J Loreau

A Faure

F Lique

S Demes

P J Dagdigian

Hyperfine collisional excitation of ammonia by molecular hydrogen

Keywords: radiative transfer -molecular data -molecular processes -scattering -ISM, molecules -ISM, abundances

Ammonia is one of the most widely observed molecules in space, and many observations are able to resolve the hyperfine structure due to the electric quadrupole moment of the nitrogen nucleus. The observed spectra often display anomalies in the satellite components of the lines, which indicate substantial deviations from the local thermodynamic equilibrium. The interpretation of the spectra thus requires the knowledge of the rate coefficients for the hyperfine excitation of NH 3 induced by collisions with H 2 molecules, the dominant collider in the cold interstellar medium. In this paper we present the first such calculations using a recoupling approach. The rate coefficients are obtained for all hyperfine levels within rotation-inversion levels up to j = 4 and temperatures up to 100 K by means of quantum scattering close-coupling calculations on an accurate, five-dimensional, potential energy surface. We show that the rate coefficients depart significantly from those obtained with the statistical approach and that they do not conform to any simple propensity rules. Finally, we perform radiative transfer calculations to illustrate the impact of our new rate coefficients by modelling the hyperfine line intensities of the inversion transition in ground state para-NH 3 ( j k = 1 1 ) and of the rotational transition 1 0 → 0 0 in ortho-NH 3 .

INTRODUCTION

Since its first detection in the interstellar medium (Cheung et al. 1968), ammonia (NH 3 ) has often been used to probe the physical properties of molecular clouds due to the favourable structure of its energy levels and its high abundance in a variety of astrophysical sources. The rotational levels of NH 3 are denoted as j k (where j is the angular momentum and k its projection on the molecular axis), and the presence of metastable states for j = k that can only decay through molecular collisions makes NH 3 an excellent probe of the local temperature. NH 3 exists in two forms, namely ortho-NH 3 (for k = 3n, n integer) and para-NH 3 (for k = 3n ± 1) that cannot be interconverted through radiative or inelastic collisional processes. In addition, the inversion (or umbrella) motion causes a splitting of each j k level into a doublet (with the exception of k = 0 due to spin statistics) whose components, denoted as j + k and j - k , are separated by 0.79 cm -1 (23.68 GHz). NH 3 can thus be detected in different frequency ranges corresponding to pure inversion, rotational, or vibrational transitions.

In most astrophysical observations, the hyperfine structure due to the electric quadrupole moment of nitrogen is resolved. N nuclei possess a spin of I = 1, so that each rotation-inversion level is further split into three hyperfine components F (with the exception of ground state ortho-NH 3 ), with a separation of the order of a MHz. A rotational line is thus split into five lines: the main line (∆F = 0), as well as two inner and two outer satellite lines characterized by E-mail: jerome.loreau@kuleuven.be ∆F = ±1. The lines are further split by the spins of the H nuclei, but this interaction is much weaker and will be neglected here.

The relative intensity of hyperfine lines can be used to derive opacities if the various hyperfine components have the same excitation temperature and if the opacity is not too large (Barrett et al. 1977;Ho & Townes 1983). However, NH 3 hyperfine-resolved observations commonly display anomalies in the spectra, where the intensities within a pair of satellite lines are not equal. This indicates different excitation temperatures, meaning that the satellite components do not obey the local thermodynamic equilibrium (LTE) assumption in the interstellar medium. Such anomalies were first observed by Matsakis et al. (1977) in source DR21 and subsequently by various authors (see e.g., Matsakis et al. (1980); Gaume et al. (1996); Rathborne et al. (2008); Rosolowsky et al. (2008); Camarata et al. (2015); Caselli et al. (2017);Zhou et al. (2020)) in diverse sources. If the hyperfine levels are not populated thermally, the interpretation of observations requires radiative transfer models to account for non-LTE effects. These occur mainly because of collisional excitation by H 2 molecules, and a knowledge of accurate rate coefficients for the hyperfine excitation of NH 3 by H 2 is thus necessary. In addition, modelling the strength of the hyperfine anomalies can be used to deduce the local H 2 density. Models to interpret anomalies were developed by Stutzki & Winnewisser (1985a,b) based on rate coefficients calculated for NH 3 -He collisions as a proxy for NH 3 -H 2 collisions. Beside this work, Chen et al. (1998) also computed hyperfine-resolved rate coefficients for NH 3 -He collisions. To this day there are thus no accurate colisional data for the hyperfine excitation of NH 3 by H 2 molecules.

Various theoretical techniques exist to compute such rate coefficients. Among these, the simplest is the statistical approach, which assumes that the rate coefficients for hyperfine excitation are proportional to the hyperfine degeneracy of the final state. In this case, rate coefficients for pure rotational transitions, which are available for molecules such as NH 3 , are the only ingredient needed to obtain hyperfine-resolved rate coefficients. This method was used to interpret the observations of Caselli et al. (2017) for the 1 0 -0 0 transition in the pre-stellar core L1544. The infinite-order-sudden limit approximation, used by Stutzki & Winnewisser (1985a), ignores the rotational energy spacing and allows again to obtain the hyperfineresolved rate coefficients on the basis of the pure rotational ones. Given the nature of its assumptions, this approximation is however not expected to be valid at low temperatures, i.e. at temperatures close or lower than the rotational spacings. Finally, the method used in the present work is the recoupling approach, which was also adopted by Chen et al. (1998) but for NH 3 -He collisions and combined with the coupled-states approximation1 . The only approximation in our version of the recoupling approach is the assumption that the hyperfine levels are degenerate, i.e. their splitting is negligible compared to the temperature (Alexander & Dagdigian 1985). While computationally more demanding, it is also expected to be more accurate than the statistical or sudden methods. A summary and comparison of these different approaches can be found in the paper by Faure & Lique (2012) for the linear molecules CN and HCN.

In this work we present rate coefficients for the hyperfine excitation of ortho-and para-NH 3 in collisions with para-H 2 for all levels up to j = 4 and for temperatures up to 100 K. While the rotationinversion collisional excitation of NH 3 by H 2 has been widely studied theoretically over the past decades (Danby et al. (1987); Offer & Flower (1990); Maret et al. (2009); Bouhafs et al. (2017); Ma et al. (2015); Demes et al. (2023)) and experimentally with crossed beams (Schleipen & ter Meulen 1991;Tkac et al. 2015;Gao et al. 2019), to the best of our knowledge this represents the first effort to provide hyperfine collisional excitation rate coefficients of NH 3 (or any nonlinear molecule) by H 2 with the accurate recoupling approach.

The paper is structured as follows: in Section 2 we discuss the theoretical methods employed. In Section 3 we discuss the features of the hyperfine-resolved rate coefficients, and in Section 4 we present radiative transfer calculations to investigate the impact of these new data on the modelling of ammonia spectra. Finally, we summarize our findings in Section 5.

THEORY

The scattering calculations were performed in two steps. First, we computed the scattering matrix for rotation-inversion transitions by means of the fully quantum-mechanical close-coupling (CC) method. This method has been used to obtain accurate cross sections for rotational excitation of NH 3 induced by collisions with H 2 by several authors (see e.g., Danby et al. (1987); Rist et al. (1993)) and we refer to these papers for further methodological details. Similarly to recent theoretical works (Ma et al. 2015;Bouhafs et al. 2017;Demes et al. 2023), we employ the accurate five-dimensional rigid-rotor potential energy surface of Maret et al. (2009), which was constructed by fitting ab initio energies obtained with the coupled cluster method with single, double, and perturbative triple excitations.

The cross sections for rotation-inversion transitions can be obtained from the T-matrix elements T J j 1 k v j 2 j 12 l, j 1 k v j 2 j 12 l , where v denoted the inversion level, J is the quantum number associated with the total angular momentum of the collision complex without nuclear spin (J = j 12 + L), j 12 = j 1 + j 2 , where j 1 and j 2 are the angular momenta of NH 3 and H 2 , respectively, and L is the relative orbital angular momentum of the collision. We do not discuss here the properties of the cross sections for rotation-inversion transitions as these are identical to those reported recently by Bouhafs et al. (2017) and Demes et al. (2023).

In a second step, the recoupling approach is used to compute hyperfine-resolved cross sections. When the nuclear spin of the N atom (I = 1) is included, each level of NH 3 is split into three components, and the total angular momentum J t is obtained by the coupling J t = J + I. However, the hyperfine splittings are of the order of the MHz (i.e. about 10 -5 -10 -4 K), which is much smaller than the rotational spacings and collisional energies considered here. We thus assume that the hyperfine levels of NH 3 are degenerate. In this case the nuclear spin wavefunctions can be decoupled from the rotational wavefunctions and, as shown by Offer et al. (1994), the angular momenta can also be recoupled as

F = j 1 + I, j R = j 2 + L, J t = j R + F.
The T-matrix elements including nuclear spin, T J t , can then be obtained from the spin-free T-matrix elements T J as

T J t j 1 v F j 2 j R l, j 1 v F j 2 j R l = J, j 12 , j 12 (-1) j R + j R +l+l + j 2 + j 2 ([F][F ][ j 12 ][ j 12 ][ j R ][ j R ]) 1/2 [J] j 1 j 2 j 12 l J j R j 1 j 2 j 12 l J j R j R j 1 J I J t F j R j 1 J I J t F T J j 1 k v j 2 j 12 l, j 1 k v j 2 j 12 l (1)
where {• • • } denotes a 6-j symbol and [x] = 2x + 1.

The state-to-state hyperfine cross sections for NH 3 -H 2 can be then computed from the T-matrix according to the expression (Offer et al. 1994):

σ j 1 k v F j 2 , j 1 k v F j 2 = π k 2 i (2F + 1)(2 j 2 + 1) J T (2J T + 1) ll j R j R |T J t j 1 k v F j 2 j R l, j 1 k v F j 2 j R l | 2 (2)
The scattering calculations were carried out with the Hibridon program (Alexander et al. 2023) for total energies between 0 and 600 cm -1 on a grid of 700 energies for ortho-NH 3 and 660 energies for para-NH 3 . These calculations were performed independently for ortho-NH 3 and para-NH 3 as ortho-para conversion is forbidden in inelastic collisions. The energy step was 0.5 cm -1 at the lowest energies and increased gradually up to 4 cm -1 at the highest energies. We used the log-derivative propagator in the short range, and the Airy propagator in the long range (R > 25a 0 ). Transitions between levels up to j 1 = 6 with j 2 = 0 were converged with a rotational basis including all levels up to j 1 = 8 and j 2 = 0, 2. The rotational constants of NH 3 were taken as A = B = 9.9402 cm -1 , C = 6.3044 cm -1 with an inversion splitting of 0.7903 cm -1 , while the rotational constant of H 2 was taken as 59.3801 cm -1 as in Bouhafs et al. (2017). Tests were performed to ensure convergence of the cross sections with respect to J t in Eq. ( 2) at all energies, as well with respect to the parameters of the radial grid.

Rate coefficients were then calculated by averaging cross sections for initial and final states i and f , σ i f (E c ), over the Maxwell-Boltzmann distribution of collision energies as expressed :

k i f (T ) = 8 π µ β 1 2 β 2 ∞ 0 E c σ i f (E c )e -β E c dE c (3)
Where β= 1 k B T and k B , T and µ denote the Boltzmann constant, the kinetic temperature and the NH 3 -H 2 reduced mass, respectively.

HYPERFINE RATE COEFFICIENTS

Hyperfine excitation rate coefficients were calculated for temperatures up to 100 K and for transitions involving all levels up to j 1 = 4 for both ortho-NH 3 and para-NH 3 , corresponding to a total of 625 transitions for ortho-NH 3 and 2304 transitions for para-NH 3 . Only excitation due to p-H 2 ( j 2 = 0) is considered here since excited H 2 levels are not significantly populated at the temperatures of molecular clouds where the hyperfine structure is resolved.

Rate coefficients for hyperfine transitions from the rotational state 2 + 1 down to level 1 - 1 are presented in Fig. 1 as a function of temperature up to 100 K. The rate coefficients are seen to be weakly dependent on T, a trend that follows directly from the behaviour of the rate coefficients for pure rotational transitions. On the same figure, we also present for comparison the statistical hyperfine rate coefficients. The statistical approach, also known as the M J randomizing limit (Alexander & Dagdigian 1985), is a commonly used approximation to compute hyperfine-resolved excitation rate coefficients, which assumes that the cross sections or rate coefficient for hyperfine excitation are directly proportional to the degeneracy of the final hyperfine state and independent of the initial hyperfine state. For NH 3 -H 2 the statistical rate coefficients are expressed as

k stat j k v F, j k v F = 2F + 1 (2I + 1)(2 j + 1) k j k v, j k v (4)
where k j k v, j k v are the nuclear spin-free rate coefficients calculated with the CC method and I = 1. To simplify the notation, in the remainder of the paper, we use j to denote the rotational quantum number of NH 3 instead of j 1 without ambiguity since j 2 = 0.

As can be seen from Fig. 1, for all three final hyperfine states of the transition 2 + 1 F → 1 - 1 F the statistical rate coefficients fall between the recoupling values, but a strong dependence on F can be observed and large deviations from the M J randomizing limit are seen to occur. It is thus clear that the statistical method cannot be used to obtain accurate hyperfine rate coefficients in the case of NH 3 -H 2 collisions.

From Fig. 1, it can be seen that the known propensity rule ∆F = ∆ j (Alexander & Dagdigian 1985) holds for the transitions within 2 + 1 → 1 - 1 . However, this is not a general trend. In Table 1 we report the rate coefficients for transitions from levels 2 + 1 F down to levels 1 + 1 F and 1 - 1 F at a fixed temperature of 30 K. For the case of 2 + 1 F to 1 + 1 F transitions (as well as for a large number of transitions involving higher lying rotational levels), the ∆F = ∆ j propensity rule is not respected. In addition, Table 1 also shows that the largest rate coefficients are not necessarily those with the largest F , as would be expected from a statistical behavior.

Of interest are also the quasi-elastic rate coefficients, i.e. rate coefficients for hyperfine excitation within a single rotational level. We present in Table 2 the quasi-elastic rate coefficients j k vF → j k vF at 25 K for the first three rotational states of para-NH 3 , and for the antisymmetric inversion level (v = v = -). The corresponding F -F transitions involving the symmetric inversion levels (v = v = +) are similar although not identical, with differences of up to a few percent (not shown). In addition, we also report in the same table the rate coefficients corresponding to a change of inversion level (-→ +; the reverse rate coefficients can be obtained by detailed balance). It should be noted that those are not strictly quasi-elastic, since the -→ + pure inversion transition correspond to an energy transfer of 0.79 cm -1 . It can be noticed that the rate coefficients strongly depend on the final inversion level, with differences that reach a factor 5 between v = -and v = +. Interestingly, the rate coefficients for inversion-conserving transitions j - k F → j - k F , which would be elastic if the nuclear spin were neglected, are in many cases smaller than those of the corresponding inversion-changing transitions j - k F → j + k F . Finally, the last column of Table 2 compares our values to the results of Stutzki & Winnewisser (1985a), which are independent of the inversion level, and show important discrepancies of up to an order of magnitude for some quasi-elastic transitions. It is worth emphasizing that those results were obtained using a different method, namely the infinite order sudden (IOS) approximation, which is not expected to be accurate at low temperatures. In addition, Stutzki & Winnewisser (1985a) used a potential for NH 3 -He instead of NH 3 -H 2 , which leads to an additional error. Finally, we note that our quasi-elastic rate coefficients are of the same order of magnitude as those for inelastic transitions.

j k v -j k v F = 0 F = 1 F = 2 2 + 1 -1 + 1 F = 1 5.
j k F -F v = -, v = - v = -, v = + Ref. 1 1 0 -2 2.
From their IOS calculations, Stutzki & Winnewisser (1985a) compute relative factors, defined as the ratio of hyperfine rate coefficients divided by the rotational excitation rate coefficient,

g F F = k j k v F, j k v F k j k v, j k v (5)
While the absolute IOS rate coefficients are not expected to be accurate at low temperature, the method should provide relative factors that are better than the statistical ones. These relative factors obtained with the recoupling approach are presented in Table 3 and compared to the IOS results of Stutzki & Winnewisser (1985a) for the 1 + 1 -2 + 1 and 1 + 1 -2 - 1 excitation transitions. We observe that in most cases the IOS method does predict the correct propensities, and performs better than the statistical approach, as expected. On the other hand, it is clear that IOS is not sufficient to obtain accurate rate coefficients, since discrepancies of up to 50% are seen for the dominant rate coefficients and up to a factor 10 for the smallest rate coefficients. Moreover, both the IOS and recoupling relative factors are almost independent of the temperature in the range explored here (see Fig. 1), which shows that the discrepancies discussed above remain up to temperatures of at least 100 K.

RADIATIVE TRANSFER CALCULATIONS

In order to investigate the impact of the hyperfine rate coefficients on the non-LTE modelling of NH 3 spectra, we present below simple Table 3. Relative factors (Eq. ( 5)) for the transitions 1

+ 1 F → 2 + 1 F and 1 + 1 F → 2 - 1 F of NH 3
induced by H 2 at a temperature of 25 K. The values in parentheses are from Stutzki & Winnewisser (1985a). RADEX program (van der Tak et al. 2007) using the large velocity gradient (LVG) formalism for an expanding sphere. The code was employed to compute the radiation temperature T R of low-lying hyperfine transitions in both para-and ortho-NH 3 . The hyperfine energy levels and radiative rates were taken from the CDMS catalog (Müller et al. 2005). The kinetic temperature was fixed at 10 K, which is typical of cold molecular clouds, while the hydrogen density was varied in the range 10 2 -10 9 cm -3 . The para-NH 3 (or ortho-NH 3 ) column densities were selected in the range 10 12 -10 15 cm -2 with a line width (full width at half maximum) of 1 km s -1 . No radiation field was considered except the cosmic microwave background (CMB) at 2.73 K. We have also checked with the MOLPOP code (Asensio Ramos & Elitzur 2018) that line overlap effects, which are ignored by RADEX, are mostly negligible with the above parameters, allowing us to focus on the impact of the collisional data.

F = 1 F = 2 F = 3 1 + 1 F -2 + 1 F F = 0 0.
We first consider the inversion transition 1 - 1 → 1 + 1 of para-NH 3 at 23.68 GHz. This transition has been widely observed in the interstellar medium since its first detection by Cheung et al. (1968). Anomalies in the hyperfine intensities of this line were reported in the 1970s (Matsakis et al. 1977) and these anomalies are now known to be very common in star forming regions. In the anomalous spectra, the intensities of the outer satellite lines are not equal: the F = 0 → 1 component is stronger and the F = 1 → 0 component is weaker than expected. Radiative transfer models have shown that these anomalies can be reproduced by non-LTE effects induced by photon trapping of selected hyperfine transitions in the rotational transition 2 +/-

1 → 1 -/+
1 , as explained by Stutzki & Winnewisser (1985b) (see in particular their Fig. 2). Systematic motions, such as expansion and collapse, are also expected to play a role and the relative contribution of these effects remains a matter of debate (see Zhou et al. 2020, and references therein). In any case, because non-LTE populations depend on the competition between the radiative and collisional rates, the accuracy of hyperfine selective collisional rate coefficients should be critical for a correct interpretation of the anomalies in the 1 - 1 → 1 + 1 inversion transition. In Fig. 2, we have plotted the intensity ratio of the outer hyperfine satellite line F = 0 → 1 relative to the strongest component F = 2 → 2, for the two sets of collisional rates (recoupling and statistical). At low hydrogen density, the ratio is found to increase above the "natural" intensity ratio (0.266) as soon as the column density exceeds ∼ 10 14 cm -2 . The ratios then increase with increasing density and plateaus are reached when n(H 2 ) is larger than about 10 5 cm -3 , where LTE conditions are fullfilled. For column densi- ties above ∼ 10 14 cm -2 , we can notice the impact of the collisional dataset: the intensity ratios are decreased by up to 20% when the statistical dataset is employed. This demonstrates the importance of the hyperfine propensity rules when the total opacity of the line is larger than ∼ 1. A similar result was actually observed by Stutzki & Winnewisser (1985a) using hyperfine collision rates computed with the IOS approximation and using Helium instead of H 2 as the collider.

As concluded by these authors, a precise knowledge of the hyperfineresolved rate coefficients is thus required for a reliable modelling of observational data.

A second example is the ortho-NH 3 ground-state rotational transition 1 + 0 → 0 - 0 at 572.5 GHz. The electric quadrupole hyperfine structure of this line was resolved for the first time in space thanks to Herschel observations towards the cold pre-stellar core L1544 (Caselli et al. 2017). The three hyperfine components were found to be heavily self-absorbed and a sophisticated radiative transfer model was employed to reproduce the observed profile and to deduce the NH 3 radial abundance in the core. In that work, hyperfine selective collisional data were generated from the rotational rate coefficients of Bouhafs et al. (2017) by applying the statistical approach to the electric quadrupole and magnetic hyperfine splittings. In Fig. 3, we have plotted the intensity ratio of the hyperfine component F = 1 → 1 relative to the strongest component F = 2 → 1, again for the two sets of collisional rates (recoupling and statistical). We note that our statistical set is identical to that used by Caselli et al. (2017) except that the magnetic splittings (< 100 kHz) are neglected here. At low hydrogen density, the ratio is found to lie slightly below the natural intensity ratio (0.6) for column density above ∼ 10 12 cm -2 . When the H 2 density exceeds ∼ 10 5 cm -3 , however, the intensity ratios rise above 0.6 and the LTE plateaus are reached at H 2 densities larger than ∼ 10 8 cm -3 , as expected from the large radiative rates (A ∼ 1.58 × 10 -3 ). The impact of the rate coefficients is smaller than in the case of the para-NH 3 transition (Fig. 2) with a maximum difference of ∼ 10% in the intensity ratios. This is not really surprising since the recoupling and statistical rate coefficients for the hyperfine transitions 1 + 0 F → 0 - 0 F = 1 are identical and equal to the pure rotational rate coefficients (by construction since the rotational 0 - 0 level has a single hyperfine sub-level F = 1). Thus, only the rate coefficients corresponding to hyperfine transitions within the rotational level 1 + 0 , the so-called quasi-elastic rate coefficients, are different in the two datasets: they are set to zero in the statistical set while they are comparable to the pure rotational rate coefficient in the recoupling set. We can conclude that the quasi-elastic rate coefficients have a significant although moderate impact in the radiative transfer calculation.

Finally, we wish to stress that the above results are only illustrative: more sophisticated radiative transfer calculations are clearly needed to explore a much larger parameter space and to include line overlap as well as systematic motion effects. Preliminary calculations with MOLPOP show, for instance, that for kinetic temperatures larger than 10 K and column densities above 10 15 cm -3 , line overlap effects can increase the intensity ratio

F = 0 → 1/F = 2 → 2 in the 1 - 1 -1 + 1
inversion transition by more than 50%. The combination of the present recoupling rate coefficients with one-dimensional radiative transfer codes able to handle hyperfine overlaps, such as ALICO (Magalhães et al. 2018), will be particularly valuable to revisit the NH 3 hyperfine anomalies and to more firmly establish the role of non-LTE effects.

CONCLUSION

We have presented a full set of rate coefficient for the hyperfine excitation of ortho-and para-NH 3 in collisions with para-H 2 for all levels of NH 3 up to j = 4 and for temperatures up to 100 K. The calculations were performed by means of the accurate recoupling technique, which relies on quantum-mechanical close-coupling scattering calculations. In general, the rate coefficients are weakly dependent on the temperature, but depend strongly on all quantum numbers of NH 3 . We found no clear propensity rules, and a comparison with statistical rate coefficients (which are independent of the initial hyperfine level) showed important deviations from this approximation. In addition, we also found large differences with previously published results on NH 3 -He collisions.

Using a simple non-LTE model, we illustrated the difference between the accurate recoupling rate coefficients and the approximate statistical rate coefficients on the inversion transition in ground state para-NH 3 (1 - 1 F → 1 + 1 F ) at 23.68 GHz and on the rotational transition 1 + 0 F → 0 - 0 F in ortho-NH 3 . These simple calculations demonstrate that the two sets of rate coefficients lead to significant differences. Moreover, non-LTE effects are predicted for column density in excess of ∼ 10 12 cm -2 . An important extension of the present work will be to perform radiative transfer calculations using a model that can treat line overlaps which play a key role in NH 3 hyperfine anomalies. This would allow the investigation of non-LTE effects at higher temperatures, thus involving higher-lying rotational states of ammonia.

As a reminder, the present calculations only considered the hyperfine splitting due to the N nucleus but did not take into account the (much smaller) further hyperfine magnetic splitting due to the three protons, which is sometimes resolved in astronomical spectra (Rydbeck et al. 1977). We suggest that the corresponding rate coefficients can be obtained from those presented here using the statistical approximation.

Finally, it should also be noted that the recoupling calculations presented in the present work are accurate but computationally expensive. At the other end, statistical rate coefficients can be immediately obtained provided the rate coefficients for pure rotational excitation are available. An intermediate approach is the IOS method, which has been used with some success for the hyperfine excitation of linear molecules by H 2 (Goicoechea et al. 2022). Extending the IOS method and recoupling theory to asymmetric polyatomic molecules would be beneficial in order to generate sets of hyperfine-resolved rate coefficients for other molecules such as deuterated ammonia (NH 2 D, NHD 2 ) or H 2 CO.

Figure 1 .

 1 Figure 1. Rate coefficients for the transitions 2 1 F → 1 - 1 F of NH 3 induced by collisions with H 2 as a function of temperature. Panel (a), F = 0; panel (b), F = 1; panel (c), F = 2, and comparison to the statistical rate coefficients given by Eq. (4).

Figure 2 .

 2 Figure2. The intensity ratio of the hyperfine satellite line F = 0 → 1 of the para-NH 3 j k = 1 1 inversion transition relative to the strongest component F = 2 → 2 as function of the hydrogen density for a set of column densities. The kinetic temperature is fixed at 10 K and the line width at 1 km s -1 .

Figure 3 .

 3 Figure3. The intensity ratio of the hyperfine component F = 1 → 1 of the ortho-NH 3 j k = 1 0 → 0 0 rotational transition relative to the strongest component F = 2 → 1 as function of the hydrogen density for a set of column densities. The kinetic temperature is fixed at 10 K and the line width at 1 km s -1 .

Table 1 .

 1 Rate coefficients for the transitions 2+ 1 F → 1 + 1 F and 2 + 1 F → 1 - 1 F of NH 3 inducedby H 2 at a temperature of 30 K. Brackets indicate powers of 10.

Table 2 .

 2 Quasi-elastic rate coefficients for the excitation j k v F → j k v F

		16[-13] 1.98[-11] 7.27[-12]
		F = 2 8.62[-12] 3.11[-12] 1.58[-11]
		F = 3 1.85[-13] 8.99[-12] 1.84[-11]
	2 + 1 -1 -1	F = 1 3.31[-11] 2.57[-11] 4.97[-12] F = 2 3.96[-13] 4.58[-11] 1.76[-11]
		F = 3 7.19[-13] 1.83[-12] 6.12[-11]
	for rotational levels 1 1 , 2 2 , and 2 1 of NH 3 by H 2 at 25 K. Comparison with
	the data of Stutzki & Winnewisser (1985a) (last column). Brackets indicate
	powers of 10.

© 2023 The Authors

We note that the results ofChen et al. (1998) are questionable and should not be used because some of their rate coefficients are unphysically large (up to 10 -7 cm 3 s -1 ).

MNRAS 000, 1-6 (2023)

This paper has been typeset from a T E X/L A T E X file prepared by the author.MNRAS 000,1-6 (2023)

ACKNOWLEDGEMENTS

Brian Svoboda is acknowledged for useful discussions. J.L. acknowledges support from KU Leuven through Grant No. 19-00313. Moreover, we acknowledge financial support from the European Research Council (Consolidator Grant COLLEXISM, Grant Agreement No. 811363). F.L. acknowledges financial support from the Institut Universitaire de France and the Programme National "Physique et Chimie du Milieu Interstellaire" (PCMI) of CNRS/INSU with INC/INP cofunded by CEA and CNES.

DATA AVAILABILITY

The data underlying this article are available as supplementary material to the article and the full set of rate coefficients will be made available through the EMAA database (https://emaa.osug.fr and https://dx.doi.org/10.17178/EMAA).


Krishna Maity
email: krishna.maity@ipcms.unistra.fr

Jean-François Dayen

Marlène Palluel

Nathalie Daro

Guillaume Chastanet

Bohdan Kundys

Bernard Doudin
email: bernard.doudin@ipcms.unistra.fr

Elucidating the effect of spin crossover materials on graphene sensing devices

they are insulating in nature, limiting their integration into electronic devices [START_REF] Prins | Room-Temperature Electrical Addressing of a Bistable Spin-Crossover Molecular System[END_REF] . On the other hand, graphene is a highly conducting material, highly sensitive to its environment [START_REF] Han | Graphene-based flexible electronic devices[END_REF][START_REF] Zhang | Capacitive Sensing of Glucose in Electrolytes Using Graphene Quantum Capacitance Varactors[END_REF][START_REF] Dhinakaran | Review on exploration of graphene in diverse applications and its future horizon[END_REF] . Pioneering the combination of these two unique materials, Dugay et al. investigated self-assembled layers of SCO nanoparticles on top of graphene transistor devices and observed how their hysteretic temperature change of conductance related to the spin transitions of the SCO [START_REF] Dugay | Phase Transitions in Spin-Crossover Thin Films Probed by Graphene Transport Measurements[END_REF] . They proposed that it resulted from phonon modes coupling to the charge carrier of graphene that enhanced scattering, with a resulting change of carrier's mobility. The charge transport of graphene/SCO devices have also been studied on a model SCO single crystal, separated from graphene by a thin polymer spacer. Here the authors found experimental and theoretical arguments molecular phase-dependent electrostatic coupling between the two materials. [START_REF] Geest | Contactless Spin Switch Sensing by Chemo-Electric Gating of Graphene[END_REF] . The electrical transductions of both thermal and light induced spin transition (LIESST) at low temperature have been demonstrated on graphene sensor interfaced with sublimated SCO thin film, and interpreted as shift of the Fermi energy level of the graphene layer upon the switching of the molecular state [START_REF] Konstantinov | Electrical read-out of light-induced spin transition in thin film spin crossover/graphene heterostructures[END_REF] . A model of strain-induced change of This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0163784 graphene conductivity has also been recently put forward to explain the influence of SCO thin films change of phase. [START_REF]Electrical Sensing of the Thermal and Light-Induced Spin Transition in Robust Contactless Spin-Crossover/Graphene Hybrid Devices -Gavara-Edo -2022[END_REF] However, most reported graphene transistor devices were studied near the neutrality point of the graphene, where it is most difficult to untangle changes of mobility and number of carriers (indicative of a change of Fermi level). There is therefore a need of clarifying the issue, check if the mechanism depends on the type of SCO materials used: nanoparticles, single crystal, or thin films, and test if any proposed mechanisms defined for such ordered SCO materials can be extended to a more disordered and discontinuous material. This could then guide design constraints for large scale integration of SCO systems for device applications as well as validate the robustness of the proposed strategy for making graphenemolecular materials hybrids.

Better understanding can also be put to question possible higher frequency operation of hybrid graphene

devices. An important bottleneck in molecular electronics applications is their considered limited speed capabilities when compared to standard semiconductor device technologies. Even though a few reports of fast switching molecular devices can be found in the literature [START_REF] Garcia-Amorós | Fastest molecular photochromic switches based on nanosecond isomerizing benzothiazolium azophenolic salts[END_REF][START_REF] Han | Molecular Platform for Fast Low-Voltage Nanoelectromechanical Switching[END_REF] , their integration into scalable circuitry remains elusive. The switching of SCO materials can occur at the ns time scale when their size is reduced down to the nanoscale, at which one can still take advantage of the molecules collective behavior. [START_REF] Ekanayaka | Dynamics of Spin Crossover Molecular Complexes[END_REF] This makes nanoparticles size and shape of utmost importance if operation speed relevance for memory device applications is mandatory. The well-established capability of graphene for fast electronics 17 makes hybrid molecular nanoparticles-graphene devices possible contenders for RF molecular electronics applications.

Impedance spectroscopy (IS) is a widely used technique of choice to better understand relaxation of charge carriers and charge transfer at interfaces in materials ranging from bulk to nanoscale. The interfacial behaviour in graphene-silicon [START_REF] Yim | Characterization of graphene-silicon Schottky barrier diodes using impedance spectroscopy[END_REF] , interlayer charge transfer in dielectric with graphene [START_REF] Jordan | Trapped charge densities in Al 2 O 3based silicon surface passivation layers[END_REF][START_REF] Ki Min | Electrical Double Layer Capacitance in a Graphene-embedded Al2O3 Gate Dielectric[END_REF][START_REF] Mills | Direct Determination of Field Emission across the Heterojunctions in a ZnO/Graphene Thin-Film Barristor[END_REF] and the effect of interface modification [START_REF] Min | AC-Impedance Spectroscopic Analysis on the Charge Transport in CVD-Grown Graphene Devices with Chemically Modified Substrates[END_REF] between graphene and SiO2 were explored with IS. In this article, we propose to get better insight into the coupling mechanism of SCO particles drop-casted on graphene devices using IS data. This will also go beyond the reported data, currently limited to DC measurements in the literature, and provide insight into broader band electronics device capabilities.

CVD grown single layer graphene on SiO2/Si substrate was bought from Graphenea. Before patterning, the devices were kept at 250 o C for 12 h inside a glovebox to remove moisture and residue in the CVD grown and transferred graphene. Two types of devices were then patterned and connected with Au electrodes using optical lithography and Ar etching.

This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0163784 SCO [Fe(Htrz)2trz) + (PF6) -] particles were synthesized by following a procedure previously reported [START_REF] Palluel | Rational direct synthesis of [Fe(Htrz)2(trz)](BF4) polymorphs: temperature and concentration effects[END_REF] . They were then dispersed in ethyl alcohol by ultrasonication for 5 minutes and drop casted on the graphene device. The SEM image in Figure 1 (b) shows elongated particles of typical 3000 nm by 500 nm size, strongly agglomerated, defining the graphene-nanoparticles hybrid, further denominated Gr/SCO. These devices were then heated for 12 hours at 80 • C in vacuum to limit moisture and solvent residues. Figure 1b illustrates the typical coverage of all samples studied, that we estimate in the 50-70% range. Adding particles was found to have little influence on the sample properties, not exceeding reproducibility in measurements.

Two electrodes' devices for impedance spectroscopy studies (1800 µm×500 µm, Fig. 1 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0163784

The Gr/SCO device exhibits resistance versus temperature value following the SCO spin state change of phase, with hysteretic variation with temperature (Fig. 1e) indicative of the collective behavior reported for nanoparticles ensembles. This is in strong contrast with the resistance of pristine CVD graphene varying little and monotonically in the measured temperature range [START_REF] Konstantinov | Electrical read-out of light-induced spin transition in thin film spin crossover/graphene heterostructures[END_REF] . The Gr/SCO resistance increases significantly when cooling and SCO particles changing their spin state from HS to LS with a transition temperature of 𝑇 !/# ↓ = 256 K that matches the one measured by magnetometry on the powder initial sample. [START_REF] Dugay | Phase Transitions in Spin-Crossover Thin Films Probed by Graphene Transport Measurements[END_REF][START_REF] Geest | Contactless Spin Switch Sensing by Chemo-Electric Gating of Graphene[END_REF][START_REF] Konstantinov | Electrical read-out of light-induced spin transition in thin film spin crossover/graphene heterostructures[END_REF] When heating up, the expected hysteresis of the SCO transition is observed, with a measured 𝑇 !/# ↑ = 306 K, significantly larger than the one found by magnetometry (278 K) and in our own previous experiments. 24 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0163784

While the cooling curve transition was reproducible (on more than 10 samples tested) with a variation of resistance upon transition in the 5 to 20% range on 80% of the samples, the frequently observed sample resistance drifts under temperature sweeps, increasing when increasing the temperature, made the heating curve less reproducible. Electric transport measurements are therefore limited to the cooling part of the hysteresis, at 280 K where most molecules are expected to be in the HS state and at 240 K where most molecules are expected in the LS state. IS data is presented in Fig. 2, where we compare the spectra of bare graphene (at 280K, with little temperature variation) and Gr/SCO at 240K and 280 K in the expected LS and HS majority states (Fig. 2 a,b). Measurements are performed at zero gate bias, with gate Si substrate and source electrode grounded, to ensure long term stability of the impedance, key for reliable data acquisition. At such gate value, graphene is p-doped, quite far from its Dirac point, in agreement with our previous findings on similar samples with smaller particles, [START_REF] Dayen | Room temperature optoelectronic devices operating with spin crossover nanoparticles[END_REF] as well as test measurements of full transfer curves (SI, Fig. S1). The real part of the impedance (Z ' ), or resistive part of the device, remains approximately constant in the low frequency range and decreases with increasing frequency. A Nyquist plot (Fig. 2c) shows a typical semi-circle aspect of a simple equivalent circuit, made of a series resistor in series with two parallel resistor and capacitor (inset Fig. 2c). The setup leads capacitance due to cabling and micromanipulators tips was previously measured using calibrated sample resistors and estimated at 170 pF, a value we subtracted from the fitted total capacitance of the circuit in the 500 pF range. We therefore suppose that the sample resistance RG in Fig. 2c relates to the intrinsic conductivity of the graphene film and that the total capacitance Ct sums the several surrounding capacitances of the graphene film. The quantum capacitance of graphene, and the capacitance of the SCO over-layer cannot exceed the aF range, more than four orders of magnitude lower than the fitted values. Ct therefore corresponds to the SiOx substrate capacitance, separating the circuit from the conductive Si/bottom ground electrode in this bottom gate FET device configuration. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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Resulting fitted values are tabulated in Table 1. The contact resistance Rc is almost constant and doesn't vary significantly compared to the graphene channel in the range of measured temperature, as expected.

The resistance of graphene RG increases when covered by SCO material, further increasing when cooling down and going from the HS to the LS phase of SCO. Note the errors indicated in Table 1 result from the fitting procedure and are overestimated for RG as measurements are performed in the low frequency range only, as illustrated in the data below 10 kHz in Fig. 2a and2b. The capacitance of our CVD grown graphene device (of the order of several tens of nF/cm 2 ), on-par with the literature on CVD graphene over SiO2 substrates, [START_REF] Min | AC-Impedance Spectroscopic Analysis on the Charge Transport in CVD-Grown Graphene Devices with Chemically Modified Substrates[END_REF] originates from trap sites at the interface of silicon oxide and graphene. Within 4-8 nm, impurity and defects create trap sites which are most commonly positively charged (N + D) [START_REF] Kim | Focused-Laser-Enabled p-n Junctions in Graphene Field-Effect Transistors[END_REF] , capable of electron trapping (N + D + e -à ND) and release (ND à N + D + e -) by Schottky emission [START_REF] Lundström | Tunneling to traps in insulators[END_REF] . Graphene becomes hole doped due to the trapping of electrons by these trap sites [START_REF] Peng | Controllable p-to-n Type Conductance Transition in Top-Gated Graphene Field Effect Transistor by Interface Trap Engineering[END_REF] , as previously measured on similar samples. [START_REF] Dayen | Room temperature optoelectronic devices operating with spin crossover nanoparticles[END_REF] One can model the total Gr/Si capacitance as the sum of the interface capacitance, here mostly due to the trap sites, with the bulk 300 nm thick SiO2 substrate dielectric capacitor. The latter is however much larger than the interface capacitor, which will then dominate the fitted value. The number of trap charges can be then estimated as 𝑛 ≈ 𝑉 ;< 𝐶 =>?@ABCD@ /𝑒 (𝑉 ;< = 0.1 V). Impedance data therefore indicates a change of graphene intrinsic conductivity combined with a change of the number of trap sites when adding SCO particles and modifying their spin state. This trend was observed on all measurements (on four samples), with however significant variations in the values and their relative change. Another example is given in the SI (Fig. S2).

Further insight is gained by performing Hall measurement on similar material patterned with a Hall bar geometry, to better pinpoint the origin of the changes of conductance of the graphene. While IS was performed on several samples, Hall data was measured on one. We chose a device with stable enough resistance at 280 K under magnetic field sweeps experiments and compare the IS data in Fig. 2 from a sample patterned and covered with SCO particles in the same lithography and baking batches. The longitudinal Hall resistance value RL follows the same trend as RG (Table 1) and confirm the reproducibility of the trend of the IS analysis. The Hall resistance 𝑅 & varies linearly with magnetic field and is shown in Fig. 3. The sign of the Hall coefficient confirms that holes are the majority charge carrier in graphene [START_REF] Karpiak | Gate-tunable Hall sensors on large area CVD graphene protected by h-BN with 1D edge contacts[END_REF] .

The Hall coefficient, which is directly related to the charge carrier density (𝑅 & = -! >@ ; n is charge carrier density and e is electron charge 1.6×10 -19 Coulomb), increases significantly in the device after deposition of the SCO particles and further increases by the changing of spin states, with the carrier densities listed in Table 1. The Hall mobility is calculated using the formula 1 shows how mobility increases remarkably when covering graphene with SCO particles, confirming the IS results. However, there is no significant change of mobility under SCO transition. This is consistent with our previously This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0163784

𝜇 & = 𝑅 & 𝑅 E ⁄ . Table
reported results with field effect transistor measurements. [START_REF] Dayen | Room temperature optoelectronic devices operating with spin crossover nanoparticles[END_REF] Note that the Hall measurements provide a more reliable insight into mobility values, as they should not be influenced by trapping-untrapping of interfacial states, possibly sensitive to applied electric fields.

The change of graphene properties under SCO transition values are consistent with previously reported data on similar graphene devices [START_REF] Konstantinov | Electrical read-out of light-induced spin transition in thin film spin crossover/graphene heterostructures[END_REF][START_REF] Amit | Role of Charge Traps in the Performance of Atomically Thin Transistors[END_REF][START_REF] Sarkar | Graphene Oxide as a Dielectric and Charge Trap Element in Pentacene-Based Organic Thin-Film Transistors for Nonvolatile Memory[END_REF] . The SCO particles have intrinsic electrical polarization due to either the molecules electrical dipoles or to their combinations with counterions co-crystallized in the material [START_REF] Geest | Contactless Spin Switch Sensing by Chemo-Electric Gating of Graphene[END_REF][START_REF] Bousseksou | Observation of a thermal hysteresis loop in the dielectric constant of spin crossover complexes: towards molecular memory devices[END_REF] .

This electrostatic gating de-trap electrons from the trap sites with a resulting increased number of positively charged trap sites, and hence increased total capacitance of the device. When adding SCO particles, a change of around 70 % of number of graphene charge carriers is mirrored by the same order of magnitude of change of trap sites, around 50%. This trend is confirmed by the same relative change of around 10% when switching the SCO state from HS to LS. The emerging picture is that SCO material changes the population of (charged) trap sites, increasing when releasing e -, which in turn diminish the graphene layer doping. We recall here that we are investigating an intrinsically highly doped graphene material, where the This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0163784 graphene carrier's mobility is expected to vary little with the number of carriers. As adding SCO particles impacts by 10-20% the graphene resistance (RG or RL), a very significant increase of mobility, directly resulting from Hall measurements, is observed. This is in contrast with most studies of doping effect of molecular adsorbates on graphene [START_REF] Lee | A review of doping modulation in graphene[END_REF][START_REF] Singh | Recent Developments in Chemical Doping of Graphene using Experimental Approaches and Its Applications[END_REF] , where results systematically report a decreasing mobility when adding molecules on graphene.

The observed change of graphene properties under the transition of the SCO state is however quite different in origin. Here, the 10-20% change in graphene resistance comes from a similar change of carrier densities, without noticeable change in the measured Hall mobility. Even though 𝜇 ! is sensitive to the presence of SCO, it is invariant under the molecular phase transition. Therefore, the graphene detects the occurrence of the spin transition through a simple shift of Fermi energy. The origin is again most probably electrostatics, where we expect that the structural change experienced by the SCO material modifies its electronics clouds configuration, especially at its surface, and therefore changes the number of carriers in graphene through the substrate trapping sites. We do not foresee any strain contribution, as the SCO coverage of the graphene is discontinuous and partial, and large strain values would be needed to significantly modify the graphene conductivity.

In summary, we found that adding SCO nanomaterial to graphene impacts the density of carriers, mostly through the de-trapping of the silicon oxide substrate states. Surface charges of the SCO particles screen the local electrostatics, which can possibly explain the increase of mobility when adding the particles, unusual for molecular adsorption on graphene. The observed changes of density of carriers and intrinsic graphene device resistance confirms the electrostatics nature of the detection of the SCO presence as well as its phase. Our results indicate that such a mechanism can apply to disordered adsorbed film, with a large size dispersion, without a need of well-organized single crystal, thin film, or self-assembled layer of identical small particles. Impedance measurements confirm the intrinsic resistive nature of the graphene hybrid device. The cut-off frequency is driven by the substrate capacitance, limited to around 500 kHz for the large-scale mm 2 area devices studied here (reaching 1 MHz for the sample in the SI, Table S1). Such limitation originates from intrinsic properties of the silicon oxide and graphene. For µm 2 sized devices, smaller by 5-6 orders of magnitude, the capacitance should scale down similarly. One can therefore expect the cutoff frequency to increase by orders of magnitude. Gr/SCO devices involving adsorbed nanoparticles material have therefore high-frequency electronics capabilities that should match those of CVD graphene transferred on SiO2. Even though such graphene substrates are not usually chosen for FET highest frequency studies, we remind here that the main approach for isothermal switching for SCO materials is the use of light, making then the top SOC film a 'top gating' material, best suited for fast FET switch operation.

Moreover, this SOC material offers room temperature operability by means of light-triggered switching This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0163784

  (a)) and four electrodes Hall-bar devices (1400 µm× 300 µm, Fig. 1(d)) were defined by laser lithography and 10 nm/40 nm Ti/Au interconnects lift-off over the etched graphene areas. IS measurements were performed by lockin technique in a frequency range from 100 Hz to 5 MHz in a variable temperature probe station under low pressure N2 atmosphere, and Hall measurements performed in a Quantum Design PPMS system under low pressure He atmosphere.

Fig. 1 :

 1 Fig. 1: Graphene devices with drop casted SCO material used for measurements. (a) two-probes design for impedance spectroscopy measurements; (b) SEM picture of the SCO coverage made of ensembles of elongated particles; (c) sketch of the Fe(Htrz)2trz + molecule (with (PF6) -counterions) with dashed lines indicating the intermolecular binding creating polymer-like ensembles; (d) four-probes design for longitudinal and transverse (Hall) resistance measurements; the patterned graphene area is emphasized by the dashed lines area, of size of 1.8 mm X 0.5 mm and 1.4 mm X 0.3 mm respectively; (e) temperature hysteresis of the Gr/SCO device resistance.

Fig. 2 .

 2 Fig. 2. Variation of real part Z ' and imaginary part -Z " with frequency: (a) comparison between bare graphene and Gr/SCO at 280 K (HS); (b) comparison between two different spin states of SCO/Gr, measured at 240 K and 280K (points in Fig. 1e); (c) Nyquist Plots (-Z " vs Z ' ) fitted with the circuit model in the insert.

Fig. 3 :

 3 Fig. 3: Variation of Hall resistance with magnetic field comparing the initial graphene FET with the same device with SCO nanoparticles in their two different spin states, measured at temperatures shown in Fig. 1e and used for the data of Fig. 2.

Table 1 :

 1

		Rc (Ω)	RG (Ω)	Ct (nF/cm 2 )	Trapping	charge	RH (Ω/T)	Carrier	RL (Ω)	𝜇 !
					density (cm -2 )		±1%	density	± 1%	(𝑐𝑚 " 𝑉. 𝑠 ⁄	)
								(cm -2 ) ±1%		±2%
	Graphene	46±7	550 ± 30	35 ± 5	(2.2±0.3) ×10 10	61	10.2×10 12	3140	194
	Gr/SCO at 280 K	39±6	600 ± 20	49 ±3	(3.0±0.2) ×10 10	192	3.3×10 12	4013	478
	Gr/SCO at 240 K	37±5	710±23	52±3	(3.3±0.2) ×10 10	212	2.9×10 12	4411	480

Table for fitted parameters contact resistance (Rc), graphene channel resistance (RG), total capacitance (Ct), Hall coefficient (RH), deduced charge carrier density, measured longitudinal resistance in Hall-bar device (RL) and deduced Hall mobility (𝜇 & ) in the graphene channel.

This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0163784

Acknowledgements

We thank Fabien Chevrier and the staff of the STnano nanofabrication facility for daily support and J. Robert for PPMS measurements. This project has received funding from the Agence Nationale de la Recherche (HEROES ANR-17-CE09-0010-01) and the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement MaMi No. 766007 and QUSTEC No. 847471, the Interdisciplinary Thematic Institute QMat, as part of the ITI 2021 2028 program of the University of Strasbourg, CNRS and Inserm, was supported by IdEx Unistra (ANR 10 IDEX 0002), and by SFRI STRAT'US project (ANR 20 SFRI 0012) and EUR QMAT ANR-17-EURE-0024 under the framework of the French Investments for the Future Program. B.D. aknowledge the support of the Institut Universitaire de France.

Data Availability

Data is available upon request to the corresponding authors.

operation [START_REF] Dayen | Room temperature optoelectronic devices operating with spin crossover nanoparticles[END_REF] . The far-fetched GHz operation speed, matching the expected optically induced transition time scale, [START_REF] Ekanayaka | Dynamics of Spin Crossover Molecular Complexes[END_REF][START_REF] Hu | Photo-Thermal Switching of Individual Plasmonically Activated Spin Crossover Nanoparticle Imaged by Ultrafast Transmission Electron Microscopy[END_REF] might therefore be possible, under the hypothesis that the particles surface charges configuration will follow their light-triggered switching. While there are still unknowns in the 3-4 orders of magnitude needed to reach GHz operation, we consider that graphene sensors of SCO nanoparticles are ideally suited for this goal, with the perspective of easy to process ideal and robust contenders for fast molecular electronics devices.

Supplementary Informations

Complementary information is provided: trandfer curve of a two-terminals device (Fig. S1), another example of IS data, showing a 1 MHz cutoff, ( Fig. S2), with table summarizing the equivalent circuit values (Table S1) that compare to Table 1. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

Author Contributions

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0163784


Jonas Aparicio
email: jonas.aparicio@enpc.fr

Tien Hoang

Gwendal Cumunel

Gilles Forêt

Yannick Jeanjean

Saint Julien Castres

Gilles Foret

Mechanical impact of wires break in grouted external prestressing tendon

Keywords: Prestress loss, Wire break, Seven-wire strand, Interwire friction, Posttensionned bridge

I. INTRODUCTION

Prestressing concrete is widely used on long-spanned bridges as it is an efficient method for reducing structural mass and improving integrity. This paper focus on prestressing tendon made of strands of steel wires laid in a plastic sheath then grouted. The strands in this paper each consist of one straight core wire layered by six helical wires. The strands are put to a tension of about 70% of their fracture strength which is the usual remaining tension in prestressing tendon after creep and relaxation. The strands are protected against exterior aggression by a plastic sheath mostly HDPE. The sheath was injected with grout to protect the steel from corrosion until the 2000s and is now injected with wax or fat. Issues during construction or exploitation can lead to loss of protection and steel corrosion. Loss of wire due to corrosion is the main durability issue of prestressing tendon [START_REF] Godart | La Pérennité Du Béton Précontraint[END_REF]. Wire breaking leads to stress redistribution due to friction of a wire with the grout and the other wires. A cut wire will get its original tension back at a distance from the cut. That distance is called re-anchoring or transfer length. It can lead to catastrophic tendon breaking as individual wires reach fracture strength leading to a cascade of breaking. Tendon breaking occurs after around 30% of section loss. This is the section loss where remaining steel surface is not sufficient to endure the remaining tension, thus causing the tendon to break. Damaged tendons usually show wire breaks distributed on multiple strands. Behaviour of damaged tendon were previously thought to behave with the strand re-anchoring phenomenon. When a strand breaks, if there is sufficient bond between the strand and the surrounding concrete and that the tendon is sufficiently long, the strand will get back to its original stress a distance to the break. This distance is called transfer, transmission, re-anchoring or anchorage length depending on the papers. This damage model keep the symmetry of the strand and the phenomenon is well understood as it is central to how stress is transferred from the strand to the concrete in internal prestressing. The Poisson effect cause the ruptured end of the broken strand to form a wedge and create friction with the grout. Furthermore, the loss of tension in the wires cause the wire to unfold and thus increase contact with the concrete, this is called the Hoyer effect [START_REF] Briere | Dilation behavior of seven-wire prestressing strand -The Hoyer effect[END_REF]). Hoyer and Poisson effect combine to create friction between the wires and between external wire and the grout which cause stress to be transferred back in the cut strand. [START_REF] Abdelatif | Modeling and Parametric Study of the Reanchorage of Ruptured Tendons in Bonded Posttensioned Concrete[END_REF] show that the transfer length for a grouted strand is of the order of a meter in a review of different studies about this phenomenon. The mechanical characteristic of steel and concrete are shown to be the main parameter influencing the re-anchoring length. A sensibility analysis can be found on the article [START_REF] Watanabe | Evaluation for Flexural-load Capacity of Prestressed Concrete Girders with Broken Tendons[END_REF] following an experimental campaign of cutting strand in different configuration. The article [START_REF] Asp | Bond and re-anchoring tests of post-tensioned steel tendon in case of strand failure inside cement grouting with voids[END_REF] shows that in case of inefficient grouting the transfer length will grow bigger, with 1.7m with incomplete grouting to 1m with good grouting. The re-anchoring of a whole strand is presented in multiple construction codes such as American Concrete Institute 2011, the Eurocode EN-1992-1-1(2004) ( §8.10.2), and the British code CS 455 ( §8.3.1). According to codes, the transfer length of a cut strand is of the order of 1 meter. Detected damage on tendon show that damage occurs on a wire by wire basis with damaged strand often breaking only after multiple wires have already ruptured.

The response of a undamaged strand subjected to tension and couple has been explored in numerous papers, reviewed by [START_REF] Ghoreishi | Validity and limitations of linear analytical models for steel wire strands under axial loading, using a 3D FE model[END_REF] where steel wires are modeled as Love thin rod and interaction between wire are expressed in the helical Fresnet coordinate system (Figure 1). The articles reviewed differ about how they solve contact, model traction or take into account Poisson Effect and wire flattening but they all rely on symmetry which makes then no longer valid for damaged strand which has seen less theoretical and experimental works. There are fewer studies about the breaking of individual wire. [START_REF] Macdougall | Tests of Unbonded Seven-Wire Tendon with Broken Outer Wires[END_REF] investigated the re-anchoring of one or two wire in a strand. Cutting of individual wire in a strand with a concrete deviation lead to transfer length of more than 10 meters. The transfer is partial as tension in the strand go down by 2-4% each cut, in a 7 wire strand each wire account for around 14% of the tension. This paper shows that stress redistribution in the wires is symmetric but not homogeneous. Tension after the cut in each wires depends on its relative position to the cut wire (Figure 2a). [START_REF] Beltrán | Static response of asymmetrically damaged metallic strands: Experimental and numerical approach[END_REF] use analytical and FE computation to derive the static response of a strand with a broken wire, the strand in unbounded.

Both of this article assume that damage of broken wire induce a helical displacement of the strand to respect equilibrium. A tentative confinement has been done on the The numerical and experimental study conducted by [START_REF] Abdullah | Experimental and Numerical Evaluation of Unbonded Posttensioning Tendons Subjected to Wire Breaks[END_REF] (Figure 2b) with unstressed strands surrounding the stressed strand in which wires are cut. Prestress loss is only 10% less with the confinement than without. Stress transfer is worse than in MacDougall deviated strand. Transfer length has been measured as the 30 meters between the cut and the end of the bench.

The review has shown that while the behaviour of the healthy strand is well known, the behaviour of a damage strand is only partially understood and there are no study on the behaviour of damaged strands with a confinement as strong as the one in grouted external prestressing tendon and neither with force transfer due to wire damage between multiple strand to the best of the authors knowledge. This behaviour is useful in order to better the assessment the remaining capability of a detected damaged tendon. Furthermore, the improved knowledge on a damaged tendon could be use to improve control and detection of default in the tendon.

In this paper, an experiment is presented which was to build realistically sized tendons then break them by cutting wires one at a time. The aim of this experiment is to better the understanding of stress transfer inside the damage tendon and to have an estimation of the length affected by the damage. The experiments has been repeated using variable number of strand, brittleness of sheath and cutting schedule in order to perform a sensibility analysis on these different parameter. Mechanical results has been obtained from a cell force, photograph and film. Department (MAST). The installation (Figure 4) was carried out by the Freyssinet company. Three tendons containing one strand each and two with 5 strands each were tested. The strands were passed through the anchoring system (step 1 on Figure 3) and then through the HDPE sheath (63x5.8mm), whose segments are welded together. In the first two experiments, the center part of the sheath was made of transparent PVC to allow visually following of the crack propagation in the grout, with the PVC and HDPE sheath connected by a screw joint. A grouting vent was provided for better grouting quality. Anchoring systems were installed at both ends with a force cell at the passive end. Tension was applied with a hydraulic jack at the active end at 70% of the tendon's braking strength (step 2 on Figure 3) . The tendon was then injected with Holcim Superstresscem grout, which is widely used for this type of application (step 3 on Figure 3). The 5 tendons had a length of 46.6 m between the two anchor blocks. After allowing the grout to sit for 7 days, a 20-cm trapdoor was made in the sheath to access the wire and the grout was removed in the trapdoor so that the wires could be cut one by one with a Dremel mini rotary saw during testing (Figure 5). Great precautions were taken to protect the operators and in particular the operator in charge of cutting the wires by taking care to protect him from whiplash effect of the tendon during its rupture. Many instrumentation systems have been installed on the tendon by different teams. This article focuses on the results obtained using a high frequency camera, the force cell and photograph of the central trapdoor. The tension in the tendon decrease over the course of the experiment. It shows that stress transfer is incomplete. Due to the loss of tension, the number of wires cut until the tension rupture will be greater than the number that would have to be done for the tendon for to reach its tensile limit, assuming that the tension in intact wires is uniform (Table 1). The tensile limit of the steel is 1.96GPa according to the certificate given by the strand manufacturer. Most of the stress that is in a wire is transferred when this wire is cut as tension loss is only of a few kiloNewtons while each wire hold initially a tension on the order of 25-30kN. However, in cases where the grout is severely damaged (see Figure 4b) or when an entire strand is broken (Square in Figure 6a), the tension loss can be much higher, exceeding the value of 30kN.

II. EXPERIMENTS

III. RESULTS

TABLE 1. List of experiments

A. Sensibility analysis on the quality of stress transfer according to the confinement and schedule of cutting When a wire is cut, some of the tension it was carrying is lost and some is transferred to other wires in the tendon. Furthermore, the transfer in the sheath and grout is not taken into account because they were removed to perform the cutting of the threads. However, this is unlikely to introduce a significant bias because the tensile strength of the grout and the stiffness of the sheath are very low compared to those of the steel. The stress transfer efficiency then appears to be directly correlated to the bond between the wires and the grout. When the confinement of the steel decreases due to excessive damage to the grout, the stress transfer decreases to almost zero (Figure 6a). Sheath failure, which occurs with the brittle PVC sheath in the first two experiments, causes the grout to rupture and the strand and grout to split over a distance of several meters (Figure 4b). In general, grout degradation due to multiple ruptures tends to make stress transfer less and less efficient as damage increases in the tendon (Figure 6a and6b). Monostrand The experiment display two different behaviours for monostrands depending of the sheath used. On the one hand, when a PVC sheath is used for the first two strands, the wire breaks cause the sheath to rupture followed by large cracks in the grout. There is then a loss of bond between the grout and the tendon over a length that increases with each cut. When the tendon is debonded from the grout, which occurs at the 4th cut for monostrand 1 and at the 2nd cut for monostrand 2, each cut causes the tension in the tendon to drop by a value only slightly less than the initial tension of the cut wire, showing that the tension transfer during wire cutting is minimal, with most of the tension lost (Figure 6a).

On the other hand, when a HDPE sheath is used, only small cracks are observed on the grout which reflects the fact that the bond between the grout and the strand is not affected. Tension loss is only a small fraction of the cut wire tension. Most of the tension from the cut wire is distributed to the other wires. As the grout and sheath continuity has been cut in order to free access to the strand, the cut wire tension is transferred to the others wires.

The relationship between bond strength and confinement is coherent with the slip pip test made by [START_REF] Hyett | The effect of rock mass confinement on the bond strength of fully grouted cable bolts[END_REF] In the first one, wires were cut in the same strand until the strand break before cutting wires in another strand. This was called the sequential cutting (Figure 7a). The second strand broke on the sixth cut on this strand (in total the thirteenth), with the entire tendon breaking on the second cut in the third strand. In the second, one wire was cut on 3 different strands. The original plan was to cut 3 wires on each strand, then 2 and finally 1. But accessibility constraints made it impossible to access some wires and an adjustment had to be made as a last resort (Figure 7b). There is little tension loss at each cut until the very end albeit tension loss of individual cut tends to get bigger as the damage of the tendon increase and when a strand finally breaks. There is strong evidence suggesting that stress of the cut wire is not only redistributed in the strand which it belongs but also in the others strands. Sequential cutting see failure of strand after 6 and 7 cuts. Failure would have occurred earlier if stress was only redistributed in the strand being cut. But the redistribution of the tension is not homogeneous in the strands. The high-frequency film of the rupture of the tendon subjected to alternative cutting shows that the outer 4 strands fail before the central one, even though only 3 of the outer strands were cut. Which lead to the four following categories 1. Monostrands with a weak confinement. The weak confinement lead to the failure of the bond. Most of the stress in the cut wires is then lost rather than transferred. 6 to 7 cuts are needed to break the strand as tension decreases at each cut by a amount close to the tension in the cut wire. In the experiment monostrand 1 and 2.

TABLE 2. Impact of grout bond and geometry of damage

2. Monostrand with a strong confinement. The holding of the bond leads to a good stress transfer with very little tension loss. The number of cuts that was needed to break the tendon is 4 instead of 3 if no stress were lost. In the experiment monostrand 3.

3. Multiple strand when strands are cut wire by wire sequentially. There is little loss of tension. Strand need the cutting of 6 to 7 wires to break which show that an important share of the stress is transferred in the others strands.

4. Multiple strand when wire breaks are distributed over several strands while not allowing strands to break. There is little loss of tension. The strand can hold with only two wires remaining, showing that the tension of the cut wires is transferred into the other strands. The kinematics of the break with the outer wire breaking before the center wire shows that the tension distribution is not homogeneous in the strands with the outer strands, even intact, more impacted.

The experiment from [START_REF] Abdullah | Experimental and Numerical Evaluation of Unbonded Posttensioning Tendons Subjected to Wire Breaks[END_REF] shows that when a strand is unbounded tension fall after a cut by the value of tension that was in a wire. When the strand is bounded by other strand, there is a small transfer, close to the result of the first category when bond is weak but with no degradation of bond. Tension loss were probably bigger in our experiments than on the field due to the need of cutting a windows to reach the strands. The authors argue that depending of bond quality, the strand will go to perfect stress transfer to no stress transfer, the tension of the cut wire is lost if the strand is totally unbounded. The bond depends of grouting quality and sheath quality, as the breaking of the sheath will cause the grout to break regardless of its quality due to the bad traction resistance of grout. No tendon with multiple tendon and poor bond was tester but it would have probably lead to close to no stress redistribution and each individual strand would act follow the first category.

There is uncertainty about the exact value of stress transfered and tension loss at each wire cut that will translate into an uncertainty about the number of wire loss that will cause the tendon to break. This is because this transfer depends on stong dynamic effect, the cracking of the grout, the exact position of the wire and the relative position of every other wires, in a real tendon the dispersion of tension between the tendon. This will lead to a small discrepedency of result even trough the experiments done show the first two monostrand and the two tendon with 5T15 having close behaviour, eventhrough the 5T15 was submited to two very different cutting schedule (Figures 6a and6b).

B. Estimation of transfer length using sliding of the wire

The transfer length is the distance to the cut where the impact of the cut has vanished. The following section introduce a model which gives an estimation on the transfer length based on the measurement of the sliding of the wire that has occurred after the cut. This sliding is due to the relaxation of the strain of the cut wire.

x ε 0 ε 0 L 0 2L 0
Linear evolution of strain and minimal distance

x ε 0 ε 0 L 0 2.3L 0 4.6L 0
Exponential evolution of strain and minimal distance FIGURE 8. Strain in the cut wire according to the distance to the cut Let g the sliding of a cut wire i. e. the distance between the two cut end and x c the distance to the cut end. The strain is assumed to be symmetric according to the cut thus sliding on each side is g 2 . The transfer length will be defined as the distance were the strain in the wire would stop its variation. This model is purely cinematic. Strain variation is assumed monotonous. Different expressions can be selected for strain.

The strain in each helical wire is cos 2 αε [START_REF] Foti | Modeling the axial-torsional response of metallic strands accounting for the deformability of the internal contact surfaces: Derivation of the symmetric stiffness matrix[END_REF]) with ε being the deformation along the axis of the strand ∆L L . A length l on the strand translate into a length l/ cos 2 α on the helical wire due to the curvature of the wire, with α being the lay angle of the helyx which is 7.8°for the experiment.

The model does not take into consideration the rotation of the strand. Assuming symmetry of the strain and L h being the length from the cut to the anchoring, the sliding would be

g 2 = L h 0 δε(x c )dx c = L h 0 ε 0 -ε(x c )dx c (1)
with δε(x c ) being the variation of the strain, projected on the axis of the strand, due to the cuting. On the x c axis 0 being the position of the cut end and L h the axial distance between the cut on the position of the anchorage. Let ε 0 the generalised strain in the strand far from the cut where the strain has converged. It is also the original strain of the strand computed with the ratio of displacement of the strand to length of the strand. It is assume that ε 0 is constant which is valid only when tension loss is very small compared to the tension of the tendon and if wire tension is homogeneous far from the cut. Both assumption would become problematic as the number of cut grow. The smallest transfer length that account for the sliding is L 0 = g 2ε0 (dashed curve Figure 8) assuming strain monotonous, non-negative and of value ε 0 far from the cut. It is achieve with

ε(x c ) = 0 if x c < L 0 ε 0 if x c ≥ L 0 (2)
With the assumption of a linear evolution of the strain from 0 to ε 0 the strain after the transfer length then the transfer length is 2L 0 (left solid curve Figure 8)

ε(x c ) = xc 2L0 if x c < 2L 0 ε 0 if x c ≥ 2L 0 (3)
If exponential evolution of the strain is assumed. Which would be the result of a very simple model assuming exponential friction between the cut wire and central wire. Which is achieved by introducing a Coulomb friction law in the model where friction force between the wires depends of contact force in the wires which is dominated by the tension. Then the evolution of the strain with the distance to the cut is (right solid curve Figure 8).

ε(x c ) = ε 0 (1 -e -xc L 0 ) (4)
With exponential evolution of the strain, the wire get backs to 90 % of its strain at a distance to the cut of 2.3L 0 and 99% at 4.6L 0 . The table 3 shows the relation between sliding and transfer length with a generalised strain assumed to be 7mm/m, the strain created in the strand by having a tension of 70% of the tensile strength.

Let ε i (x)
is the longitudinal strain of a wire, i from 1 to the number of wire N wire and ε g (x) for the grout (Figure 9), where x is the position on the tendon. The contribution of the sheath to the tension is insignificant due to very low Young modulus. L is the total length of the tendon in the reference configuration before jacking with x such as 0 ≤ x ≤ L represent the whole length. The extension of a tendon is given by (Figure 10)

∆L = L 0 ε(x)dx (5)
This extension is initially, before any cutting occur ∆L 0 . Because the two ends of the tendon are fixed, the sliding length of each cut wire is calculated by

∆L 0 =g i + L 0 ε i (x)dx (6) g i =∆L 0 - L 0 ε i (x)dx (7)
The loss of tension can be written as

i E i S i g i = ∆L 0 i E i S i - L 0 i E i S i ε i (x)dx (8)
where S i is the section of the wire and ∆L 0 the initial elongation of the strands. The tension T can be written depending of the strain

T = i E i S i ε i (x) + E g S g ε(x) g
with the initial tension T 0 being

T 0 = ∆L 0 L i E i S i Thus, i E i S i g i = L(T 0 -T ) = L∆T (9)
The aforementioned equations can be written as follow

∆T = E g S g g g + E s i S i g i L (10)
where E g , E s are the Young's modulus of the grout and the steel, S g , S i and g g , g i are the section and the sliding length of the grout and the cut wire i.

At each cut, a new wire slide and sliding of the already cut wires increase. This phenomenon account for the quasi-quadratic evolution of the total tension loss (∆T n ) for the first few cut (Figure 11). There is no estimation for the 5T15 with alternate cutting schedule as the angle of the camera during this experiment makes it very hard to get an estimation of the sliding with the chosen precision. Figure 11 shows the comparison between measured tension loss ∆T n and estimated tension loss due to sliding according to the number of cut n. The sliding of the grout was not used for the estimation as the grout broke and felt into the bench during the experiment. It leads to an underestimation of the tension loss, which makes it conservative. The error bar is done assuming sliding is known ± 1cm. This uncertainty is due to the measurement of sliding done using photo of the tendon and having to account to angle of view effect and inaccuracy due to end of some wires being out of the field of view, hidden by the strand or the sheath. The sliding has to be extrapolated using the wires where the two end are visible. Only sliding of the wire is considered as sliding of the grout can't be measured on the two monostrand due to grout breaking and falling into the bench. The loss of tension due to rotation is not considered in the model. As rotation is not permitted during tension, an important torque is created in the strand (see [START_REF] Ghoreishi | Validity and limitations of linear analytical models for steel wire strands under axial loading, using a 3D FE model[END_REF] for models linking elongation to torque). As damage of the strand cause relaxation of the torque, a twist is created on both side of the strand.

This twist will cause the tension in the strand to further decrease. The transfer of tension to the grout and the loss of tension due to the freeing of strand rotation are supposed to be responsible for the difference between the model and loss of tension measured by the strength cell.

IV. CONCLUSION

An experimental campaign has been done on multiple prestressing tendon in order to enhanced the understanding of damage mechanism. With the results of campaign impact of grouting, multiplicity of strands and wire breaking influence on the behaviour of a damage tendon has been explored. Tension has been shown to conserve with damage because of the bond between the strand and the grout. When this bond fail, it cause most of the tension supported by a wire to be lost when it eventually breaks.

A model was developed using cinematic of the wire in order to get an estimation of the transfer length depending on assumption on the strain and the displacement of the cut wires on the damaged tendon. A close formula linking sliding of the wire, assumption on the cinematic form of the strain and transfer length has been derived. This model may prove useful for assessing the reach of damage on industrial tendon.

Another model was created to get a lower bound of the tension loss which translate into an upper bound of the tension. It thus gives another tool for assessing the remaining durability of damaged tendons.

The models presented still need more testing to be fully validated. Only part of the tension loss is accounted for in the second model, a way to account for the rotation of the strands in the tension loss expression should be derived.

FIGURE 1 .

 1 FIGURE 1. Force and moment generalised on the strand and on a helical wire extracted from Foti & Martinelli (2019)

FIGURE 2a .

 2a FIGURE 2a. Strain distribution in a strand with a broken wire extracted from MacDougall & Bartlett (2003) FIGURE 2b. Numerical model of a strand with wires breaks extracted from Abdullah et al. (2016)

  FIGURE 3. Sketch of the experimental set-up for a tendon with one strand

FIGURE

  FIGURE 5.a Second cut of the third mono strand

  FIGURE 6a. in the monostrand according to the number of wire cut

  FIGURE 7a. Sequential cutting schedule

FIGURE 9 .

 9 FIGURE 9. Up photos of damage on opened tendon on a bridge © IFFSTAR for Sanef. Down photos of broken wire during the experiments

FIGURE

  FIGURE 11.a Monostrand 1

Strand Sheath Initial tension Theorical/Experimental wire loss needed for the rupture

					Others
	1T15	PVC	186 kN	3/6 (43%/86%)	Brittle sheath
	1T15	PVC	204 kN	3/7 (43%/100%)	Brittle sheath
	1T15	HDPE	204 kN	3/4 (43%/57%)
	5T15	HDPE	1018 kN	10/16 (29%/46%)	Sequential cutting
	5T15	HDPE	988 kN	12/15 (34%/43%)	Alternate cutting

TABLE 3 . Sliding and transfer length for

 3 ε 0 = 7mm/m

	Damage	Typical sliding			Transfer length
			Minimal Linear Exp (90% strain) Exp (99% strain)
	1-2 wire cut	1cm	1.4m	2.8m	3.2m	6.4m
	4-5 wires cut	5cm	7m	14m	16m	32.2m
	1 strand cut	10cm	14m	28m	32.2m	64.4m

Acknowledgement

This experiment has been funded by the ASFA.