o d = 1 5 µm U n i f o r m C 0 ϕ d Z j / d e g r e e s f / H z D i s k g e o m e t r y (a)

Figure 10 :

 10 Figure 10: Imaginary-impedance-derived phase angle for a disk electrode within an insulating plane: a) phase angle as a function of frequency; b) phase angle as a function of dimensionless frequency based on the averaged capacitance.

r f a c e D i s t r i b u t i o n D i s k g e o m e t rFigure 11 :

 11 Figure 11: Imaginary-impedance-derived phase angles values calculated from the impedance data with the disk radius as a parameter: a) imaginary-impedance-derived phase angle as a function of frequency; b) imaginary-impedance-derived phase angle as a function of dimensionless frequency K = ω C0 πr0/4κ.

2 ϕFigure 12 :

 212 Figure 12: Imaginary-impedance-derived phase angle calculated from impedance data on disk electrodes with a radial distribution of capacitance with the amplitude of the square wave as a parameter: a) phase angle as a function of frequency; b) phase angle as a function of dimensionless frequency based on the averaged capacitance and the period of the square wave as the characteristic length.

Figure 13 :

 13 Figure 13: The ratio of the calculated effective capacitance and the surface-averaged input capacitance as a function of dimensionless frequency K4P/πr0 with the period of the distribution as a parameter for disk electrodes within an insulating plane.

Figure 14 :

 14 Figure 14: The frequency K4P/πr0=1 at which the surface heterogeneity influences the impedance as a function of distribution period and disk radius with κ/ C0 as a parameter.

Acknowledgements

M. Orazem acknowledges partial support from the University of Florida

Research Foundation Professorship.


M Glass-Maujean

Ch Jungen

G D Dickenson

N De Oliveira

W Ubachs

The excited J = 0 1 Σ + u levels of D 2 : Measurements and ab initio quantum defect study

The DESIRS beamline of the SOLEIL synchrotron facility, equipped with a vacuum ultraviolet Fourier-transform spectrometer has been used to measure P (N = 1) (N -N = -1) absorption transitions of the D 2 molecule. Some 44 P -lines were assigned and their transition frequencies determined up to excitation energies of 134000 cm -1

above the ground state, thereby extending the earlier work by various authors, and considerably improving the spectral accuracy (< 0.1 cm -1 ). The assignments have been aided by first principles multichannel quantum defect theory (MQDT) calculations. These calculations also provide predictions of the autoionization widths of the upper levels which agree well with the observed resonance widths.

1 Graphical abstract 2 Highlights

7 v = v = v = v = v = v = v = v = v = v = v = v = v = v = 2
• 44 photoabsorption transitions to npσ 1 Σ + u N = 0 upper levels of D 2 measured and assigned.

• 27 previously unknown spectral lines observed and assigned.

• Autoionization widths determined.

• Calculation from first principles of perturbed spectra in D 2 .

• Quantitative agreement between observations and ab initio theory.

Introduction

Recently there has been renewed interest in the spectroscopy of the D 2 molecule because the isotopomers D 2 and H 2 are key molecules in the ongoing high-precision tests of quantum theory [1]. A detailed comparison of the spectroscopic properties of D 2 and H 2 must necessarily involve the study of non-adiabatic coupling of the electronic and nuclear degrees of freedom, a type of coupling which is strongly dependent on the nuclear mass. D 2 has been less systematically studied than H 2 . In the sixties and seventies of the preceding century only the lowest excited states that can be reached by photoabsorption have been studied in detail, namely the B, C, B and D states, mainly in Herzberg's spectroscopy group at the National Research Council in Ottawa [2,3,4,5]. More recently the B, C, B and D states have been reinvestigated at improved accuracy by classical emission spectroscopy [6,7], vacuum-ultraviolet (VUV) Fourier-transform absorption [8,9], and by Doppler-free VUV-laser excitation [10]. Higher electronic states of D 2 have been studied in photoabsorption in the seventies, in a single paper, as far as we are aware, by Takezawa and Tanaka [11] whose observations went up to ≈ 130000 cm -1 .

Theoretical calculations on D 2 taking account of non-adiabatic effects in excited states began as early as 1977 when ab initio MQDT calculations of levels of the B, C, D and B states based on the then best available quantumchemical potential energy curves were published [12]. More recent studies are those of Refs. [6] and [7] where the coupled equations (CE) approach was used. A comparative study of the CE and MQDT methods as applied to the levels of the B and C states was carried out in Ref. [13] a few years ago. The MQDT approach used in that work was similar to that employed in Ref. [12] but included various improvements and extensions such as the systematic inclusion of the energy dependence of the quantum defects (see Ref. [START_REF] Ch | Elements of Quantum Defect Theory[END_REF]) and the calculation of transition intensities in addition to the level energies. The same approach was applied for the calculations contained in the very recent combined theoretical and experimental studies of excited 1 Π - u states of D 2 [START_REF] Glass-Maujean | [END_REF]16].

The purpose of the present paper is to extend the studies of Refs. [START_REF] Glass-Maujean | [END_REF] and [16] to the N = 0 excited levels of pure 1 Σ + u symmetry, which are investigated in absorption. These levels do not undergo any rotational-electronic non-adiabatic interactions, but they are significantly affected by vibrationalelectronic (vibronic) coupling -much more so than their 1 Π - u counterparts. The MQDT calculations reported here have been carried out in exactly the same manner as those reported in Ref. [17] for the N = 0 1 Σ + u levels of the H 2 isotopomer, except that the proton mass has been replaced by the deuteron mass in the computations. The reader is referred to that paper for all the details concerning the calculations. We shall use the calculated level positions and spectral line intensities for the further analysis of the absorption spectrum recorded at the SOLEIL synchrotron on the DESIRS beam line [8].

Experimental

The absorption spectrum of D 2 has been recorded using the Fourier-transform spectrometer installed on the DESIRS beamline at the SOLEIL synchrotron [20,21]. Before entering the spectrometer, the synchrotron radiation passes through a cell that is 10 cm long and contains the D 2 sample which is maintained at a quasi-static pressure of ≈ 0.1 mbar. The continuum background provided by the undulator corresponds approximately to a Gaussian of 7000 cm -1 width. The spectrum from 117000 -135000 cm -1 has therefore been partitioned into four overlapping sections. The Fourier-transform spectrometer possesses its own internal calibration derived from the spatial step-size of the interferometer, which is controlled by a frequency stabilized He-Ne laser. Nevertheless small alignment errors may induce errors in the calibration that may vary from one recording to another. De Lange et al. [9] introduced corrections for the different energy intervals under investigation, which account for the small deviations in the travel of the moving mirror. By applying these derived corrections it is possible to improve the accuracy of the measurements, and, for higher energies for which no corrections have been derived, it is possible to extrapolate these corrections and again improve the accuracy of the measurements. This procedure however induces a residual error which increases as the energy increases [START_REF] Glass-Maujean | [END_REF].

The continuum background has been fitted to a Gaussian and Beer-Lambert's law applied to obtain a signal proportional to the absorption cross section. A section of this quantity is plotted in Fig. 1 along with the calculated P (1) transitions and their assignments. The individual observed peaks thus obtained have been fitted to Gaussians, from which the natural widths were determined as follows.

Each observed absorption transition represents the result of the convolution of three effects: (i) the apparatus function which has the form of a sinc function, (ii) the Doppler width which is a Gaussian, and (iii) the natural width which typically corresponds to a Lorentzian. As the Doppler width (0.4 cm -1 ) is much larger than the apparatus function (0.16 cm -1 ), the convolution of these two functions is well represented by a single Gaussian function of width 0.4 or 0.5 cm -1 depending on the spectral recording con- sidered. We have verified this point by test calculations made by measuring the effective widths of lines known to produce fluorescence radiation, and whose natural widths therefore are far less than the Doppler width and the instrument width [16]. Next, we have calculated for each transition the line profile resulting from the convolution of the known Gaussian representing the effective Doppler width and an assumed Lorentzian profile representing the natural width. This profile was fitted itself by a Gaussian, which was then compared to the experimentally determined effective width. For each transition the assumed natural width was then varied until the observed effective width could be reproduced. The uncertainty of the natural width thus determined was taken to be three times the uncertainty obtained in the fit. Despite the high spectral resolution achieved in this experiment, we have found that many transitions are not completely resolved so that we had to represent the profiles by several overlapping Gaussians. The uncertainty of the natural width determination in these cases is substantially larger.

1 2 2 0 0 0 1 2 3 0 0 0 1 2 4 0 0 0 n = n = n = n = n = n = n = n = n = n = n = n = n = n = 4 4 6 5 7 4 8 6 5 9 1 0 1 1 4 7 v = v = v = v = v = v = v = v = v = v = v = v = v = v = 2

Calculations

As has already been stated the calculations have been carried out in exactly the same fashion as in our previous paper on the P (1) transitions H 2 [17], except that the reduced nuclear mass and the value of the Rydberg constant have been adapted to the D 2 molecule. The approach assumes that the Rydberg spectrum corresponds to excitation of a single electron which may be described by a single partial wave, = 1 which yields npσ and npπ Rydberg configurations. Higher partial waves and doubly excited configurations are not explicitly taken into account. However the quantum defect functions derived from ab initio potential energy curves account for the effect of such configurations in an effective manner (cf. Ref. [17] for details and the general comments made in Ref. [12]). In practice, two sets of calculations have been carried out. In the first set of calculations all open ionization channels have been artificially removed so that a spectrum of discrete lines was obtained which was used for the purposes of spectral assignments and from which the line intensities were obtained. In the second set of calculations the open ionization channels were retained so that the MQDT calculations yielded a continuous spectral intensity distribution from which, via a fit to a Lorentzian shape, the width for each autoionization resonance could be extracted. Unlike in the previous Ref. [17] we have neglected the effect of predissociation in the present work as it significantly affects only few of the observed upper levels (cf. the discussions in Sec. 6.2 below and in the following paper [18]).

Results

The present results are summarized in Table 1. We discuss the various aspects consecutively.

Energy levels

We have been able to assign 44 P (1) lines unambiguously in the range 117300 -134300 cm -1 . These correspond to upper level npσ 1 Σ + u with n ≥ 4. The 2pσB 1 Σ + u and 3pσB 1 Σ + u states have been studied previously and their discrete level structure lies at lower energies than we consider here [9]. The uncertainty of the energy levels has been taken as three times the statistical error plus the uncertainty of the calibration (up to 0.04 cm -1 ). In the case of very weak lines an additional uncertainty arises which amounts to 0.07 cm -1 and corresponds to the interval of the noise wiggles (i.e. the energy difference between successive wiggles) present in the recordings.

The deviations observed-calculated are generally less than 1 cm -1 , but are clearly larger than the experimental uncertainty and therefore reflect the short-comings of our calculations. The histogram of the deviations observedcalculated corresponds in a good approximation to a Gaussian centered at -0.29 ± 0.03 cm -1 and a width of 0.6 cm -1 . Figure 2 is a plot of these deviations versus the transition energy which shows that the accuracy of the calculations deteriorates as the energy increases. The higher excitation energies lead to higher electronic as well as higher vibrational excitation. In higher vibrational levels the molecule explores larger internuclear distances, where the assumptions underlying the theoretical approach begin to break down. Specifically, the B B 1 Σ + u state possesses a potential maximum near R ≈ 6 a.u. at an energy ≈ 130000 cm -1 [5,19], which is disregarded in our approach, but whose effects clearly show up in the plot of Fig. 2. A fuller treatment would require explicit inclusion of core-excited Rydberg channels and of the long-range H + +H -ion-pair channel. These developments are outside of the scope of the present paper. The intensities of the npσ P (1) transitions are expected to decrease with increasing principal quantum number n in accordance with the n -3 Rydberg scaling law. This is true overall, but locally the effects of vibronic interseries perturbations disrupt the regular decrease. At the same time the autoionization widths are rather substantial in the pure 1 Σ + u symmetry upper levels of the P (1) transitions -several cm -1 in some cases -with the result that for n ≥ 10 the resonances tend to merge into the background continuum and cannot be detected in our experiment. Further, as v increases, the density of states increases while the line intensities strongly decrease for v ≥ 4 due to unfavorable Franck-Condon factors. The widths also tend to increase. As a result, again, definitive assignments become more difficult at higher energies. Nevertheless it has been possible to follow the upper levels of 4pσ up to v = 13, 5pσ and 6pσ up to v = 10, 7pσ up to v = 3, 8pσ up to v = 7, and 9pσ up to v = 4, respectively. The P (1) transitions to 4pσ, v = 0 -9 have previously been observed and assigned by Takezawa and Tanaka [11]. We have remeasured them here with higher accuracy and extended the vibration progression to v = 13. Similarly, Takezawa and Tanaka had observed the 5pσ and 6pσ progressions up to v = 4 and v = 3, respectively, whereas we have remeasured all of these transitions and extended the assignments up to v = 10. It turns out that the 5pσ, v, N = 0 and 6pσ, v -1, N = 0 levels are completely mixed by ∆v = 1 vibronic interaction for v = 3 and v = 4. By labeling each level according to the largest channel contribution given by the MQDT calculation we find that we have to invert the assignments made by Takezawa and Tanaka [11], by formally assigning the P (1) transition at 124325.7 cm -1 to 5pσ, v = 3 ← X, v = 0 instead of 6pσ, v = 2 ← X, v = 0, and the transition at 125667.9 cm -1 to 5pσ, v = 4 ← X, v = 0 instead of 6pσ, v = 3 ← X, v = 0, and changing the assignments of the transitions at 124569.2 cm -1 and 125930.9 cm -1 accordingly. Recall however that these assignments to specific upper levels npσ, v have only a limited meaning and are useful mainly for bookkeeping purposes. It is the full MQDT calculation which accounts for the vibronic interactions and produces mixed states that allows the spectrum to be analyzed. Another reassignment concerns the P (1) 4pσ, v = 9 ← X, v = 0 transition which is identified here as corresponding to the line observed at 127968.2 cm -1 , whereas previously [11] it had been assumed to be blended by the R(0) 4pπ, v = 7 ← v = 0 transition that appears at 127974.2 cm -1 , i.e. about 6 cm -1 higher.

Overall, out of 18 P (1) line assignments corresponding to 4 -6pσ published previously [11], 3 have been reassigned, one has been invalidated (4pσ, v = 9), while the remaining transitions have been remeasured here with higher accuracy. 5pσ, v = 0 is an exception as it is too weak in our spectra to be unambiguously assigned. 26 P (1) transitions npσ with n = 4 -9 are reported here for the first time.

Widths

The natural decay widths of the upper levels of the P (1) transitions have been determined as described in Sec. 4 and are included in Table 1. The uncertainties given in Table 1 correspond to three times the 1σ statistical errors due to the measurement of each resonance and to the apparatus function (≈ 0.15 cm -1 ). These values are compared in Table 1 and Fig. 3 with the calculated ionization widths. In most cases the calculated width is less than the uncertainty of the measuremement, which therefore provides an upper limit for the natural width. All 4pσ resonances are calculated to have widths less than the exper-imental uncertainty. Exceptions arise for v = 11 and v = 13 where the measured values are larger than calculated. We suspect that these widths are due mainly to dissociation, a decay channel that we have not included in our calculations. Indeed we know from the study of H 2 that these levels primarily decay via predissociation [22]. Fig. 3 shows that while the experimental widths of the 5pσ levels with v values around 6 agree with the calculated ionization widths in the sense specified above, there is marked disagreement for v = 3 and v = 9. The line intensity in the latter case is quite weak and hence the experimental determination has a large uncertainty, but nevertheless the calculated ionization width is distinctly larger than the observed value, a fact that is not understood at this stage. The line widths of the 6pσ progressions again agree with the calculations. The large calculated width of v = 9 is to be noted -however it has proved impossible to locate the corresponding transition in the experimental spectrum. The few transitions to 7pσ levels that have been identified experimentally, up to v = 3 only, yield widths that are again in line with the calculations. The same is true for the few observations that we have made in the 8pσ progression, cf. Fig. 3.

Only two transitions belonging to the 9pσ progression have been observed, namely v = 2 and v = 4. The observed and calculated large widths for v = 2 are in nice agreement, 5.9 ± 3.6 and 6.2 cm -1 , respectively, and this is also the case for the contrastingly small v = 4 widths: 0.00 ± 0.16 and 0.03 cm -1 , respectively. The latter case is interesting as the 9pσ, v = 4, N = 0 level is part of a 'complex' resonance [23], as it is immersed in the dense manifold of Rydberg levels npσ, v = 3 with n ≈ 43 with which it interacts. These are spaced by about 2.8 cm -1 and have very small ionization widths in such a way that the present experiment is able to resolve the comb-like fine structure of the 'complex' resonance, see Fig. 4. For this reason the observed value for the width given in Table 1 is the genuine width of the 9pσ, v = 4 interloper and does not reflect the width of the whole group of interacting levels.

In conclusion, it appears that the observed widths of the P (1) transitions are for the most part due to autoionization, whereas predissociation contributes significantly only for n = 4 (4pσ progression). The ionization widths of all the upper levels with n ≤ 8 are basically quite small as they cannot proceed via the ∆v = 1 propensity rule since the v -1 ionization continuum is closed for this class of upper levels. Their widths are erratic as they may be enhanced only by the accidental local perturbations, which involve levels that are able to ionize via the preferred ∆v = 1 ionization route. These perturbing levels then play the role of 'doorway' states as they are known in non-radiative transitions in large molecules or in nuclear reactions [26,27].

Summary and Conclusion

We have observed the VUV Fourier-transform absorption spectrum of D 2 and analyzed in the present paper the P (1) transitions with the help of firstprinciples MQDT calculations. The lower states correspond to X 1 Σ + g , v = 0, N = 1, while the N = 0 upper states of these transitions have positive total parity and pure 1 Σ + u electronic symmetry. We have identified 44 such transitions with an experimental uncertainty of better than 0.1 cm -1 . The upper states correspond to Rydberg series npσ with 4 ≤ n ≤ 9. 18 transitions have been observed previously but are determined here with improved accuracy, while the remaining ones are reported here for the first time. On the basis of our calculations we have revised a few previous assignments. The observed and calculated line frequencies agree to within 0.3 cm -1 on the average. The ionization widths have also been evaluated and agree fairly well with the natural widths extracted from the experimental spectra. Overall, however, the agreement between experiment and theory deteriorates noticeably as the energy increases and levels are excited whose vibrational motion explores internuclear distances up to and beyond ≈ 5 a.u.. Values in italics are calculated transition energies, given when there is no corresponding observation. The positions of the upper state energy levels above the v = 0, N = 0 ground state level are obtained by adding the ground-state rotational energy 59.78 cm -1 (N = 1) [1] to the transition energy.

  o r p t i o n c r o s s s e c t i o n s ( a . u . ) t r a n s i t i o n e n e r g y ( c m -1 )

1 )Figure 1 :(

 11 Figure 1:(Color online) Absorption cross section (in arbitrary units) of D 2 between 121400 and 124000 cm -1 taken at the DESIRS beamline of the SOLEIL synchrotron. Calculated P (1) transitions leading to npσ, v, N = 0 Rydberg levels are indicated by filled circles (red online). Their intensities have been adjusted to the experiment by a global adjustment factor.

1 )Figure 2 :

 12 Figure 2: Deviations observed-calculated for the P (1) transitions assigned in the D 2 absorption, plotted versus the transition energy.

Figure 3 :(

 3 Figure 3: (Color online) Observed (black filled squares) and calculated (open blue squares) natural widths of P (1) transitions assigned in the D 2 absorption spectrum, plotted versus the vibrational quantum number. The calculated widths correspond to the ionization partial widths obtained from the MQDT calculations.

  o r p t i o n ( a . u . ) t r a n s i t i o n e n e r g y ( c m -

Figure 4 :

 4 Figure 4: Section of the absorption spectrum of D 2 near 129200 cm -1 showing the 9pσ, v = 4, N = 0 resonance immersed in the npσ, v = 3, N = 0 Rydberg series with n ≈ 43.

Table 1 :

 1 P (1) transitions in the photoabsorption spectrum of the D 2 molecule

	state v	ν/c obs a	± b	obs-calc Γ(obs) c	± b	Γ i (calc) d
	4pσ	0 117137.40 (0.04)	-0.53	0.00	(0.16)
	4pσ	1 118628.25 (0.03)	-0.51	0.00	(0.16)
	5pσ	0 119964.61
	4pσ	2 120052.72 (0.04)	-0.51	0.00	(0.15)
	4pσ	3 121393.89 (0.03)	-0.28	0.01	(0.15)
	6pσ	0 121427.85 (0.04)	-0.07	0.00	(0.18)
	5pσ	1 121544.22 (0.04)	-0.62	0.01	(0.15)
	7pσ	0 122344.84
	4pσ	4 122680.82 (0.03)	-0.01	0.02	(0.15)
	8pσ	0 122897.97
	6pσ	1 122927.33 (0.07)	-0.10	0.00	(0.19)
	5pσ	2 123077.98 (0.05)	-0.70	0.15	(0.16)
	9pσ	0 123300.46
	4pσ	5 123890.95 (0.07)	-0.32	0.09	(0.15)
	7pσ	1 123901.50 (0.08)	-0.22	0.00	(0.24)
	5pσ	3 124325.74 (0.88)	-0.08	0.38	(0.92)
	8pσ	1 124468.14
	6pσ	2 124569.22 (0.04)	-0.64	0.00	(0.15)
	9pσ	1 124870.46				2.9
	4pσ	6 125026.34 (0.05)	-0.34	0.00	(0.15) 0.000002
	7pσ	2 125404.49 (0.09)	0.00	0.37	(0.17)	0.17
	5pσ	4 125667.90 (0.05)	0.03	0.00	(0.15)	0.00029
	6pσ	3 125930.93 (0.05)	-0.30	0.12	(0.16)	0.003
	8pσ	2 126025.17 (0.04)	-0.39	0.09	(0.15)	0.013
	4pσ	7 126086.71 (0.04)	-0.33	0.04	(0.15)	0.0004
	9pσ	2 126372.69 (0.69)	-2.02	5.90	(3.62)	6.2
	7pσ	3 126816.55 (0.10)	-0.20	0.06	(0.23)	0.27
	5pσ	5 126937.56 (0.06)	0.04	0.07	(0.16)	0.001
	4pσ	8 127071.15 (0.04)	-0.37	0.05	(0.16)	0.0002
	6pσ	4 127310.23 (0.05)	-0.54	0.06	(0.15)	0.032
	8pσ	3 127443.05 (0.04)	-0.28	0.00	(0.15)	0.014
	9pσ	3 127814.57				8.9
	4pσ	9 127968.82 (0.06)	-0.35	0.08	(0.16)	0.0001
	5pσ	6 128153.20 (0.05)	-0.12	0.12	(0.15)	0.084
	7pσ	4 128196.95				0.34
	6pσ	5 128616.05 (0.04)	-0.66	0.10	(0.15)	0.045

a Transition energy in cm -1 . b Uncertainty in cm -1 . c Observed line width in cm -1 . d Calculated ionization width in cm -1 .

Acknowledgments

The authors thank D. Joyeux for his help in the data acquisition on the FTS and L. Nahon for his constant support. Ch. J. thanks the Miescher Foundation (Basel, Switzerland) for partial support.


Arnaud Ginestet
email: arnaud.ginestet@noveltis.fr

Christine Letrou

Gilles Beauquet

Emna-Amira Fnaiech

Adaptability of Parabolic Wave Equation and Gaussian Beam Shooting methods to Electromagnetic Propagation in Urban Configurations

In spite of well-known limitations, ray based methods are the quasi-unique tool for modelling the electromagnetic propagation in urban areas. As an alternative to these methods, two models based on the resolution of the Parabolic Wave Equation, and on Gaussian Beam Shooting, are proposed. Both are specifically adapted to meet the challenges of urban areas propagation: three-dimensional, wide angle approximation and considering the backscatter propagation for the first one; taking into account ground and grazing angles for the second one. A preliminary test case is presented and configurations of interest under processing are detailed.

I. INTRODUCTION A. Context

The knowledge of electromagnetic propagation is critical to increase the efficiency and need for control of communication systems. It improves detection and discrimination capabilities of radar systems in presence of dense obstacles.

Nowadays cities become a major configuration, since for the first time in the history of mankind the urban population outnumbered rural population in 2007. More, this ratio is expected to reach 60 % by 2030. This change has to be taken into account by the scientific community, regardless of their speciality.

Indeed, the presence of obstructing buildings in an urban areas causes shadowing of the emitter from the receiver and can lead to severe loss of signals. Thus, electromagnetic propagation models need to be adapted to the specific urban configurations, as far as three-dimensional processing becomes mandatory.

B. Configurations of interest

When dealing with electromagnetic propagation, configurations of interest will support the efficiency assessment of modelling tools regarding two objectives: 1) Analysis of urban areas and semi-urban areas as electromagnetic scenes; 2) Production of enough results to help the deployment of a system (radar or stations).

The interest lies in the theoretical forecast of performances as close as possible to reality: it concerns attenuation between Base Station (BS) and User Equipment (UE) for broadband mobile or it concerns complete monitoring of radar detection loss due to obstacles in civil, military or security domains.

Masking and multiple reflections must be taken into account in three-dimensional electromagnetic scenes. Elementary mechanisms in play could be:

• Obstruction due to erected obstacle(s) close to the emitting antenna or close to the receiving antenna; • Lateral deviation on large distant walls; • Influence of relief (at a large scale): beam is guided in a turning valley or dispersed around a hill.

Configurations of interest can be illustrated in the two following cases.

1) Microcell: There are a lot of Line-Of-Sight (LOS) and Non-Line-Of-Sight (NLOS) geometrical situations encountered in urban area as shown in Fig. 1. Various multiple propagation mechanisms occur when BS and UE are operating from and close to tall buildings, in particular those relayed by roofs. Table I lists parameters and their ranges when considering typical configurations.

2) ATC radars: A Primary Surveillance Radar (PSR) system is deployed on an airport. The beam set is rotated. The forward way is sought as not obstructed, except for certain beam directions where building or tower can affect the bottom of the radar beam and could also reflect backward a part of energy. For monopulse radar, signal processing can be more

C. Literature review

The study of electromagnetic wave propagation in urban areas has led to numerous studies, mostly since the advent of mobile telephony. Thus, in reference to the work done within the COST (European Co-operation in the field of Scientific and Technical research) and more precisely the action 231 (Evolution of Land Mobile Radio Communications), a large number of models have been developed. The following models can be highlighted as the most significant ones: the "COST-Hata-Model" [START_REF] Okumura | Field strength and its variability in VHF and UHF land-mobile service[END_REF], the "COST-Walfish-Ikegami-Model" [START_REF] Walfisch | A theoretical model of UHF propagation in urban environments[END_REF] [3], the "CNET ray lauching model" [START_REF] Rossi | A ray Shooting model in urban area[END_REF], the "ASCOM-ETH micro-cell model" [START_REF] Dersch | Physical characteristics of urban micro-cellular propagation[END_REF] [6], the "Villa Griffone's Lab (VGL) model" [START_REF] Daniele | Field prediction tools for wireless communications in outdoor and indoor environments[END_REF], the "Uni-Stuttgart 3D micro-cell model" [START_REF] Gschwendtner | Adaptive Wellenausbreitungsmodelle fr die Funknetzplanung[END_REF] [9] [10] and the "Uni-Karlsruhe model 3D-URBAN-MICRO" [START_REF] Cichon | Strahlenoptische Modellierung der Wellenausbreitung in urbanen Mikro-und Pikofunkzellen (in German, Ray optical modelling of wave propagation in urban micro-and pico-cells)[END_REF] [12] [13] [START_REF] Cichon | Comprehensive ray optical propagation models for indoor and outdoor environments: Theory and applications[END_REF]. However, these models are either empirical models computing only attenuation in two dimensions and consequently neglecting three-dimensional effects; or models based on ray tracing and thus implying the limitations of this method (i.e. very large number of rays and thus unsustainable computation time, presence of caustic(s), far field approximation... specifically in urban configurations).

Our objective is to overcome the limitations of previous models, developing new formulations and new algorithms for modelling the electromagnetic field propagation in urban areas, based on the three-dimensional Parabolic Wave Equation and the three-dimensional Gaussian Beam Shooting.

1) Parabolic Wave Equation: Parabolic Wave Equation (PWE) remains for many years one of the most effective method to simulate electromagnetic propagation in complex environments. However, it is based on two restrictions: the paraxial approximation reducing the validity of the method to a cone around the main propagation direction and the ignoring the back-propagation.

In urban or semi-urban areas, we often encounter the scenario of an obstacle (i.e. a building) obstructing or interfering with the electromagnetic propagation. This will generate reflections on the sides or diffractions by the edges of this obstacle, or in the worst case a back-propagation of the EM signal. Given the limitations of the Parabolic Wave Equation method, urban or semi-urban configurations are disadvantageous using this method.

Nevertheless, it is possible to enlarge the cone of validity of the Parabolic Wave Equation when using a development of higher order (i.e. Feit and Fleck formulation [START_REF] Feit | Light propagation in graded-index fibers[END_REF]). Recently, a Parabolic Wave Equation resolution taking into account both the forward and backward propagation, called "two-ways", was developed. The idea (behind this resolution) is to consider both directions of propagation by reversing the main direction in the resolution of the Parabolic Wave Equation [START_REF] Ozgun | Recursive two-way parabolic equation approach for modelling terrain effects in tropospheric propagation[END_REF] [17] [START_REF] Ozgun | Two-way Fourier Split Step Algorithm over Variable Terrain with Narraow and Wide Angle Propagators[END_REF]. In [START_REF] Apaydin | A Novel Two-Way Finite-Element Parabolic Equation Groundwave Propagation Tool, Tests With Canonical Structures and Calibration[END_REF], a two-ways finite element resolution is presented for the modelling of ground wave propagation over irregular terrain and in the presence of an inhomogeneous atmosphere.

It is interesting to note that all the two-ways resolutions of the Parabolic Wave Equation currently published are restricted to two-dimensional areas. Our ambition is therefore to implement a three-dimensional wide-angle 2-ways resolution of the Parabolic Wave Equation in order to overcome its current limitations and to make it adapted to urban configurations.

2) Gaussian Beam Shooting: Although the Gaussian Beam Shooting (GBS) method is particularly well suited to 3D field simulations in multi-reflecting contexts, few works are currently devoted to the use of Gaussian Beams (GB) to model electromagnetic propagation in urban or semi-urban areas.

In the millimetric range, "simple" versions of Gaussian Beam Shooting have provided results equivalent to that obtained by ray tracing techniques. In these "simple" algorithms, no specific formulation is developed to account for diffraction. If a beam axis is incident on an obstacle, the beam is considered as if incident on an infinite surface. Conversely, if the beam axis does not intercept the obstacle, the beam is considered as if propagating in free space. This methodology was validated by measurements around 60 GHz in indoor environments [START_REF] Fluerasu | Gaussian Beam Shooting for 3D physical modeling of propagation channels[END_REF] [START_REF] Fluerasu | Modélisation de champs dans le domaine spatiotemporel par une méthode de frame de Gabor. Application à la caractérisation du canal indoor millimétrique[END_REF]. An ultra-wide band version of this kind of methodology has been implemented for building areas at non-millimetre frequencies, but to our knowledge result validation has not been published [START_REF] Timchenko | Gaussian beam summation algorithm for ultra wide band indoor channel characterization[END_REF].

More accurate simulations of diffracted fields, yet keeping their representation as a sum of Gaussian beams (formulations sometimes called "B2B", beam to beam), is however a prerequisite for an efficient use of Gaussian Beam Shooting in multipath propagation environments at frequencies of conventional communication systems or radar systems. As mentioned above, only few recent studies have focused on this problem, and the resulting formulations depend on the method used to represent the radiated or scattered fields as discrete sums of Gaussian beams: empirical sampling and normalisation at arrival [START_REF] Darras | Diffraction by a sharp edge for an electromagnetic field described by summation of Gaussian beams: the Ufimtsev way[END_REF], empirical sampling in the angular domain and frame decomposition along the edge [START_REF] Katsav | Gaussian beam summation representation of beam diffraction by an impedance wedge: a 3D electromagnetic formulation within the Physical Optics approximation[END_REF], frame decomposition and frame based truncation of fields on obstacle surface [START_REF] Ghannoum | Gaussian beam shooting algorithm based on iterative frame decomposition[END_REF] [22] [START_REF] Ghannoum | Études d´outils de calcul de propagation Radar en milieu complexe (milieu urbain, présence de multi-trajets) par des techniques de Lancer de Faisceaux Gaussiens[END_REF]. The latter approach is used in this work and yields Physical Optics like representation of diffracted fields.

II. FORMULATION

A. Parabolic Wave Equation

The vectorial wave function Ψ (x, y, z) can be obtained as the solution of the following three-dimensional Helmholtz equation:

∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 + k 2 0 n 2 (x, y, z) Ψ (x, y, z) = 0 (1)
Where n (x, y, z) is the refractive index, k 0 = 2π/λ is the free space wavenumber (λ is the wavelength), and x, y and z are the longitudinal (range), the transverse and the height coordinates, respectively. Ψ (x, y, z) corresponds to either the electric and magnetic field components for horizontal and vertical polarizations.

Considering that the direction of wave propagation is predominantly along the +x-axis paraxial direction, the Parabolic Wave Equation is written by separating rapidly varying phase terms (Ψ (x, y, z) = exp (ik 0 x) u (x, y, z)). That is, the Parabolic Wave Equation in terms of the slowly varying amplitude function u (x, y, z) is expressed as follows:

∂ ∂x + ik 0 1 - √ 1 + T ∂ ∂x + ik 0 1 + √ 1 + T u (x, y, z) = 0 (2)
Where

T = k -2 0 ∂ 2 /∂y 2 + ∂ 2 /∂z 2 + n 2 -1.
The first and second parts of (2) correspond to the forward and backward propagation waves, respectively. If the backward propagation is ignored (2) reduces to:

A 0 ∂ ∂x + A 1 ∂ ∂x ∂ 2 ∂y 2 + ∂ 2 ∂z 2 + A 2 + A 3 ∂ 2 ∂y 2 + ∂ 2 ∂z 2 = 0 (3)
with the help of

√ 1 + T ≈ (a 0 + a 1 t) / (b 0 + b 1 t) ap- proximation; therefore, A 0 = b 0 + b 1 n 2 -1 , A 1 = b 1 k -2 0 , A 2 = ik 0 (b 0 -a 0 ) + (b 1 -a 1 ) n 2 -1 , and A 3 = ik -1 0 (b1 -a1).
If the angle of propagation measured from paraxial direction is less than 15 • , the standard Parabolic Wave Equation is obtained with the help of square root approximation √ 1 + t ≈ 1 + t/2. If the propagation angle is more than 15 • , the Claerbout equation is obtained by using the first-order Padé approximation √ 1 + t ≈ (1 + 0.75t) / (1 + 0.25t) to satisfy propagation angles up to 40 • . Thus, the contribution of a wideangle resolution can be investigated.

A Finite-Difference methodology is then adopted in order to "introduce" the Leontovich boundaries condition into the propagation algorithm.

A comparison between the mixed-implicit scheme and an alternating method will be performed and the contribution of a 2-ways resolution will be quantified.

B. Gaussian Beam Shooting

In Gaussian Beam Shooting (GBS) algorithms, propagated fields are represented as a superposition of Gaussian beams (GB) launched from the emitting antenna and transformed through successive interactions with obstacles. The formulation developed in this work is based on frame decomposition of source fields:

• The planar distribution of source fields is expressed as a weighted sum of Gaussian windows which form a "frame". • Each Gaussian window radiates in the form of a Gaussian beam. 1) Frame of Gaussian windows: The frame windows ψ µ (y, z) used to describe a planar source distribution in the yOz plane are constructed as the product of frame windows ψ y m,n (y) and ψ z p,q (z) in L 2 (R):

ψ µ (y, z) = ψ y m,n (y)ψ z p,q (z) (4)
where µ = (m, n, p, q) is a composite translation index in Z 4 . The frame windows ψ y m,n (y) are obtained by translation of a Gaussian function ψ(y) along the spatial coordinate y and along its spectral counterpart k y :

ψ y m,n (y) = ψ(y -mȳ)e in kyy (5)

• ȳ and ky are respectively the spatial and spectral translation steps, m and n the spatial and spectral translation indices.

• {ψ µ , µ ∈ Z 2 } is a frame if and only if ȳk y = 2πν, with ν < 1 (ν: oversampling factor).

• the favorite choice for the translation steps is: ȳ = √ νL, ky = √ ν(2π/L) ("balanced" frame). The same construction is used for the frame along z, with p, q the spatial and translation indices along z. In the following, the Gaussian function ψ is taken as:

ψ(y) = √ 2/L 1 2 e -π( y L ) 2 (6)
2) Frame window radiation: Through paraxial asymptotic evaluation of plane wave spectrum integrals, the fields radiated by the source field distributions ψ µ (y, z) (either for y or z component) are put in the form of paraxial Gaussian beams. Let us denote B µ (r) the field radiated at point r. The paraxial expression of B µ is:

B µ (r)=B 0 det Γ -1 (0) det Γ -1 (x µ ) exp ik x µ + 1 2 y t µ Γ(x µ )y µ (7)
with B 0 a vector depending on the source polarization. This expression is analog to that of a geometrical optics ray along the x µ direction, with Γ the curvature matrix and det Γ -1 (0) det Γ -1 (xµ) the divergence factor. y µ = (y µ , z µ ) is the position vector of the point r in a plane transverse to the x µ axis.

The difference between geometrical optics rays and Gaussian beams comes from the fact that Γ is a real matrix in the first case, a complex one in the latter. This complex matrix accounts for the Gaussian decay of fields with increasing distance from the x µ axis in transverse planes. The x µ axis is then called the beam axis.

3) Gaussian beam tracking in a built environment: In the presence of "infinite" smooth interfaces described either as dielectric interfaces or by surface impedance models, GB can be tracked in a way similar to rays, thanks to their paraxial properties. In the case of planar interfaces, reflected or refracted beams can even be expressed as beams originating in "image" source distributions. The presence of ground is easily accounted for in this way. Yet, paraxial beam tracking cannot address problems where beams impinge on discontinuities, as shown in Fig. 4. Specific B2B algorithms are thus developed to simulate propagation in the presence of buildings:

• selection of beams contributing (above a given threshold level) to the incident field distribution on a finite reflecting surface (wall); • for the selected beams impinging on a building corner, decomposition of their incident fields on a frame of narrow windows, which is truncated to the edge of the wall; • change of frame, yielding the frame coefficients of the field of interest on a set of (spatially) wide windows, which are used to further propagate fields in the form of collimated beams. Analytical expressions have been established for incident beam field decomposition on a narrow window frame, and for frame change matrix elements [START_REF] Ghannoum | Frame based gaussian beam bouncing[END_REF], [START_REF] Ghannoum | Études d´outils de calcul de propagation Radar en milieu complexe (milieu urbain, présence de multi-trajets) par des techniques de Lancer de Faisceaux Gaussiens[END_REF]. They are under generalization in the case of successive diffractions around the corners of a building.

III. NUMERICAL RESULTS

A. Source modelling

Meaningful comparisons between the Parabolic Wave Equation (PWE) and the Gaussian Beam Shooting (GBS) methods are achievable only when the codes propagate the same source field. For the numerical simulations reported in this article, we define an Oxyz coordinate system, wherein z is the vertical axis, the ground is in the plane x0y defined by z = 0, the source is located in the plane yOz defined by x = 0.

Vertical (E y = 0) and Horizontal (E z = 0) polarizations are studied separately. The non-zero component of the source field, denoted as E(y, z), is taken as:

E (y, z) = B 0 exp -π/L 2 0 y 2 + (z -h) 2 exp i n k1 y + q k2 z (8)
Where:

• B 0 = √ 2/L 0 ; • h is the emitter height, defined as h = pz with p ∈ Z 2 and z = √ νL 0 ; • k1 = k2 = √ ν 2π/L 0 ;
• n and q are in Z 2 and define the beam pointing respectively along the y and z directions. For the results presented below, the following values have been used:

• the frequency f = 430M Hz; With these parameters, the source is centered at a height h approximately equal to 12.5m.

B. Preliminary test case

A preliminary test case is presented in Fig. 5, with the previously defined source. The polarization is horizontal and the ground is considered flat and perfectly conducting.

Simulation results obtained by both PWE and GBS methods are shown in Fig. 6. The field magnitude is represented along the vertical axis defined by y = 0m (laterally centered) and x = 400m (distance from the source plane).

We can see an excellent agreement between both methods. Due to the Finite Difference methodology used, PWE results depend on spatial domain sampling (mainly the transverse ones, i.e. in y and z dimensions). Thiner the sampling is, better the modelling is. As an example, the presented result has been obtained thanks to a λ/10 transverse sampling.

Validation by comparison with Method of Moments (MoM) [START_REF] Harrington | Field computation by Moment methods[END_REF] is also presented in Fig. 6. Due to the perfect comparison, both methods implemented can be considered as validated on this test case. Comparison between both methods and validation with MoM at a longer distance, i.e. 1km, is presented in Fig. 7.

Agreement between all methods can still be considered as excellent.

IV. FURTHER WORK

The first test case was presented, comparing the electromagnetic wave propagation over a perfectly conducting flat ground (at a distance of 400m and 1km), modelled by the Parabolic Wave Equation and the Gaussian Beam Shooting methods.

Comparisons on all components of the electromagnetic field are under progress.

The specific developments needed for each method, in order to cope with the configurations of interest presented above, are under test.

More complex test case results (cf. Fig. 8, 9 and 10), and further applied in configurations of interest (cf. Fig. I and2), will be presented at the conference.

Fig. 1 .

 1 Fig. 1. Configuration of interest no. 1 -Geometry

Fig. 3 .

 3 Fig. 3. Configuration of interest no. 2 -Dimensions projected in vertical cut

Fig. 4 .

 4 Fig. 4. Gaussian beam impinging on a building corner: 2D cut for a beam propagating in the xOy plane, launched from a frame window source centered at the origin in the yOz plane (7.36m beam width at 430MHz). "T" denotes the target.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Test case representation

Fig. 7 .Fig. 8 .

 78 Fig. 7. Comparison between PWE and GBS along z, for y = 0m and x = 1000m. Validation by comparison with the Method of Moments

Fig. 9 .

 9 Fig. 9. Lateral wall test case

TABLE I CONFIGURATION

 I OF INTEREST NO. 1 -PARAMETERS INSPIRED FROM UIT-R P.1411-6 (02/2012)

		Description	Unit	Range
	h	Height of buildings	[m]	4 to 50
	w	Street width	[m]	10 to 25
	δ	Building density	[%]	20 to 40
	h BS	Base Station height	[m]	1 to 100
	h U E	User Equipment height	[m]	A few
	d	Distance BS to UE	[m]	1000 (up to 5000)
	f	frequency	[MHz]	800 to 20000
	s	UE speed	[m.s -1 ]	0 to 30
		Fig. 2. Configuration of interest no. 2 -Perturbed beam

ACKNOWLEDGMENT

The authors acknowledge partial support by the French National Research Agency (ANR) under "Specific Support for Research Works and Innovation Defense" (ASTRID) grant ANR-12-ASTR-0013.


Ch Vasserot

H Jungen

A Schmoranzer

S Knie

A Kübler

Ehresmann

POSTAL M Glass-Maujean

A-M Vasserot

Ch Jungen

H Schmoranzer

A Knie

S Kübler

A Ehresmann

Experimental and theoretical study of the npσ 1 Σ + u (n ≥ 4) N = 0 excited states of D 2 : absolute absorption cross sections and branching ratios for ionization, dissociation and fluorescence

    Experimental and theoretical study of the npσ 1Σu+ (n≥4) N=0 excited states of D2: absolute absorption cross sections and branching ratios for ionization, dissociation and fluorescence

Introduction

This paper is the fourth in a series [1,2,3] dealing with the photoabsorption spectrum of molecular deuterium. This is a combined experimental and theoretical effort, in which multichannel quantum defect theory, MQDT, is used to interpret spectra taken at the SOLEIL and BESSY synchrotron radiation sources. The present paper is concerned with the P (1) transitions leading to N = 0 excited states of 1 Σ + u symmetry recorded at BESSY, and specifically with their absorption intensities and decay dynamics.

The absorption spectrum of molecular deuterium has been recorded at room temperature using a 10 m normal-incidence monochromator at BESSY. The measured transition energies have an uncertainty of 1 cm -1 , which is less precise than the Fourier-transform (FT) measurements recorded at SOLEIL [3]. However, by using a monochromator, we are in a position to perform absolute intensity measurements and, by monitoring simultaneously the signals for dissociation, ionization and fluorescence, we also have access to the dissociation, ionization and fluorescence partial cross sections. The two techniques therefore complement each other. While most of the transitions observed here have been measured in the preceding paper, we report here, in addition to our study of the intensities and decay dynamics, a number of additional P (1) transitions which had not been observed there.

Experiment

The experimental setup has been described in detail in previous papers [2,4,5,6]. Briefly, the VUV photons coming from the undulator beamline U125/2-10m-NIM of BESSY II were dispersed by a 10 m normal-incidence monochromator equipped with a 4800-lines/mm grating giving an effective linewidth of 0.0010 nm (2 cm -1 ) in first order [6]. This value represents the convolution of the apparatus function with the Doppler width at room temperature. The energy uncertainty is about ±1 cm -1 and results mainly from a small residual non-linearity of the monochromator's energy scale (see [5]). The intensities of the absorption transitions, recorded at the effective resolution just mentioned, were calibrated directly, based on the known absorption path length (39 mm) and the gas pressure of 10 or 20 mTorr (13.3 or 26.7 µbar) at room temperature. Our experimental setup enables the excitation spectra for photoionization, photodissociation and visible molecular fluorescence to be recorded simultaneously. The photodissociation spectrum was recorded by monitoring the Ly α emission, while the visible molecular fluo-rescence was monitored in the 400-650 nm spectral range. The photoionization and photodissociation spectra were calibrated on the basis of absorption peaks that are known to be fully ionized or fully dissociated. The molecular fluorescence excitation cross section was obtained indirectly, by subtracting the photoionization and photodissociation cross sections from the photoabsorption cross section. These procedures are described and justified in detail in Ref. [5]. Fig. 1 displays a section of the experimental spectra obtained with the BESSY setup. Note how a given line may appear with very different intensities in the different detection channels. It is this information which allows the decay dynamics to be quantified.

Theory

The theoretical multichannel quantum defect approach used here reproduces step by step the approach described for H 2 in Ref. [7] and used in the preceding paper [3], with the only difference that the values of the reduced nuclear mass and the Rydberg constant have been adapted to the deuterium molecule. As previously the quantum defects and channel dipole transition moments were extracted from the highly accurate theoretical clamped-nuclei (Born-Oppenheimer) potential energy curves [8] and dipole transition moments [9], so that the input data for our calculations are fully ab initio.

The calculations were carried out in two steps. In a first set of computations all open ionization channels were omitted from the treatment and discrete energy level positions and Einstein spontaneous emission coefficients for the P (1) transitions were evaluated and used for the analysis of the observed spectrum, cf. Fig. 1. In a second set of calculations all ionization channels, open and closed, were included and ionization profiles were evaluated, from which the autoionization widths were obtained. As in the preceding paper [3] we have omitted dissociation channels from our theoretical treatment.

Results

Spectra have been recorded in the wavelength range from 81.3 to 74.6 nm corresponding to 123000 -134000 cm -1 . This range is roughly equivalent to the range from the ionization threshold to the third dissociation limit of diatomic deuterium. The results are summarized in Table 1. Many of the transitions listed in the Table have also been observed in the preceding paper [3]. The recommended values for the observed transition energies are those given in the preceding paper except for the values given in boldface in the present Table 1 which are new. Before discussing the various aspects consecutively, we address the problem of determination of line positions, intensities and widths.

Determination of energies, intensities and widths

Each observed transition profile has been fitted to a Gaussian profile which yields the peak position, the peak area and peak width. The peak positions agree to within the uncertainty of the BESSY measurements with the more precise FT values from SOLEIL where these are available. A difficulty in determining line intensities, and particularly in applying Eq. (1) to the observed data, see Sec. 6.3 below, is that spectral lines are often blended. This difficulty may be overcome in many cases by exploiting the information obtained when several decay channels are monitored simultaneously. For instance, a spectral line may appear blended in one channel while appearing fully resolved in another, so that the number of unknown parameters in the fitting procedure of a given line profile can be kept to a minimum.

For each intensity determination, the uncertainty was taken as three times the statistical 1σ error, to which was added an estimated calibration uncertainty of 10% [2]. The width value obtained from each Gaussian fit represents the convolution of the Doppler width, the apparatus function and the natural width of the upper level of the transition. The de-convolution procedure applied to each profile in order to obtain the natural width is the same which we have used in the preceding paper [3]. The combined width of the Doppler broadening and the apparatus function amounts to 1.5 -2.5 cm -1 depending on the experimental condition of each given run at BESSY. These values are of course larger than those observed in the FT study on SOLEIL, where the spectral resolution is higher and the temperature was lower. Nevertheless, where a comparison can be made, the natural widths extracted from the BESSY spectra are compatible with those obtained from the SOLEIL spectra. Fourteen new values could be added based on the BESSY measurements.

Energy levels

We have identified 46 P (1) transitions. Out of these, 30 have already been listed in the preceding paper [3]. Three P (1) transitions observed in the FT spectrum could not be resolved in the BESSY spectrum. These are P (1) 9pσ, v = 4, 8pσ, v = 5 and 5pσ, v = 9. On the other hand, 14 additional weak transitions could be detected in the BESSY spectra, since here the sensitivity is higher and the background noise lower, particularly in the ionization and dissociation excitation spectra. These are: 10pσ, 12pσ and 15pσ for v = 0, 9pσ, 10pσ and 21pσ for v = 1, 11pσ, 15pσ and 18pσ for v = 2, as well as 7pσ, v = 4 and v = 5, 6pσ, v = 9, and 4pσ and 5pσ v = 12. These newly identified transitions are printed in boldface in Table 1.

Line intensities

The peak area of each measured transition is related to the Einstein A coefficient for the same transition by

p σdω = 1 8πcω 2 A n,v,N →X,v =0,N 2N + 1 2N + 1 n N , (1)
where N = 1 and N = 0 for the P (1) transitions studied here. In Eq. ( 1), p is the integral of the measured absorption cross section σ over the given line profile p, ω = (E n,v,N -E X,v =0,N )/hc is the incident photon energy in cm -1 , and n N is the fraction of molecules in the rotational state N of the vibrational ground state at the given temperature of 300 K (n N = 0.20 in the present case). An experimental value for the Einstein coefficient for each transition is obtained from Eq. ( 1) by solving for A. Table 1 and Fig. 2 compare the experimental Einstein A coefficients determined here for 42 P (1) transitions with the values calculated theoretically for transitions to v = 0. In the majority of cases the measured and calculated A values are in agreement with one another. However, in a number of instances the measured intensities turn out to be significantly larger than predicted by theory. This appears clearly from inspection of the statistical histogram of Fig. 2, and may in part be due to the uncertainty of our measurements. However, more likely, this arises from the fact that the spectral lines are overlapped. This hypothesis is corroborated by the fact that no calculated value has been found to be higher than the corresponding measured value including its error bar. Whenever a transition is blended by another transition which itself has been assigned, the corresponding value has been omitted from Table 1. Fig. 3 provides a detailed comparison between the measured and calculated Einstein A coefficients for v = 0 -14 and n = 4 -8. The figure also in-cludes the intensities expected in the situation where the Born-Oppenheimer approximation would hold and the Franck-Condon principle could be applied: the substantial deviations of the observed A values from those predicted in this approximation demonstrate the omnipresence of vibronic interactions. By contrast, the MQDT calculations are seen to correctly predict the irregular variations of the observed line intensities. Note that the vibronic interactions are far stronger in the 1 Σ + u upper states of the P (1) transitions than in the 1 Π - u upper states of the Q(1) transitions [2], a fact that has been known for a long time [10], and which is related to the different orientations of the pσ and pπ orbitals with respect to the molecular frame. Fig. 3 shows that all of the 7pσ, v transitions are weakened by vibronic interaction, a de facto suppression of the 7pσ ← X electronic transition that causes the upper state to be difficult to be identified in the spectra. Fig. 4 displays the observed and calculated Einstein A coefficients for higher n-values, n ≥ 8 up to n ≈ 28, and for v = 0 to 3. Observed and calculated values are overall seen to be in fair agreement. A major discrepancy arises for n = 11, v = 2 where the observed intensity is three times larger than calculated. This transition appears well isolated in the spectrum, as no P (N ), Q(N ) or R(N ) transition is predicted to be situated nearby.

Dynamical aspects

All excited levels studied in this work are situated above the second dissociation limit, D(1s) + D(n = 2), and are therefore subject to predissociation. The ionization threshold lies at 124715 cm -1 , so that below this threshold fluorescence and dissociation are the only accessible decay pathways. The radiative lifetimes of various rotational levels in the 4pσ, v = 0 state are known to be 7.6 ± 0.6 ns [11], corresponding to a radiative width of 7 • 10 -4 cm -1 . According to the n -3 Rydberg scaling law we expect smaller radiative widths for the npσ levels with n ≥ 5. Our experimental data show that there is competition between the radiative and dissociative decay channels for most of the levels situated between 123500 and 124715 cm -1 , in the sense that the partial widths are comparable. We thus conclude that the dissociation widths of these levels must be ≤ 10 -3 cm -1 as well. It may well be that the 4pσ, v = 5, N = 0 level is fully dissociated, but since the corresponding P (1) transition is blended, it is difficult to be certain.

For energies above 124715 cm -1 decay by autoionization also becomes energetically possible and is expected to take over eventually. We know that in H 2 none of the npσ levels with N = 0 situated above the ionization threshold fluoresces [5]. However, vibrational autoionization is slower in the heavier isotopomer D 2 and, indeed, in the 5pσ, v = 4, N = 0 level fluorescence remains in the competition with a share of 17% of the total decay. The ionization and dissociation processes are particularly slow in this case. As in H 2 , the 4pσ, v, N = 0 levels are all fully dissociated in D 2 . An exception occurs for v = 10 where an ionization yield of 27% is observed. As already stated, ionization takes over completely in H 2 above the ionization threshold for N = 0, with the exception of the 4pσ levels which are dissociated. In D 2 dissociation subsists also in the 5pσ, v, N = 0 manifold, with yields ranging from 0 to 100 %. Further, some of the 6pσ levels, v = 2 and 3, are also partially dissociated and the same is true even for 8pσ, v = 2, N = 0. These observations, and particularly the differences between the decay dynamics in H 2 and D 2 , are somewhat surprising as both, predissociation and autoionization, are non-adiabatic effects that are expected to be the weaker the heavier the isotope. We shall come back to this point in Sec. 7 below.

Line widths

For fluorescence to be able to compete with dissociation and/or ionization, the predissociation and/or autoionization widths must be very small, i.e. 0.01 cm -1 . For instance our calculation yields an ionization width of 3 • 10 -4 cm -1 for the 5pσ, v = 4, N = 0 level, which is indeed of the same order as the radiative width. For all N = 0 levels where fluorescence has been observed, we find, as expected, that the measured width is far smaller than the experimental uncertainty.

For dissociation to be able to compete with ionization, the autoionization width must also be sufficiently small. For all N = 0 levels that appear in dissociation, the measured value of the width is smaller than its uncertainty, and the calculated autoionization width is ≤ 0.06 cm -1 . Moreover, the calculations predict ionization widths for the 4pσ, v, N = 0 levels up to v = 12 that are smaller than 10 -3 cm -1 . An exception is again the v = 10 level (cf. Sec. 6.4), for which we calculate an autoionization width as large as 0.03 cm -1 . The ionization yield observed in this case is 17 %, implying a dissociation partial width of ≈ 0.15 cm -1 , which again is too small to be detected in our experiment. Dissociation is particularly slow for this level, and in fact much slower than one could expect based on the previous study of H 2 [12].

Apart from the exceptions just discussed, the npσ N = 0 levels of D 2 are found to be fully autoionized so that the experimental natural widths may be compared directly with the calculated autoionization widths. This is done in Fig. 5, which is seen to show good overall agreement of experiment and theory. Fig. 5 is a good example demonstrating the validity of the well-known ∆v = -1 propensity rule for vibrational autoionization [10,13,14,15]: the calculated and observed ionization widths shoot up by several orders of magnitude at the point where the v + = v -1 threshold is passed, in other words, where the v + = v -1 decay channel becomes accessible. After this spectacular onset, the widths decrease more or less smoothly as function of n, following the n -3 Rydberg scaling rule and in accordance with the two-channel autoionization formula given in Ref. [10]. This decrease is somewhat irregular due to local perturbations that affect the Rydberg series. For instance, 14pσ, v = 5, N = 0 interacts with the perturber 5pσ, v = 9, N = 0 which cannot autoionize via ∆v = -1; the result is that the width of the perturbed level is reduced. This type of effect is of course accounted for by the multichannel treatment employed here.

For the purpose of comparison we have included in Fig. 5 the calculated values from our previous study of N = 0 Rydberg levels in H 2 [7]. The ∆v = -1 autoionization widths are known [2,16] to scale according to M -1/2 where M is the nuclear mass factor relevant for the process. Therefore the expectation is that the vibrational autoionization widths in H 2 should be by a factor 1.4 larger than in D 2 . The maximum values shown in Fig. 5 correspond, very roughly, to a ratio of 2 for the first two series and to 1.3 for the third series. There is, therefore, qualitative albeit not exact agreement with the expectations based on the two-channel approximation.

v = 1 v = 2 v = 3 w i d t h s ( F W H M ) ( c m -1 ) v = 4 v = 5 n

Conclusion

Using a 10 m normal-incidence monochromator at BESSY we have measured the positions, intensities and widths of 42

npσ 1 Σ + u , v, N = 0 ← X 1 Σ + g , v = 0, N = 1 P (1)
transitions of D 2 between 123000 and 134000 cm -1 . The BESSY measurements are complementary to the preceding Fourier-transform study carried out at SOLEIL [3], as they provide absolute absorption cross sections as well as a wealth of information on the decay dynamics. Further, due to the higher sensitivity, 14 additional P (1) transitions have been observed that had not been identified previously.

An interesting result concerns the decay dynamics and partial cross sections and their comparison with the analogous quantities in H 2 . We have found that while ionization on the whole takes completely over in D 2 and in H 2 as expected when the excitation energy increases above the ionization threshold, dissociation and even fluorescence persist up to higher Rydberg members in D 2 than in H 2 . This behavior is not intuitively understandable, since both autoionization and predissociation processes are expected to become slower when the nuclear mass increases. The higher density of levels present in D 2 as compared to H 2 is also an issue here, as it favors local interactions that become active when coincidences or near-coincidences occur between levels that may couple vibronically. In Ref. [2] we have presented a discussion of these effects in some detail, and come to the conclusion that the non-intuitive aspect of the differences between the D 2 and H 2 dynamics has genuine multichannel character and cannot be understood in the framework of simple models.

The role of level coincidences, leading to 'indirect' or 'accidental' predissociation or autoionization, has long been recognized in molecular physics [START_REF] Herzberg | Molecular Spectra and Molecular Structure[END_REF]. In diatomic hydrogen it has been invoked to explain predissociation [START_REF] Glass-Maujean | [END_REF] and autoionization processes [19] that would not occur in a zero-order picture. The multichannel character of some of the observed phenomena which we discuss here may be regarded as a generalization of these earlier concepts. On the other hand, we have also found that some quantities, such as the ∆v = -1 propensity-favored vibrational autoionization widths, do not deviate much from the expectations based on a two-channel model. a ν/c, observed transition energy in cm -1 . When no entry obs -calc is given, the transition has not been observed and the energy given corresponds to the calculated value.

b A, emission probability for the transition to X 1 Σ + g , v = 0, N = 1, in 10 6 s -1 .

c Γ/hc, level width (FWHM) in cm -1 . The calculated widths are ionization partial widths.

d γ, experimental decay branching ratio in %.

e The positions of the upper state energy levels above the v = 0, N = 0 ground state level are obtained by adding the ground-state rotational energy 59.78 cm -1 (N = 1) [20] to the transition energy.

f Transition energies in boldface correspond to new assignments. The recommended values for the observed transition energies are those of the preceding paper except for the entries in boldface which are new. * blended line.

  i d e n t p h o t o n e n e r g y ( c m -1 ) f l u o ( a r b . u n i t s )

Figure 1 :(

 1 Figure 1: (Color online) A section of the D 2 absorption and photoexcitation spectra recorded at BESSY. (a) (black online) absorption, σ abs . (b) (blue online) dissociation, σ diss . (c) (red online) ionization, σ ion . (d) (violet online) σ f luo = σ abs -σ diss -σ ion , flurescence signal determined indirectly. (e) (green online) fluorescence measured directly. Calculated P (1) transitions leading to npσ, v, N = 0 upper Rydberg levels are indicated by filled circles with intensities (on an arbitrary scale) corresponding to absorption.

Figure 2 :

 2 Figure 2:Histogram of the ratios of the observed and calculated Einstein A coefficients of P (1) transitions in D 2 absorption.

Figure 3 :

 3 Figure 3: (Color online) Einstein A coefficients for P (1) absorption transitions in D 2 for the npσ, n = 4 -8 states, plotted versus the vibrational quantum number v. Black squares with error bars: experimental values. Circles (blue online) connected by straight lines: MQDT calculation. Dashed lines: A values calculated using the Franck-Condon principle.

Figure 4 :(

 4 Figure 4: (Color online) Einstein A coefficients for P (1) absorption transitions in D 2 for the npσ, v, v = 0 -3, levels plotted as functions of the principal quantum number n.

Figure 5 :(

 5 Figure 5: (Color online) Upper state level widths for P (1) transitions in D 2 absorption for v = 1 -5, plotted versus the principal quantum number n. Filled circles with error bars (red online): experimental values (BESSY). Filled squares with error bars: experimental values (SOLEIL). Open squares connected by full lines (blue online): ionization widths calculated by MQDT. Open triangles connected by dotted lines: calculated widths of the same states in H 2 . The vertical arrows indicate for each level v the position of the ionization limit v + = v -1.

Table 1 :

 1 P (1) transitions in the photoabsorption spectrum of the D 2 molecule

	v	P (1) (obs) a	obs -calc	A calc	b	A obs	Γ calc	c	Γ obs	γ diss	d	γ ion	γ f luo
	10pσ	123572.6 e,f	-0.4	0.35	1.0 ± 0.8	0.00
	11pσ	123773.6		0.14	*	0.00
	4pσ	123891.7	0.4	15.15	*	0.00		0.1 ± 0.2	100 ± 5	0	0
	7pσ	123901.7	0.0	2.18	2.8 ± 0.9	0.00		0.3 ± 0.4	56 ± 5	0	44 ± 5
	12pσ	123931.9	-0.3	1.16	1.6 ± 0.6	0.00		0.0 ± 0.3	100 ± 3	0	0
	13pσ	124046.1		0.25		0.00
	14pσ	124139.2		0.15		0.00
	15pσ	124214.3	0.1	0.11	*	0.00		0.0 ± 0.3
	16pσ	124275.5		0.077		0.00
	5pσ	124326.4	0.6	3.33	8.1 ± 5.1	0.00		0.0 ± 0.3	31 ± 4	0	69 ± 4
	17pσ	124368.6		0.06		0.00
	18pσ	124404.3		0.05		0.00
	19pσ	124434.7		0.04		0.00
	20pσ	124459.7		0.10		0.00
	8pσ	124468.1		0.14		0.00
	21pσ	124484.7		0.000		0.00
	22pσ	124504.2		0.001		0.00
	23pσ	124521.4		0.000		0.00
	24pσ	124550.2		0.02		0.00
	25pσ	124562.1		0.25		0.00
	6pσ	124569.0	-0.9	14.52	27 ± 9	0.00		0.0 ± 0.1	36 ± 6	0	64 ± 6
	26pσ	124573.5		2.03		0.00
	27pσ	124582.8		0.20		0.00
	9pσ	124870.0	-0.5	1.82	2.4 ± 1.6	2.93		3.2 ± 1.4	0		100 ± 2	0
	4pσ	125026.7	0.0	15.02	*	2.0E-06
	10pσ	125143.5	0.0	0.76	0.6 ± 0.2	2.06		3.7 ± 1.9	0		100 ± 2	0
	11pσ	125341.4		0.06		1.29
	7pσ	125404.5	0.0	3.96	5.1 ± 2.3	0.17		0.5 ± 0.9	0		100 ± 2	0
	12pσ	125503.1		1.04		1.23
	13pσ	125620.1		0.55		0.95
	5pσ	125667.6	-0.3	5.44	16.9 ± 2.3	2.9E-04	0.1 ± 0.8	69 ± 3	15 ± 1	16 ± 1
	14pσ	125713.4		0.34		0.76
	15pσ	125788.6		0.26		0.61
	16pσ	125850.0		0.20		0.49
	17pσ	125900.5		0.23		0.39
	6pσ	125931.3	0.1	2.63	3.9 ± 1.8	3.0E-03	0.0 ± 0.4	24 ± 6	76 ± 6	0
	18pσ	125944.9		0.15		0.33
	19pσ	125979.6		0.04		0.28
	20pσ	126009.2		0.58		0.19
	8pσ	126025.2	-0.3	8.64	15.6 ± 4.9	0.013	0.2 ± 0.6	6 ± 2	94 ± 2	0
	21pσ	126039.0	0.3	1.86	2.3 ± 0.5	0.24		0.0 ± 0.4	0		100 ± 5	0
	22pσ	126060.3		0.39		0.21
	23pσ	126079.9		0.11		0.18
	4pσ	126087.3	0.1	16.40	24 ± 11	3.8E-04	0.0 ± 0.4	100 ± 2	0	0
	24pσ	126097.2		0.25		0.15
	9pσ	126374.7		2.48		6.20
	10pσ	126647.7		1.16		4.10
	7pσ	126816.6	-0.1	1.65	2.1 ± 0.7	0.27		0.0 ± 0.5	0		100 ± 5	0
	11pσ	126870.5	1.7	3.77	11.6 ± 1.8	2.91		3.0 ± 0.3	0		100 ± 5	0
	5pσ	126937.2	-0.4	3.00	3.7 ± 1.0	1.4E-03	0.0 ± 0.3	100 ± 5	0	0
	12pσ	127011.0		1.12		2.74
	4pσ	127071.4	-0.5	11.42	8.8 ± 4.7	2.5E-04	0.0 ± 0.6	100 ± 5	0	0
	13pσ	127128.9		0.64		2.14
	14pσ	127222.5		0.46		1.60
	15pσ	127298.1	0.7	1.01	1.5 ± 0.7	1.21		1.1 ± 0.2	0		100 ± 5	0
	6pσ	127310.9	0.1	7.21	7.0 ± 2.5	0.032	0.0 ± 0.5	0		100 ± 5	0
	16pσ	127359.6		0.002		0.97
	17pσ	127409.4		0.12		0.70

Table 1

 1

	(continued)
	v	P (1) (obs) a	obs -calc	A calc	b	A obs	Γ calc	c	Γ obs	γ diss	d	γ ion	γ f luo
	22pσ	130404.2		0.12		1.04
	29pσ	130500.5		0.40		0.00
	9pσ	130507.6		0.56		0.18
	30pσ	130514.9		0.41		0.36
	31pσ	130522.1		0.28		0.40
	32pσ	130529.0		0.20		0.39
	33pσ	130535.4		0.15		0.30
	34pσ	130541.2		0.12		0.28
	7pσ	130707.2		0.34		0.25
	10pσ	130811.8		1.79		11.65
	4pσ	130991.6	0.2	1.37	2.2 ± 0.4	3.1E-03	0.2 ± 0.3	100 ± 5	0	0
	11pσ	131002.7		0.06		6.00
	6pσ	131050.0	0.1	7.41	*	1.98
	12pσ	131159.3		0.82		6.13
	13pσ	131276.4		0.17		3.41
	14pσ	131352.0		0.000		1.00
	5pσ	131375.2		1.99		1.51
	8pσ	131403.1	-0.6	6.18	8.7 ± 4.8	2.06		2.2 ± 1.2	0		100 ± 5	0
	15pσ	131460.2		1.53		4.12
	16pσ	131519.7		0.69		3.31
	17pσ	131570.2		0.44		3.20
	4pσ	131590.3	5.2	2.55	4.5 ± 0.7	2.6E-04	0.0 ± 0.3	100 ± 5	0	0
	18pσ	131612.9		0.28		3.04
	19pσ	131648.9		0.18		2.60
	20pσ	131679.6		0.10		2.28
	21pσ	131705.6		0.04		1.27
	22pσ	131727.2		0.001		0.57
	23pσ	131743.5		0.23		0.03
	9pσ	131756.1		0.49		0.03
	24pσ	131770.6		0.33		0.57
	25pσ	131785.1		0.22		0.95
	26pσ	131798.3		0.16		0.96
	38pσ	131885.6		0.21		0.82
	7pσ	131887.5		0.40		0.00
	39pσ	131890.8		0.12		0.41
	10pσ	132058.9		0.49		12.00
	6pσ	132168.0	-0.3	3.25	3.7 ± 2.2	1.28		0.9 ± 0.9	0		100 ± 5	0
	11pσ	132267.3		0.95		19.47
	5pσ	132336.9	2.9	5.66	7.9 ± 1.3	0.30		0.3 ± 0.2	0		100 ± 5	0
	8pσ	132582.6	-0.7	2.61	3.5 ± 1.9	2.52		1.8 ± 1.0	0		100 ± 5	0
	5pσ	133159.1		0.01		0.12
	6pσ	133324.6	-1.0	7.56	6.5 ± 2.8	7.07		7.5 ± 1.0	0		100 ± 5	0
	5pσ	134028.7	8.4	0.63	1.1 ± 0.5	0.062	0.0 ± 0.3	100 ± 5	0	0

Acknowledgments

We acknowledge the Helmholtz-Zentrum Berlin -Electron storage ring BESSY II -for providing synchrotron radiation at the beamline U125/2-10m-NIM. In particular we would like to thank Gerd Reichardt and Andreas Balzer for their assistance with the experiments. The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 226716. A. K. would like to thank the Otto-Braun-Fonds of the University of Kassel for financial support. Ch. J. thanks the Miescher Foundation (Basel, Switzerland) for partial support.