G Artola

A Monzón

J Lacaze

J Sertucha

Tensile properties and fracture toughness at service temperatures of an optimized pearlitic ductile iron alloy for automotive crankshafts

Keywords: Ductile iron, fully pearlitic structure, mechanical properties, fracture toughness, automotive castings, crankshafts

Cast iron and forged steel have competed against each other for many years for different industrial applications. Crankshafts are one of the products where this competition is still active, both for heavy duty and light machinery applications. For the specific case of automotive crankshafts, high strength pearlitic cast iron grades have found a strong position in passenger car market niches. The mechanical performance data available for these specific ductile cast iron grades mostly come from room temperature tensile and impact testing. Nevertheless, the failure mode of cast iron crankshafts involves fracture mechanics, and their usual range of working temperatures goes from 20ºC for cold starts to 120ºC for top maximum lubricating oil temperature. This work covers this range by studying a high strength pearlitic cast iron employed to manufacture crankshafts both in terms of tensile, impact and fracture mechanics testing at 20ºC, 22ºC and 120ºC. A specific molding pattern was designed to produce the desired materials, namely crankshafts and Y2 and Y3 wedges, which were cast in an industrial crankshaft manufacturing line. The results indicate that there is a relevant difference between the low temperature and the high temperature mechanical response of the studied cast iron. This finding is interpreted in terms of safety, as the crankshafts should not fail in service after the engine has started running and the lubrication has reached its steady state working temperature.

INTRODUCTION AND BACKGROUND

The competition between cast irons and steels in the manufacture of certain metallic parts is a subject of industrial interest since the 1970s [START_REF] Kudryavtsev | Comparison between the strength of steel and cast iron crankshafts in large-scale routine fatigue bending tests[END_REF]. There is a specific structural difference between both materials which is related to the formation and growth of graphite particles in the metallic matrix of cast irons due to their comparatively high carbon and silicon contents. Spheroidal graphite particles in ductile cast irons strongly affect both solidification process and the austenite transformation that determine the final structure observed at room temperature [START_REF] Sertucha | Prediction of solid state structure based on eutectic and eutectoid transformation parameters in spheroidal graphite irons[END_REF]. The relevant development of these cast alloys is such that the required structures can be commonly obtained without any heat treatment by controlling the content of some alloying elements. As the main mode of failure of high-security parts for both cast iron and steel [START_REF] Fonte | Failure mode analysis of two diesel engine crankshafts[END_REF][START_REF] Moore | Crankshaft failure and why it may happen again[END_REF][START_REF] Asi | Failure analysis of a crankshaft from ductile cast iron[END_REF][START_REF] Hou | Fracture failure analysis of a ductile cast iron crankshaft in a vehicle engine[END_REF] is fatigue fracture, the combination of their cost and resistance to mechanical damage becomes the factor of competitive advantage. In this context, both cast iron grades and steel grades have evolved in an attempt to broaden their field of application, though remaining niche markets with differentiated technical-economic requirements.

From an economic point of view, there is little information available to clarify the conditions under which cast iron gains an advantage over forged steel, being the work of Nallicheri et al. [START_REF] Nallicheri | Material alternatives for the automotive crankshaft; a competitive assessment based on manufacturing economics[END_REF] in 1990s a very used reference [START_REF] Faura | Technical-economic models for materials systems analysis and their application to automotive parts fabrication[END_REF][START_REF] Zoroufi | A literature review on durability evaluation of crankshafts including comparisons of competing manufacturing processes and costa analysis, 26th Forging Industry Technical Conference[END_REF][START_REF] Citti | Current challenges in material choice for high-performance engine crankshaft[END_REF]. Fortunately, this work is still illustrative as it reflects general trends. On the one hand, these authors present a scenario in which ductile cast iron is more costcompetitive than forged steel for producing as-cast crankshafts, except for high production volumes (Figure 1a) mainly due to tooling and machining related costs. However, it is also reported that cast iron parts become the best solution in terms of cost of finished components (including all finishing operations) regardless of the volume of production (Figure 1b). Figure 1. Cost comparison from the 90's on (a) as-cast and (b) finished crankshafts adapted from Nallicheri et al. [START_REF] Nallicheri | Material alternatives for the automotive crankshaft; a competitive assessment based on manufacturing economics[END_REF]. Data from a generic quenching and tempering steel (Q&T) and a micro-alloyed steel have been also included as forged for comparison. ADI stands for Austempered Ductile Iron.

Considering the cost differences between a part produced using a cast iron alloy and by forging a conventional quenched and tempered steel, it does not seem to make economic sense to employ forged crankshafts. The use of forged parts is justified, not by economic aspects, but by the advantages they offer in terms of assurance of internal soundness and alignment of the grain fibers with the main directions of the loads applied to the component in service [START_REF] Zoroufi | A literature review on durability evaluation of crankshafts including comparisons of competing manufacturing processes and costa analysis, 26th Forging Industry Technical Conference[END_REF]. In general terms, cast iron parts cover the market governed by the performance/cost trade-off, while forged parts monopolize the niches in which the toughness and mechanical resistance prevail over the price.

Today, cast iron technology has evolved to more robust manufacturing processes, which widely ensure the internal soundness of the parts, and to the use of controlled cooling conditions or even specific heat treatments which allow achieving mechanical strengths capable of penetrating market segments of forged steel parts. On the other side, forging has adopted the use of micro-alloyed steels which allow the removal of the subsequent heat treatments and improve machinability. This has increased the cost competitiveness of the forged parts thus manufactured. The fact that Austempered Ductile Iron (ADI), which is a special type of heat-treated cast iron, and microalloyed steels are present in cost studies like the one presented in Figure 1 shows that these improvement strategies were already considered in early developments, and they remain valid today [START_REF] Citti | Current challenges in material choice for high-performance engine crankshaft[END_REF][START_REF] Khameneh | Evaluation of high-cycle bending fatigue testing and fracture behaviors in EN-GJS-2 ductile cast iron of crankshafts[END_REF].

In this scenario, high strength pearlitic ductile cast irons have become very attractive in terms of performance/cost ratio, due to the excellent combination of price and mechanical properties they offer. As a result of previous investigations [START_REF] Lacaze | Effect of Cu, Mn and Sn on pearlite growth kinetics in as-cast ductile irons[END_REF][START_REF] Serrallach | Effect of selected alloying elements on mechanical properties of pearlitic nodular cast irons[END_REF] (and references therein), a group of optimized fully pearlitic ductile iron alloys was developed in terms of chemical composition and cooling rate with ultimate tensile strength and yield strength values higher than 900 MPa and 550-600 MPa, respectively, while keeping relevant elongation at rupture (2-6%). Although, these results evidenced such materials as a cheap alternative to forged parts, these studies were carried out only in terms of tensile properties at room temperature that are not enough when it comes to exploring the application potential of these improved pearlitic ductile irons. Accordingly, this work investigates the mechanical behavior of automotive crankshafts produced with optimized material, both in terms of plastic ductile failure and of fracture mechanics, as required for high strength alloys, and also explores the effect of the temperature in extreme operating conditions, namely 20°C and 120°C.

EXPERIMENTAL WORK

The present study involves three different cast parts, namely, two standard wedges designed as Y2 and Y3 (according to the EN-1563:2012 standard) and one automotive crankshaft 12.0 kg in weight (Figure 2). This last cast part will be the reference casting in this work as it is regularly produced with pearlitic ductile iron alloys in a foundry plant which supplies crankshafts to several automotive constructors. It has been established that the most relevant areas of this crankshaft are the two diagonals along two opposite rod bearing journals as indicated in Figure 2 with dashed rectangles. These areas are considered of high interest due to their critical role during the life span of combustion engines and they were used for obtaining the tensile and impact specimens. The smallest wedge (Y2) was selected as it is widely used as standard casting for mechanical testing and this allows the comparison of the results obtained with other previous and future results. The heaviest wedge (Y3) was used to carry out the fracture toughness study which, in general, requires using a specimen thickness as close as possible to the thickness of the component of interest. In this case the selected section of the crankshaft has a thickness of 50 mm and it corresponds to the diameter of all rod and main bearing journals (Figure 2). Consequently, bending specimens of thickness B = 40 mm were all machined out from the Y3 wedges (see subsequent description) since it is the thickness closest to 50 mm for which test tools were available. The geometric characteristics of the two different wedges are shown in Figure 3. All the cast parts used in this work were manufactured and then used to obtain the corresponding tensile and toughness fracture samples without any heat treatment. All cast parts were produced with a single pattern plate which included two crankshafts, four Y2 wedges and one Y3 wedge according to the distribution shown in Figure 4 which corresponds to the drag plate. The filling system was composed of a down sprue, two filter chambers and runners with different sections. Only the two crankshaft cavities had feeders to avoid the formation of shrinkage porosities in these parts. The testing area indicated in Figure 3 was located at the bottom of the molds to ensure that contraction porosities were not present in this area of the Y2 and Y3 wedges. The base melt batch used in the present investigation was prepared in a medium frequency induction furnace (250 Hz, 12000 kW) with 16 t in capacity. The metallic charges were composed of about 45% steel scrap and 55% ductile iron returns. After completing the melting process, the chemical composition of the batch was checked and then adjusted adding the corresponding ferroalloys. The carbon and silicon contents were corrected, respectively, with a high purity graphite (>98.9 wt.%) and a FeSi75 alloy (grain size 10-90 mm, Si = 75.2, Ca = 0.26, Al = 0.86, C = 0.13 and Fe balance, wt.%). The melt temperature was then increased to 1510-1520ºC and the slag was removed. Afterwards, the prepared melt was transferred in two batches to a 3300 kg capacity ladle which contained 26-33 kg (0.8-1.0 wt.% of the batch weight) of a FeSiMg alloy (grain size 2-10 mm, Si = 45.9, Mg = 5.7, Ca = 2.42, Al = 0.39, rare earth elements = 2.24 and Fe balance, wt.%) to carry out the spheroidizing treatment by the tundish-cover method (1450-1470ºC). This FeSiMg alloy was positioned into a chamber located at the bottom of the ladle and then covered, first, with 5 kg of steel scrap and then, with 3.3 kg of a commercial inoculant (grain size 2-6 mm, Si = 75.1, Al = 1.50 and Ca = 0.96, Fe balance, wt.%) and finally with further steel scrap (grain size 15-30 mm). Once the spheroidizing treatment was finished, the two prepared batches were consecutively transferred to a 10 t capacity pressurized pouring unit from which the molds were poured at 1360-1380ºC. During pouring, post-inoculation was performed by adding 0.15-0.18 wt.% of a commercial inoculant (grain size 0.2-0.6 mm, Si = 64.1, Al = 0.85, Ca = 1.00, Mn = 3.30, Zr = 3.49 and Fe balance, wt.%) in the melt stream. A total number of 38 molds were poured to give 152 Y2 wedges, 38 Y3 wedges and 76 crankshafts.

A metal sample (medal) was obtained per batch when pouring the molds to determine the chemical composition of the melt. A combustion technique (LECO CS300) was then used for determining carbon and sulfur contents on the sample while spectrometry (ARL 3460) was utilized for other elements. Chemical compositions for all significant elements are shown in Table 1 which does not include the contribution of post-inoculation. reaching room temperature and shot blasted to clean all their surfaces. Such time range of cooling into the molds was planned to reach about 600-650ºC in the central area of the Y3 wedges when demolding started while the thinnest zones of the crankshafts (counterweights and pulley ends) were at about 300-400ºC according to simulation results.

The cast parts from which each specimen was machined out were chosen at random from both production batches, so that any effect related to production order was randomly distributed. Table 2 shows a summary of the different tests performed in the present work, the relevant cast parts, the test temperatures, and the number of specimens used and of repeated measurements in each case. Standard tensile strength specimens per ISO 6892-1 with 10 mm in diameter (cylindrical section) and calibrated length of 50 mm were machined out from the cast parts. On these specimens, the tensile strength UTS, the elongation at rupture A and the yield strength YS values were measured using a Zwick Z250 tensile testing equipment at a controlled actuator speed of 0.90 mm/min in the range where YS was determined. According to the standard ISO 6892-1, this rate was then increased to 24.12 mm/min to determine UTS and A. All tensile specimens were produced with threaded ends and elongation values were determined by using a Zwick integrated transverse extensometer with 0.10 µm in resolution. For the impact strength determination (Kv), the dimensions of the specimens were 101055 mm with an ISO V notch as per ISO 148-1 standard and they were tested in a Zwick 5111 apparatus. Finally, proportional bending specimens as per ASTM E399 standard with 40 mm in thickness and 80 mm in height (Figure 5) were prepared for the fracture toughness. The notches with a base radius of 0.25 mm were machined by wire electric discharge machining in the side facing the thickest section of the Y3 wedge. At the bottom of each notch, a crack was generated in a controlled way that meets the parameters indicated in the standard ASTM E399 to ensure that the value of the shape factor "f" is known and that there is no plasticization at the crack tip. Note here again that the thickness of the toughness specimens was similar to the diameter of the rod and main bearing journals of the crankshafts (50 mm). An Instron 300DX apparatus with an environmental chamber MTS 651.06E-03 was used to perform the fracture toughness tests. This chamber was equipped with a forced air system to ensure the homogeneity of the temperature inside.

Figure 5. Characteristics of the toughness specimens used in this work [START_REF] Astm | ASTM E399-17 Standard method for linear-elastic plane-strain fracture toughness K I c of metallic materials[END_REF]. Roughness requirements of the flat surfaces have been indicated in µm (Ra).

The most widespread approach to the mechanical design of components is based on the calculation of the conditions under which plastic collapse occurs, employing YS and UTS as key material properties for calculations. However, this design strategy is no longer valid when working with very high strength materials, relatively thick thicknesses and in the presence of cracks. In these circumstances, the risk for the component to break in a brittle way increases, which introduces the need for fracture mechanics. Toughness expressed in terms of the critical stress intensity factor, K IC , becomes the key material property. The K IC value is calculated using the shape factor "f "and the depth of crack "a" (see Figure 5) according to equation 1 [START_REF] Astm | ASTM E399-17 Standard method for linear-elastic plane-strain fracture toughness K I c of metallic materials[END_REF].

(

) 1
where K IC is the toughness of the material (MPa•m 1/2 ), F Q is the force at the point of deviation of the linear elastic behavior (N), S is the distance between the points of support of the specimen in the three point bending setup (m), "a" is the depth of the crack in the material that includes the length of the machined notch plus a sharp crack grown in a controlled manner by mechanical fatigue (m), W and B are the width and the thickness of the specimen (m), respectively, and is the shape factor to consider the modification of the stress state at the tip of the crack (dimensionless). This shape factor is given by [START_REF] Astm | ASTM E399-17 Standard method for linear-elastic plane-strain fracture toughness K I c of metallic materials[END_REF]:

As detailed in Table 2, three different temperatures were selected to investigate the effect of this parameter on mechanical results. In addition to room temperature, used as reference, a group of specimens was also tested at 20ºC and another at 120ºC which are the temperatures that represent the engine start in cold climates and in overheated oil conditions, respectively. Thus, conditioning systems were used to maintain the specimens at the selected temperatures for a while before testing. Before impact testing, the samples were hold either 20 min in a cryostat at -20°C or 60 min in a furnace maintained at 120°C. The impact tests were performed in less than 3 s after extraction of the specimen from the temperature conditioning system. For the tensile and fracture toughness tests at -20°C and 120°C, the environmental chamber was kept running for 60 min before and during all the tests (see Figure 6). Scanning electron microscopy (SEM) characterizations were carried out on the fracture surface of representative toughness specimens using a FEI Quanta 200 microscope. Microstructure characterizations of the cast alloys were made in two steps: Firstly, samples were obtained from the bottom area of 6 randomly chosen Y2 and Y3 wedges. The analyses confirmed that all these wedges were fully pearlitic ductile iron alloys without carbides and this validated the rest of produced cast parts. In a second step, the specimens obtained from both the Y2 and Y3 wedges and from the crankshafts which were used in the tensile tests at room temperature were cut to obtain metallographic sections at 5-10 mm from the fractures. Quantitative image analyses were made on each sample to determine the area fraction of the graphite particles with an area equal or higher than 25 µm 2 and to assign each particle to classes III (irregular particles), V (irregular nodules) or VI (well-formed nodules). The criteria used to classify these graphite particles into classes III, V and VI are given in Table 3. The two shape parameters circularity (CR) and aspect ratio (AR) were defined as CR = 4• and AR = where A is the area of a given graphite particle, P is its perimeter and L min and L max are, respectively, the minimum and maximum length of the graphite particle. The obtained data were then used to determine the area fractions of graphite particles assigned to each class, namely f III-A , f V-A and f VI-A . Nodule count (Nc) and nodularity (Nod) values were determined considering those graphite particles assigned to classes V and VI. Thus, Nc values were determined as the number of such particles included in 1 mm 2 while nodularity values were calculated as Nod = (f V-A + f VI-A )•100. Fractions of ferrite f α and pearlite f p were determined by comparing the microstructure obtained after etching with 5% Nital with standard reference microstructures. Three different metallographic fields per sample were examined to determine the corresponding average values of each structural feature.

Those specimens used for determination of the Brinell hardness (HBW) of the cast parts were nearly cubic samples with about 10 mm side which were obtained from the impact specimens tested at room temperature. In this case, the testing surfaces were located 15-17 mm from the fracture areas. After conditioning these surfaces, hardness measurements were performed using an Instron Wolpert apparatus with a 5 mm diameter sphere and a load of 750 kg.

Finally, a discussion is made in this section to justify the number of samples "n = 6" used in this study for a given condition, namely casting type and working temperature. This "n" value should be high enough to optimize both the certainty of the statistic mean and the experimental effort needed for obtaining such data. For this purpose, it has been considered that the optimal "n" value is the one in which the relative cost of increasing the sample size by one unit is higher than the relative improvement achieved in statistical reliability. The selection of the population size has been made taking as a reference value the so-called Confidence Interval of 95% (CI95), that is, the tolerance around the experimental average that is expected to contain the actual mean value of each property with a reliability of 95 %. Thus, the CI95 parameter can be calculated according to equation 3:

-

where "t 0.05,n-1 " is the Student's "t" for CI95, "n" is the number of samples (number of tests) indicated above and S n is the standard deviation of the dataset. From equation 3, the uncertainty improvement factor (MI n ) for using "n" instead of "n-1"samples is calculated as follows:

- -  - - (4)
The cost increment factor (IC n ) associated to increase by one the number of samples has been calculated by means of equation 5:

-

Microstructure analysis

Although a primary verification of some Y2 and Y3 wedges was made to check their fully pearlitic structures and to validate the produced castings, a detailed characterization of the microstructure in the tested specimens has been performed to correlate with mechanical properties of the castings. For this purpose, metallographic samples were obtained from the tensile specimens tested at 22ºC leading to the results shown in Table 4. In all cases, nodularity values are higher than 93% and microstructures are fully pearlitic with traces of ferrite in some samples obtained from the crankshafts and the Y3 wedges. From the change in Nc, it is seen that the Y2 wedges had the highest cooling rate during solidification, followed by the crankshafts and finally the Y3 wedges. Notice that one of the crankshafts was manufactured being located beside the Y3 wedge while the other one was in the external side of the mold (see Figure 4). Similarly, two of the four Y2 wedges were positioned near the Y3 one while the other two were in the external side of the mold. These characteristics affected the solidification rate and are considered as the main cause of the variability found in the Nc values for both Y2 wedges and crankshafts. As a matter of fact, variability of the Nc data of the Y3 wedges in Table 4 is comparatively low. Although all wedges and crankshafts could be considered as having fully pearlitic microstructures, traces of ferrite were occasionally found in the crankshafts (Figure 8a) and were frequently found in the Y3 wedges (Figure 8b), always around the graphite nodules. On the contrary, ferrite was not detected in the Y2 samples. These observations agree with the comparative solidification rates discussed above on the basis of the nodule count differences among casting types. Unexpectedly, analogous inspection on pearlite aggregates of the Y3 wedges showed that the interlamellar spacing is quite similar to the one observed in the Y2 castings (see Figures 8b and9a) being finer in both wedge types than in the crankshafts (Figure 9b). This result was confirmed in all 6 samples obtained from the Y3 wedges despite these castings have the heaviest size and the comparison of other structural feature like Nc indicates a low solidification rate. Thus, a possible explanation for this unexpected observation is the time elapsed between pouring the molds and the demolding process. It seems evident that solidification rate was the lowest in case of the Y3 wedges, however a rapid cooling just before the eutectoid transition of these castings provoked by an early demolding step would explain the structural finding. The traces of ferrite that have been mentioned must have appeared slightly before the shake out, at a temperature higher than 650°C. Thus, a schematic of the cooling rates expected on the three casting types according to the Nc and pearlite interlamellar spacing results is given in Figure 10.

Figure 10. Comparative cooling rates of the three castings used in this work according to the microstructure results.

Brinell hardness

Table 5 lists the results obtained from all hardness tests, as well as the average value and the standard deviation (S n = 6 ) for each cast part. The hardness values show some heterogeneity for a given cast part, and it is seen that the mean values obtained from the Y3 and Y2 wedges are similar and are significantly higher than those obtained for the crankshafts.

Compared to Y3 wedges, hardness variability among the Y2 wedge samples or the crankshafts is due to the effect of different cooling rates of the different cavities present in the pattern plate as described above (Figure 4). Nevertheless, the changes in hardness observed in Table 5 when comparing the three castings can be explained in terms of pearlite interlamellar spacing as described in section 3.1. Thus, the Y3 wedges samples led to high hardness values even with traces of ferrite as the pearlite aggregates present in this casting showed comparatively a quite low interlamellar spacing. On the other hand, high hardness values were also found in the Y2 samples where ferrite was not detected and the interlamellar spacing was small. Notice that values from the Y2 samples in Table 5 are slightly lower than those from the Y3 ones which denotes that pearlite aggregates could contain the lowest interlamellar spacing in these heaviest castings. This is supported by the already mentioned effect of an early demolding on the eutectoid reaction in the Y3 wedges. Finally, the samples obtained from the crankshafts showed the lowest hardness values because of the higher interlamellar spacing of pearlite. Data from Tables 1, 4 and 5 were used here to quantitatively estimate the average cooling rate values into the range 1050-715ºC (V r ) for the sampling areas of the castings. This parameter represents the cooling rate just before austenite transforms to pearlite by the eutectoid reaction and it has been evaluated by using the relationship reported by Serrallach et al. [START_REF] Serrallach | Effect of selected alloying elements on mechanical properties of pearlitic nodular cast irons[END_REF]. These authors obtained equations by statistical analyses relating composition data, cooling rates, hardness and tensile data from 85 pearlitic ductile cast irons. In the present case the reverse calculation has been made and the obtained Vr values have been used to check the cooling trends suggested in Figure 10. Thus the calculated cooling rate values are: 10.3 K•min -1 , 32.0 K•min -1 and 34.8 K•min -1 for the crankshafts, the Y2 wedges and the Y3 wedges, respectively. In the following section, these results will be discussed together with the Vr values calculated with the tensile tests findings and compared to the expectations shown in Figure 10.

Tensile tests

Regarding the tensile properties of the alloys, Table 6 shows the average values and the standard deviations of UTS, YS and A at the three working temperatures considered in the present study. The UTS and YS data have been also plotted in Figure 11 to facilitate the interpretation of the results. As expected, an increase of the testing temperature reduces both UTS and YS, while an increment of A is observed since the material increases its ductility (Table 6). Notice that UTS and YS always remain above 900 MPa and 550 MPa thresholds, respectively, for the two wedges while these parameters are reduced by about 100 MPa in the case of the crankshafts. Comparatively, it is also remarkable that the tensile strength shows a lower decrease (≈4%) than the yield strength (≈12%) between 20ºC and 120ºC. This result is observed for all three casting types and it indicates that the material maintains its overall strength during the tensile test though it begins to flow at slightly lower loads as the working temperature increases.

The use of a graphical representation of the results is also of help when comparing the strength results obtained from the three types of castings (Figure 11). The specimens from the Y2 and Y3 wedges showed the highest UTS and YS values which will be correlated later with trends previously described for microstructure and hardness. The fact that Y2 wedges exhibit UTS and YS results with an evolution which parallels that of the crankshafts implies that the Y2 wedges can be used as a valuable reference for quality control in foundry plants. Notice that the average UTS and YS values obtained at 22ºC from the crankshafts are about 870 MPa and 530 MPa, respectively. It should be considered that these values were obtained from the diagonal areas highlighted in Figure 2 which likely underwent the lowest cooling rate among the different areas of this casting. In order to further explore the effect of the cooling rate on the tensile properties of the crankshafts, four additional tensile specimens 5 mm in diameter were machined out from the counterweights where cooling rates are expected to be the fastest for this casting. The results obtained from these tests performed at 22ºC were UTS = 956-982 MPa, YS = 593-622 MPa and A = 4.0-4.9% which are similar to the average values reported in Table 6 for the Y2 and Y3 wedges. The microstructures found in the counterweights were fully pearlitic with interlamellar spacing smaller than that observed in the diagonal areas. As it has been already observed in the microstructural and hardness characterizations, the UTS and YS values in Table 6 also suggested that the tested areas of the crankshafts experienced the lowest cooling rate in the eutectoid zone when comparing to the wedges. This fact appears related to the highest interlamellar spacing of pearlite aggregates in the specimens obtained from the crankshafts. However, the most remarkable result is the similar UTS and YS values obtained at the three temperatures tested for the Y2 and Y3 wedges (see Figure 11) despite their different size. This result indicates that the similar interlamellar spacing of pearlite aggregates present in these two castings strongly affected these two parameters rendering a minor role to the ferrite traces detected in the Y3 specimens.

Figure 12 shows the evolution of elongation at rupture with temperature for the three different casting types. The corresponding scatter bands, determined as the range [A]t 0.05,5 • , where [A] corresponds to the average elongation value, are also plotted in this figure and they represent 95% confidence interval of the average values. The elongation at rupture of all three casting types drops at 20ºC, the values obtained at 22°C and 120ºC being similar to each other in case of the specimens obtained from the Y2 wedges and the crankshaft. A relevant increase of elongation at the highest temperature is found in case of the Y3 wedges which could be due to the effect of ferrite traces surrounding some of the graphite nodules and/or to the comparatively low Nc values found in this heavy casting. As the small ferrite contents did not show any significant effect on UTS and YS in the Y3 wedges, it can be concluded that the effect of nodule count is predominant over the influence of ferrite contents.

Figure 12. Effect of temperature on the average values of elongation at rupture. The greyed areas represent the 95% confidence intervals (see text).

To visualize the effect of Nc on A, these two parameters have been plotted in Figure 13 for each testing temperature and casting type. No influence of Nc on elongation is observed at 22ºC while elongation decreases in those specimens with increased nodule count in a clear way at 120ºC. A similar trend is observed at 20ºC though it is weak and scattered. A values from the Y3 wedges are higher than those from the Y2 wedges at these two temperatures, suggesting that low Nc values are beneficial as the pearlite interlamellar spacing is similar in these two wedges. In case of the crankshafts, elongation values are scattered and slightly lower than those from the Y2 wedges despite they showed the highest interlamellar spacing. These observations suggest again that Nc has a predominant effect on A.

It is also worthy to note that the evolutions of UTS, YS and A shown in Figures 111213for a given casting geometry at 120ºC do not show the usual behavior of decreasing elongation with increasing UTS and YS. More precisely, the Y3 wedges exhibited the highest average UTS, YS and A values at 120ºC as pearlite interlamellar spacing and Nc are comparatively low. These findings suggest that the effect of interlamellar spacing is dominant on ultimate tensile and yield strengths while nodule count mainly affects elongation in these fully pearlitic cast irons. This effect seems to be mainly related to the resistant metallic sections (mean-free distance) in between nodules which are larger when decreasing nodule count.

Figure 13. Evolution of elongation at rupture with Nc.

Data included in Tables 1, 4 and 6 have been used to estimate the V r parameter (see section 3.2) in the areas from which tensile specimens were obtained according to the expressions reported by Serrallach et al. [START_REF] Serrallach | Effect of selected alloying elements on mechanical properties of pearlitic nodular cast irons[END_REF]. All the calculated values are compared in Table 7 including the data from hardness results (section 3.2). Although these data appear with some scattering, the highest Vr values are obtained for the Y3 and Y2 wedges while the lowest ones correspond to the crankshafts. These results agree with the expectations shown in Figure 10 which are based on the microstructure observations. It is observed in Table 8 that impact values show low variability with low standard deviation values regardless of the type of casting and of the testing temperature. The basic statistical descriptors in Table 8 show that the impact test does not present any change allowing to discriminate the effect of the variables used in the study. The produced material is too sensitive to the notch effect and it is not possible to detect differences in the absorbed energy as usually found in cast iron grades as stated by Wallin [15]. This makes the use of K IC specimens recommendable to show relevant differences between two testing or material conditions, as it was shown with ADI grades by Artola et al. [START_REF] Artola | Austempered ductile iron (ADI) alternative material for high-performance applications[END_REF]. Thus, more advanced experimental techniques, such as the ASTM E399 method, have been selected to characterize the toughness of the crankshafts.

Table 8. Average impact values (J•cm -2 ) and standard deviation at the three working temperatures. All the K IC toughness results obtained from the Y3 wedges are listed in Table 9 and their average values with the scatter band defined by the range [K IC ]t 0.05,5 • are plotted in Figure 14. The effect of the temperature on the material toughness can be clearly assessed, what was not feasible with impact test specimens. It is observed that there is a significant increase of the toughness with the temperature, which almost doubles from 20ºC to 120ºC. Tensile tests had also shown an increase in elongation with temperature raise. For the fracture toughness test to be valid, plain strain condition should be ensured that needs the following condition to be satisfied: [START_REF] Nallicheri | Material alternatives for the automotive crankshaft; a competitive assessment based on manufacturing economics[END_REF] With the values found, the choice of a thickness B = 40 mm for the toughness test specimens has been appropriate since the crack tip was under plane strain. This condition implies that the fracture mechanism has followed linear elastic fracture mechanics in the present case. On the contrary, if 20 mm thick specimens had been chosen, the plain strain condition imposed by equation 7 would not have been fulfilled for the tests at 120ºC, meaning that the result would have not been representative of the bearing journal of the crankshaft fracture mechanics [START_REF] Astm | ASTM E399-17 Standard method for linear-elastic plane-strain fracture toughness K I c of metallic materials[END_REF].

When translating the toughness test results to design decisions, it must be considered that in most cases the crankshaft is at room temperature when the engine starts, and the regular working temperature for steady state condition is 90ºC. The least-square linear regression fit to the data in Figure 13 shows a slope of 0.144. This means that the toughness increases by 9.1 MPa•m 1/2 from cold start, at 25ºC, to stable engine regime, at 90ºC. Taking the stress intensity factors for a forged steel crankshaft modeled in [START_REF] Metkar | Evaluation of FEM based fracture mechanics technique to estimate life of an automotive forged steel crankshaft of a single cylinder diesel engine[END_REF] as an example, 1.5 mm and 2.5 mm cracks cause stress intensity factors of 40 MPa•m 1/2 and 50 MPa•m 1/2 respectively. Thus, if a crankshaft after the design from [START_REF] Metkar | Evaluation of FEM based fracture mechanics technique to estimate life of an automotive forged steel crankshaft of a single cylinder diesel engine[END_REF] and manufactured in the material studied in this work stands a cold engine start while having a 1.5 mm crack, it should not break until the crack grows up to 2.5 mm. Hypothesizing an engine speed of 2000 rpm, a peak fatigue cycle each 4 turns (significant only during the expansion cycle) and a torque ratio of 0.1 from start to average running condition [START_REF] Maurya | Reciprocating Engine Combustion Diagnostics[END_REF], an estimated crack growth rate of 10 -3 mm/min [START_REF] Lee | Fracture toughness and crack growth rate of ferritic and pearlitic compacted graphite cast irons at 25ºC and 150ºC[END_REF][START_REF] Agostino | Fatigue crack propagation in ductile cast irons: an artificial neural networks-based model[END_REF] would lead to 16 h of driving until fracture, what turns out far longer than the time between refueling stops. Thus, crankshaft fracture risk during driving, after oil warmup is greatly reduced by the temperature effect on toughness. This advantage is further improved if the cold start temperature of the engine is below zero, for example at 10ºC. The linear regression obtained in Figure 14 gives a K IC value of 37 MPa•m 1/2 for this condition, which leads to a critical crack size of 1.3 mm considering the same crankshaft geometry as above [START_REF] Metkar | Evaluation of FEM based fracture mechanics technique to estimate life of an automotive forged steel crankshaft of a single cylinder diesel engine[END_REF]. If the same crack growth rate as before is assumed (10 -3 mm/min), the time to fracture would increase to 19 h of driving.

It is worth it to recall here that the unexpected behavior found on the Y3 wedges related to their comparatively high UTS, YS and hardness values is likely due to premature demolding. If Y3 wedges would have completely undergone the eutectoid transition inside the mold, it would have led to hardness, UTS and YS values lower than those reported in Tables 5 and6. Thus, the fracture toughness response of such an alloy would be expected to show K IC values higher than the expected ones for the Y3 wedges produced in the present work, with these latter castings representing the most critical test condition between these two cooling possibilities. As a consequence, longer safe driving times could be achieved.

Not considering the intimate fracture mechanisms involved in brittle fracture (cleavage) and ductile collapse (dislocation movement), the experimental data presented in this work have been used to obtain an equation for estimating K IC by using as input variables the testing temperature values and the tensile results from specimens produced with Y2 wedges commonly used in foundry plants.

Experimental determinations of K IC values at different temperatures are expensive compared to conventional tensile testing. Therefore, such an equation may be useful to assess the fracture toughness of fully pearlitic ductile irons with similar casting sizes (cooling rates) to the ones used in this study, avoiding time consuming and costly tests.

In a first step, the correlation coefficients between K IC , as the output variable, the three tensile parameters included in Table 6 (UTS, YS and A) in case of the Y2 wedges and the testing temperature (T) were determined by means of Pearson method. The coefficients thus obtained are shown in Table 10 where the sign of each of them shows the expected positive or negative correlation between each variable and K IC . where K IC , T, UTS, YS and A parameters are given in MPa•m 1/2 , ºC, MPa, MPa and %, respectively.

Figure 16 compares the experimental and calculated values of K IC at the three testing temperatures used in this work. As it has been seen in Figure 14, the highest scatter of data in Figure 16 is found at 20ºC with an absolute average error of 7.8%. At low temperature, it is assumed that the fracture toughness of the alloy is comparatively more sensitive to the presence of inclusions and/or small porosities (pinholes or isolated microshrinkage) than at room or high temperatures. This fact is probably the cause of the higher variability of the data at low temperature.

CONCLUSIONS

As stated in the results presented above, both thermal moduli and demolding times caused property variations between the Y2 wedges, the Y3 wedges and the bearing journals of the crankshafts from the same mold. These variations were reflected in microstructure (nodule count, pearlite interlamellar spacing and presence of residual ferrite) and in tensile properties. Nevertheless, the effect of temperature shows trends that are not geometry dependent for the tensile results, thus rendering valuable results on Y2 and Y3 wedges. For a given temperature, UTS, YS and hardness of the investigated pearlitic ductile iron appeared mainly affected by pearlite interlamellar spacing while nodule count shows a predominant effect on elongation at rupture.

The Y2 wedge results reported here can be used for performing comparisons with other past and future research works related or not to crankshafts, as Y2 wedge is a widespread cast iron sample manufacturing geometry. Nevertheless, the Y2 wedges could not be used for performing fracture toughness tests aimed at being representative of the bearing journal because the required plain strain condition would not be satisfied for a specimen machined out of them. This is especially critical in case of testing at high temperature.

This limitation has been overcome with Y3 wedges. The fracture toughness results obtained from these wedges represent better than yield strength or tensile strength the mechanical performance of the bearing journals because cleavage is reported as the failure mode for cast iron crankshafts. Furthermore, the use of yield strength and ultimate tensile strength for studying the mechanical response of the material underestimates the benefit of the heating up of lubricating oil to the working temperature on the performance of the material.

The interpretation of the toughness change with temperature that has been measured is valuable for the competition between casting and forging. As the toughness of the cast iron used to produce the crankshafts increases when the engine warms up and because the worst situation in terms of torque occurs when starting the vehicle, the fracture of a cast iron crankshaft should not happen once the vehicle is running if it did not happen right when the vehicle started. This leads to a relevant conclusion in terms of part safety in the competition between cast iron and forged steel crankshafts: although the pearlitic ductile cast iron of a crankshaft shows a brittle failure, this fact does not increase the risk of a failure once the vehicle started running.

Finally, an experimental fit has been proposed between the tensile test results from the Y2 wedges and the toughness measured on the Y3 wedges. This relation is expected to apply to any fully pearlitic ductile iron crankshafts whose manufacturing and microstructural conditions are similar to the ones employed in the study. This equation is intended to perform quick estimate of K IC for design purposes by using tensile test results on commonly used Y2 wedges.

Figure 2 .

 2 Figure 2. Crankshaft used in the present work. The critical areas have been marked with dashed rectangles and they have been used for obtaining the testing specimens from this casting type.

Figure 3 .

 3 Figure 3. Dimensions of the wedges used in this work (adapted from EN 1563:2012).

Figure 4 .

 4 Figure 4. Distribution of the cast parts in the drag pattern plate used to produce the molds.

Figure 6 .

 6 Figure 6. Testing rig employed to perform the fracture toughness tests.

Figure 7 .

 7 Figure 7. Evolution of MI n , IC n and the t 0.05,n-1 •n -1/2 quantity with the number of samples.

Figure 7

 7 Figure 7 has been built from equations 4 and 5 and it shows the change of MI n and IC n parameters and of the t 0.05,n-1 •n -1/2 quantity with the number "n" of samples. Despite uncertainty improvement and cost factor curves cross each other at 5.4 value and 5 samples would have been a good choice, when n = 6 the value becomes 1.05, what is very convenient to calculate CI95, since it allows using the standard deviation (S n = 6 ) directly as CI95 according to equation 6:

Figure 8 .

 8 Traces of ferrite present in (a) crankshafts and (b) Y3 wedges.Another relevant structural feature is the interlamellar spacing of pearlite aggregates, which was qualitatively evaluated in the 18 samples. Examples of representative pearlite lamellas in the samples obtained from the Y2 wedges and from the crankshafts are compared in Figure9. It is seen that the interlamellar spacing appears smaller in the former case than in the latter, supporting again the correlation of the cooling rate estimates based on the Nc values.

Figure 9 .

 9 Detailed view of the interlamellar spacing of pearlite in: (a) Y2 wedges and (b) crankshafts.

Figure 14 .

 14 Figure 14. Effect of temperature on the average K IC values for 40 mm thickness specimens extracted from Y3 wedges. The dashed line shows the least-square linear regression on the average values.

Figure 16 .

 16 Figure 16. Comparison of calculated and measured values of K IC .

Table 1 .

 1 Chemical composition of the melt batch used for producing all the cast parts (wt.%).

	Batch	C	Si	Mn	P	S	Cr
	1	3.61 2.26 0.44 0.022 0.006 0.042
	2	3.65 2.23 0.45 0.025 0.004 0.044
	Batch	Ni	Al	Cu	Sn	Ti	Mg
	1	0.021 0.009 0.84 0.024 0.024 0.036
	2	0.023 0.011 0.81 0.021 0.027 0.041
	60-70 min after pouring, all cast parts were removed from the molds, cooled down in open air until

Table 2 .

 2 Outline of hardness and mechanical tests and conditions used in this work.

	Test	Y2	No. of cast parts Y3 Crankshaft	Temperature (ºC)	No. of repeated measurements	No. of specimens
	HBW Tensile Impact (Charpy) Fracture toughness	6 18 6 ---	18 18	6 18 9 ---	22 20, 22 and 120	6 per cast part and temperature	18 54 54 18

Table 3 .

 3 Criteria used for classifying the graphite particles.

	Class	CR	AR
		≤0.70	≥2.0
	III	<0.50	1.5-2.0
		≤0.35	1.0-1.5
		>0.70	≥2.0
	V	≥0.50	1.5-2.0
		0.35-0.77 1.0-1.5
	VI	≥0.77	1.0-1.5

Table 4 .

 4 Microstructure data of the tensile specimens (Y2 and Y3 wedges and crankshafts).

	Casting	Sample Nod (%) Nc (mm -2 ) f III-A	f V-A	f VI-A	f 	f p
		1	97.9	224.1	0.02 0.16 0.82	0.00	1.00
	wedge Y2	2 3 4 5	96.1 98.5 96.4 97.9	168.2 203.9 188.0 228.5	0.02 0.40 0.58 0.01 0.18 0.81 0.04 0.19 0.77 0.01 0.16 0.83	0.00 0.00 0.00 0.00	1.00 1.00 1.00 1.00
		6	97.3	251.9	0.02 0.15 0.83	0.00	1.00
		1	96.3	96.0	0.01 0.36 0.63 Traces	1.00
	wedge Y3	2 3 4 5	95.3 98.4 100 98.7	101.7 94.2 96.4 103.0	0.01 0.45 0.54 Traces 0.00 0.34 0.66 Traces 0.00 0.34 0.66 0.00 0.00 0.34 0.66 Traces	1.00 1.00 1.00 1.00
		6	97.5	96.0	0.01 0.44 0.55 Traces	1.00
		1	94.3	166.4	0.04 0.37 0.59 Traces	1.00
	Crankshaft	2 3 4 5	93.2 97.5 93.1 96.4	166.0 192.0 171.7 186.3	0.06 0.27 0.67 0.02 0.31 0.67 0.05 0.24 0.71 0.03 0.26 0.71 Traces 0.00 0.00 0.00	1.00 1.00 1.00 1.00
		6	95.4	164.7	0.04 0.43 0.53 Traces	1.00

Table 5

 5

	. HBW results obtained at room temperature (22ºC).
	Sample	Y2 wedge Y3 wedge Crankshaft
	1	298	301	268
	2	282	311	274
	3	295	309	276
	4	288	307	275
	5	300	303	258
	6	304	311	267
	Average	295	307	270
	S n = 6	8	4	7

Table 6 .

 6 Main statistical descriptors for the tensile tests at the three working temperatures.Figure 11. Evolution of UTS and YS with temperature for the three casting types.

	Casting	T (ºC)	UTS (MPa) Average S n = 6 Average S n = 6 Average S n = 6 YS (MPa) A (%)
		20	963	12	652	9	2.5	0.5
	Y2 wedge	22	969	16	598	6	4.2	0.3
		120	922	7	572	9	4.2	0.5
		20	983	20	675	14	3.2	0.5
	Y3 wedge	22	959	8	610	13	4.3	0.5
		120	944	9	591	5	5.4	0.2
		20	856	15	573	7	3.1	0.3
	Crankshaft	22	870	7	529	9	4.7	0.3
		120	839	6	497	7	4.6	0.3

Table 7 .

 7 Estimated V r values (K•min -1 ) according to tensile results at 22ºC and to the equation reported by Serrallach et al.[START_REF] Serrallach | Effect of selected alloying elements on mechanical properties of pearlitic nodular cast irons[END_REF].

	Property Y2 wedges Y3 wedges Crankshafts
	HBW	32.0	34.8	10.3
	UTS	48.7	44.0	12.5
	YS	41.9	46.8	19.0

Table 9 .

 9 Fracture toughness testing results on the Y3 wedges at different temperatures.

	T (ºC)	Sample	K IC (MPa•m 1/2 )	T (ºC)	Sample	K IC (MPa•m 1/2 )	T (ºC)	Sample	K IC (MPa•m 1/2 )
		1	32.4		1	43.9		1	58.3
		2	30.5		2	40.8		2	54.4
		3	34.6		3	46.4		3	52.2
	20	4 5	38.1 36.2	22	4 5	43.8 44.6	120	4 5	56.3 55.5
		6	33.0		6	45.3		6	54.9
		Average	34.1		Average	44.1		Average	55.3
		S n = 6	2.7		S n = 6	1.9		S n = 6	2.0

Table 10 .

 10 Pearson correlation coefficients referred to K IC of the tensile parameters related to the Y2 wedges and of the testing temperature.As all Pearson coefficients were over 0.70, they are considered relevant and the four input variables from the Y2 wedges were used in a second step to carry out a multivariable linear regression using the 18 rows of data available and to determine K IC according to equation 8. This analysis gave a R 2 regression coefficient of 0.94: K IC = 65.66 + 0.107•T  0.002•UTS  0.046•YS + 1.205•A[START_REF] Faura | Technical-economic models for materials systems analysis and their application to automotive parts fabrication[END_REF]

	Variable referred to Y2 wedges	UTS	YS	A	T
	Pearson coefficient	0.714 0.926	0.752	0.954

Acknowledgements

The authors want to thank Draxton-Teruel Foundry for all the collaborating efforts made to produce the cast parts used in this study. This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors.


Bruno Concina

Guillaume Montagne

Serge Martin

Christian Bordas

Kinetic Energy Released in the Vibrational Autodetachment of Sulfur Hexafluoride Anion

The kinetic energy release distribution (KERD) in the vibrational autodetachment (VAD) from sulfur hexafluoride anion SF6 -has been measured in a velocity map imaging spectrometer for delays in the range of a few tens of microseconds. The experimental KERD is analyzed within the framework of the detailed-balance: first using the standard Langevin model, subsequently using a more refined and realistic model based on the experimental attachment cross-section. A discussion on the processes involved in the attachment and the VAD is presented based on an empirical fit of the attachment cross section. The lifetime derived from the model is in good agreement with the experimental time window, strengthening this theoretical approach for this model system.

I. Introduction

Sulfur hexafluoride (SF6) is of particular interest for industry: it is widely used as a gaseous dielectric and in plasma etching. [1][2][START_REF] Christophorou | Fundamental Electron Interactions with Plasma Processing Gases[END_REF] This molecule is also known to be a greenhouse gas with an extremely long lifetime in the Earth's atmosphere, which reinforces the interest of studying the formation and the decay of the SF6 -anion. [START_REF] Maiss | [END_REF][5][6] Non-dissociative electron attachment to SF6 at low energy is characterized by an exceptionally large cross section. 7 It proceeds by s-wave electron capture and forms mostly longlived SF6 -anions by non-dissociative attachment. 7 Vibrational autodetachment (VAD) from SF6 -is a statistical process and is closely related to the (non-dissociative) electron attachment to SF6 through the detailed-balance principle. 8 Detailed theoretical studies have been devoted to both processes, SF6 being considered as a model system. [8][9][10] An accurate description of the attachment and detachment processes requires an accurate determination of the adiabatic electron affinity (AEA) of SF6. The latter has been the subject of a considerable theoretical and experimental work over the years. 11,12 On the experimental side, the direct AEA measurement is not possible. Indeed, the 0-0 line is invisible because of the poor Franck-Condon overlap due to the significant geometry change between the anion and the neutral. 13 On the theoretical side, the anion was predicted to have the same Oh symmetry as the neutral, the S-F bond being elongated by about 0.2 Å. 14 However, recent coupled-cluster calculations led to a C4v distorted geometry for the anion 15 and an AEA of 1.0340+/-0.03 eV after corrections. 16,17 Electron affinity may be determined from a third-law analysis of the equilibrium constant Kc using experimental data for thermal attachment/detachment and, among others, the vibrational partition function. 18 The new geometry of SF6 -considerably modifies the latter and inspires a reanalysis 19 leading to EA=1.03+/-0.05 eV, in agreement with the corrected coupled-cluster results. In the following, this value will be considered as the most accurate to date. SF6 -lifetime was investigated in an electrostatic storage ring showing neutralization of hot SF6 -by VAD on the microsecond to the millisecond timescale. 20 The decay exhibits a power law dependence due to the broad distribution of vibrational energy. A cryogenic ion-beam trap with extremely low residual gas densities provided observation of the neutralization rates due to VAD over times up to 100 ms. 21 The statistical modeling of the experimental decay relies on the accurate description of the attachment process and of the vibrational level densities. In this frame, the distorted C4V geometry of the anion is essential for achieving a consistent description of the experimental data. The deduced EA value is smaller, but still consistent with the one determined from third-law analysis. 19 In the present work, we report on the first measurement of the Kinetic Energy Release Distribution (KERD) of VAD of SF6 -. It has been achieved on the timescale of a few tens of microseconds. When compared to the neutralization rate, the KERD brings additional information and allows a more stringent test of the VAD statistical modeling. In the frame of the detailed balance formalism, a crucial point is the choice of the cross section of the reverse process, i.e. the electron attachment to the neutral. The anion formation may be pictured as a two-step process: electron capture in the polarization potential of the neutral followed by the transfer of the electron excess energy to the vibrational modes leading to a stable anionic state, the latter process being called Intramolecular Vibrational Redistribution (IVR). The classical Langevin model provides an estimate of the capture cross section at the most basic level of approximation. 22 As a classical model, it is not well suited to the description of the s-wave capture to SF6. Besides, it does not deal with the IVR process. A more realistic detailed-balance modeling of VAD is proposed, based on the fit of the experimental attachment cross section 7 by an empirical expression 10 including s-wave electron capture and IVR. The experimental KERD is fitted with the one derived from the model, using a single parameter, the microcanonical daughter (SF6) temperature. 23,24 The vibrational energy of the parent SF6 -is deduced and used to calculate the electron emission rate, which is finally found in good agreement with the experimental timescale.

This work is part of a long term study on the delayed electron emission of negatively charged molecules and small clusters, interpreted as thermionic emission. Our approach is based on the measurement in a velocity map imaging spectrometer of the delayed electron KERD after photoexcitation and on the development of statistical modeling relying on the detailed-balance principle. Measurements on small negatively charged tungsten clusters showed deviations from the Langevin classical capture model. 22 The KERD was observed sharper than expected. When compared to the Langevin model, the attachment cross section magnitude deduced from the experiment was found lower by up to two orders of magnitude. 22 Measured KERD of fullerene anions showed significant variations as a function of the molecular size. 25 The quantum Vogt and Wannier's formalism was introduced to describe the electron capture and allowed to write an empirical expression for the IVR. Information on the IVR were deduced from the fit of the experimental spectra and were interpreted using the molecular symmetries. 25 A thermal tunneling detachment was evidenced in the study of the fullerene dianions. 24,26 SF6 -allows a much more stringent test of the thermionic emission modeling than the tungsten clusters 22 or the fullerene anions. 25 Indeed, there is little uncertainty on the attachment cross section of this model system, which has been measured with high resolution. 7 The article is organized as follows. In Sec. II, the experimental set-up is briefly described and the experimental KERD is compared to the Langevin model. Sec III is devoted to the presentation of the detailed balance model using an experimental attachment cross section. In Sec. IV, the experimental KERD is compared to model outputs, allowing a discussion on the physical processes. Sec V. shows that the VAD rate derived from the model agrees with the experimental timescale. Finally, Sec. VI is devoted to the conclusion.

II. Experimental and comparison to the Langevin model

Hot SF6 -anions are produced in an Electron Cyclotron Resonance (ECR) ion source by electron attachment to the SF6 gas. 27,28 A continuous beam of negative ions (and electrons) is extracted from the plasma by an acceleration voltage of a few kV and is mass analyzed by a 90° magnetic sector. The mass selected anion beam is pulsed using a parallel plate deflector leading to ion bunches of a few µs duration at 1-kHz rate. The ion bunches pass through a drift tube and enter a velocity map imaging (VMI) spectrometer orthogonally mounted to the ion drift tube. 29 An inhomogeneous static electric field perpendicular to both the ion trajectory and the laser beam path accelerates the electrons towards a position-sensitive detector (PSD) composed by a microchannel plate detector and a phosphor screen. The voltages applied to the PSD are gated in order to reduce the background. Electron impacts are recorded by a CCD camera. The angularly resolved velocity distribution is calculated from the raw image by numerical inversion. 30 The KERD is deduced after angular integration. The VMI spectrometer is calibrated with the photoelectron spectrum of F -at 349 nm laser wavelength. 31 Unlike the previous studied systems (tungsten clusters 22 and fullerene anions 25 ), photoexcitation in the NIR-UV-VIS does not lead to thermionic emission (measurements have been attempted at 1064, 532 and 349 nm laser wavelength). Only the direct detachment is detected and will be separately discussed in a forthcoming article. Nevertheless, a spontaneous electron emission is visible and can be measured without laser excitation. This decay is due to the internal energy that the SF6 -anion has acquired during its production in the ECR source. It is expected that a wide range of internal energy is populated in the ion source, producing a set of excited species whose lifetime covers a broad range. The spontaneous electron emission should depend slightly on the time of flight from the ion extraction to the VMI spectrometer, since hotter species are expected to decay faster. Two acceleration voltages (4 and 6 kV) have been used leading to two slightly different times of flight (48 and 39 µs). No significant differences have been noticed in the measured KERD, certainly because of a too small change in time of flight. The electron raw image and the electron KERD of the vibrational autodetachment of sulfur hexafluoride anions are reported in Fig. 1. The electron distribution on the PSD is isotropic in agreement with the picture of a statistical electron emission.

The geometry of our VMI spectrometer is such that, at least for the low kinetic energies we are currently interested in, all the electrons emitted in the VMI collection zone fulfill the correct operating conditions of the VMI, which allows us to reconstruct with a satisfactory resolution the KERD of the emitted electrons, as verified by two experimental procedures. First, the settings must lead to an isotropic image, which is the only possible geometry when there is no laser beam. Second, it has been checked that the KERD does not change when all VMI voltages are rescaled by the same factor, which changes the size of the image while preserving the spatial focusing conditions. Additionally, the VMI voltages are very low to prevent from a large deflection of the ions and to measure in good conditions the very low energy of this KERD: this distribution peaks at 12 meV (an extremely low value only accessible thanks to specific VMI capabilities) and vanishes completely above 200 meV. Using the photodetachment spectrum of F -at 349 nm laser wavelength, a resolution Δ𝜀 = 8 𝑚𝑒𝑉 (FWHM) is found at 𝜀 = 101 𝑚𝑒𝑉. The scaling law in √𝜀 leads to Δ𝜀 = 3 𝑚𝑒𝑉 for 𝜀 = 12 𝑚𝑒𝑉.

A first approach consists in fitting the experimental KERD with the distribution derived from the Langevin model: 22

𝑓(𝜀) ∝ 𝜀 !∕# 𝑒 $%∕& ! ' "
(1) with 𝜀 the electron kinetic energy and 𝑇 ( the microcanonical daughter (SF6) temperature. 24 The fit reproduces nicely the experimental data. The optimized temperature 𝑇 ( is 319 K. An empirical extension of this fit function is frequently used:

𝑓(𝜀) ∝ 𝜀 ) 𝑒 $%∕& ! ' "
(2) The agreement with the experimental curve is slightly improved and is obtained for the parameters 𝛾 = 0.44 and 𝑇 ( = 346 𝐾. The two fit functions reproduce quite well the experimental data but a straightforward question arises: are the fitted temperatures meaningful? The Langevin model is based on the classical capture cross section, which is far from the realistic attachment scenario, composed of the s-wave capture and the IVR. As the empirical expression including the adjustable 𝛾 parameter is not derived from a model, the interpretation of its parameter 𝑇 ( as the daughter temperature is problematical and therefore it should not be regarded as a true temperature. The interest of this type of fitting function is rather to characterize the KERD shape with the 𝛾 parameter. 32 Let us point out that our experimental result is at variance with detailed statistical rate calculations combined with electron capture theory and kinetic modeling reported in ref. 8 , which led to an exponent 𝛾 ≈ 0.15. All these findings call for a more realistic modeling of VAD, as described in the following section.

III. Detailed-balance modeling of VAD using an experimental attachment cross section

The vibrational autodetachment (VAD) of the SF6 -anion 𝑆𝐹 * $ → 𝑆𝐹 * + 𝑒 $ (3) may be studied in the frame of detailed balance. 33 As the anion and the neutral are both assumed to be in the electronic ground state, the electron binding energy 𝐸 + is equal to the electron affinity 𝐸𝐴 of SF6 (𝐸𝐴 = 1.03 𝑒𝑉). 19 The differential rate for emission of an electron of kinetic energy 𝜀 from a sulphur hexafluoride anion of vibrational energy 𝐸 reads: 24

𝑘(𝐸, 𝜀) = 𝑚 , 𝜋 # ℏ - 𝑔 . 𝑔 $ 𝜀𝜎(𝜀) 𝜌 . (𝐸 -𝐸 + -𝜀) 𝜌 $ (𝐸) (4)
with 𝑚 , the electron mass, 𝑔 $ the electronic degeneracy of anion and 𝑔 . the electronic degeneracy of the neutral times the spin degeneracy of the free electron. 𝜌 . and 𝜌 $ denote the vibrational density of states respectively for the neutral and the anion. 𝜎 is the cross section of the reverse process, i.e. the (non-dissociative) electron attachment to SF6. We introduce the daughter temperature 𝑇 ( , defined as the microcanonical temperature of the neutral SF6 with vibrational energy 𝐸 -𝐸 + : 34

1 𝑘 / 𝑇 ( = 𝑑 𝑑𝐸 𝑙𝑛𝜌 ( (𝐸 -𝐸 + ) (5)
𝑙𝑛K𝜌 . (𝐸 -𝐸 + -𝜀)L may be expanded through a Taylor expansion in powers of 𝜀 using the daughter temperature 𝑇 ( . 24 The first term leads to an Arrhenius factor. The expression of the second and the third terms are given in the Appendix I and depends on the heat capacity 𝐶 ( (temperature derivative of the energy) of the neutral (daughter) SF6 defined for 𝑇 = 𝑇 ( . Finally, the differential rate may be written:

𝑘(𝐸, 𝜀) = 𝑚 , 𝜋 # ℏ - 𝑔 . 𝑔 $ 𝜀𝜎(𝜀) 𝜌 . (𝐸 -𝐸 + ) 𝜌 $ (𝐸) 𝑒𝑥𝑝 P- 𝜀 𝑘 / 𝑇 ( -⋯ R (6)
with the dotted line denoting the second and the third terms. In the case the level densities of the neutral and the anion can be assumed to be equal, their ratio may be replaced by an exponential term depending on the so-called emission temperature 𝑇 , . 23 In the case of SF6, they differ so much (due to the change of geometry) 15 that the level densities are kept in place 8,35 and are calculated (in the harmonic approximation) by the Whitten-Rabinovitch expression 8,36,37 as a function of the internal energy (see the Appendix II). An analytical expression is derived for the caloric curve (i.e. the vibrational energy as a function of the microcanonical temperature, see eq. (A.8)) and thus the heat capacity 𝐶 ( is also calculated, as needed for the second and the third terms of the expansion in powers of 𝜀. Finally, the kinetic energy distribution reads in this theoretical approach:

𝑓(𝜀) ∝ 𝜀𝜎(𝜀)𝑒𝑥𝑝 P- 𝜀 𝑘 / 𝑇 ( -⋯ R (7)
The full expression of the KERD is available in Appendix I, eq. A.1. The KERD depends on two essential ingredients: the cross section 𝜎(𝜀) of the reverse process and the daughter temperature 𝑇 ( appearing in the exponential factor. In the case of a KERD with low values of

% & ! ' "
, the exponential factor has a simple expression: it contains only the first term and the knowledge of the heat capacity 𝐶 ( is not required. If the KERD is characterized by higher values of

% & ! ' "
, it is necessary to take into account the higher-order terms, which requires the knowledge of 𝐶 ( (see Appendix I).

Additional modeling is needed for expressing the attachment cross section 𝜎(𝜀). The anion formation may be pictured as a two-step process: electron capture in the polarization potential followed by transfer of the electron excess energy to the vibrations leading to a stable anionic state. The quantum description of the first step was developed by E. Vogt and G. H. Wannier. 38 The capture cross section is the sum of the contributions of the partial waves of given orbital momentum ℓ and reads in atomic units as a function of the reduced wave number 𝜅: 25

𝜎(𝜅) = 𝛼 𝜋 𝜅 # V(2ℓ + 1) ℓ 𝑃 ℓ 12 (𝜅) (8)
with 𝑃 ℓ 12 (𝜅) the orbital momentum-dependent capture probability and 𝛼 the polarizability of SF6 (𝛼 = 44.1 𝑎 .

-). [START_REF]CRC Handbook of Chemistry and Physics[END_REF] The reduced wave number 𝜅 reads in atomic units : 𝜅 = √2𝛼𝜀. In the case of SF6, the attachment proceeds only by s-wave capture: 7

𝜎(𝜅) = 𝛼 𝜋 𝜅 # 𝑃 . 12 (𝜅) (9)
A simple and accurate analytical expression is available for 𝑃 . 12 (𝜅): [START_REF] Dashevskaya | [END_REF] 𝑃 . 12 (𝜅) = 1 -0.25𝑒 $!.4!5 -0.75𝑒 $4.6*5 (10) The cross section of the non-dissociative electron attachment to SF6 was measured by Braun et al. with high resolution on a large electron energy range (0.1-500 meV). 7 It is represented as a function of 𝜅 in Fig. 2. It allows to determine empirically the second step of electron attachment. In this simple picture, the non-dissociative attachment cross section reads:

The factor 𝑃 718 (𝜅) characterizes the intramolecular vibrational redistribution (IVR), i.e. the transfer of some electronic energy to the vibrations to lead to a stable anion; this factor may be given by the empirical expression: 9,10

𝑃 718 (𝜅) = 𝑒 $; # 5 $ (12)
The parameter 𝑐 ! is deduced from the fit with the experimental cross section. Additionally, some electron captures may lead to vibrational excitation (VEX) and then electron release. The energy threshold for such an inelastic collision is given by the quantum of the symmetric stretch mode A1g (96.6 meV) 7 leading to 𝜅 . = 0.560. The factor 𝑃 19: (𝜅) may be written as:

𝑃 19: (𝜅) = 1 for 𝜅 ≤ 𝜅 . ( 13
) 𝑃 19: (𝜅) = 𝑒 $; $ <5 % $5 & % =
for 𝜅 ≥ 𝜅 . Finally, the two free parameters 𝑐 ! and 𝑐 # are deduced from the fit to the experimental cross section: 𝑐 ! = 2.5 and 𝑐 # = 3.4 (see Fig. 2).

One deduces from Eq. ( 7) the detailed-balance expression of the 𝜅 distribution: 25

𝑓(𝜅) ∝ 𝜅 -𝜎(𝜅)𝑒𝑥𝑝 ]- 𝜅 # 2𝛼𝑘 / 𝑇 ( -⋯ ^ (14)
Finally the 𝜅 distribution used to fit the experimental one reads:

𝑓(𝜅) ∝ 𝜅𝑃 . 12 (𝜅)𝑃 ,> 718 (𝜅)𝑃 19: (𝜅)𝑒𝑥𝑝 ]- 𝜅 # 2𝛼𝑘 / 𝑇 ( -⋯ ^ (15)
It may be noticed that, in the factor

5 $ #?& ! ' "
, 𝛼 is given in atomic units and 𝑇 ( in Kelvin, leading to the Boltzmann constant 𝑘 / in Hartree by Kelvin : 𝑘 / = 3.1668 × 10 $* HK $! . 25

IV. Discussion on the reduced-wave-number 𝜿 distribution of VAD

The experimental κ distribution is reported in Fig. 3. It is compared with several fits respectively based on different assumptions regarding the attachment cross section. In all cases, the fit expressions include the second order term in 𝜀. A large variation of the daughter temperature Td may be observed whereas in all cases the agreement with the experimental curve is quite satisfactory. The small differences between the three fit curves occur in the high κ region where the signal is quite low and consequently may not be significant. Assuming a pure s-wave capture leads to a daughter temperature Td equal to 479 K. It is significantly higher than the temperature deduced from the Langevin capture cross section (Td = 319 K). The temperature increases when the cross section is reduced by assuming a second step in the attachment process: IVR alone leading to Td = 652 K, or IVR and VEX leading to Td= 667 K. The influence of the IVR is dominant in this increase. Only a change of 2% is induced by VEX.

As mentioned above, the expansion of the density of state has been cut at the second order. The position of the cut has been determined by calculating the fits at different orders assuming s-wave capture followed by IVR and VEX. A fit with only the first order term leads to 𝑇 ( = 620 𝐾, showing a significant difference (7 %) with the second order result (𝑇 ( = 667 𝐾). By contrast, including the third order term changes by less than 1% the daughter temperature (𝑇 ( = 672 𝐾), which justifies to cut the expansion at the second order. Finally, as enough terms are given in the expansion and a realistic model is used to describe the reverse process, the temperature 𝑇 ( = 667 𝐾 is meaningful.

As outlined in ref. 8 , detailed balance should link truly reverse processes between, on one side, the anion in the electronic ground state and, on the other side, the neutral and the electron infinitely separated. The competition with the vibrational excitation (VEX) has to be included in the attachment cross section: first the electron is captured and then the IVR takes place whereas VEX does not occur. But the detachment is presumably not concerned with VEX: the inverse process of IVR takes place and then the inverse process of electron capture occurs. Finally when comparing with experiment, this point is of little concern: most of the κ distribution is defined below the VEX threshold (𝜅 . = 0.560, materialized by a red arrow on Fig. 3), which explains why VEX has a so little influence on the temperature. Besides, above the VEX threshold, the fit of the attachment cross section allows only the determination of the product 𝑃 718 (𝜅)𝑃 19: (𝜅). Another expressions for both 𝑃 718 (𝜅) and 𝑃 19: (𝜅) might be assumed as much as the product remains the same. This uncertainty on the empirical factors prevents from an exact determination of the cross section employed in the detailed-balance expression of VAD.

In order to reinforce our approach, the VAD rate derived from the model is compared with the experimental lifetime in the next section.

V. Discussion on the VAD rate

As outlined in section II, the internal energy distribution of the SF6 -anion is broad. As a consequence, the time dependence of the decay rate follows a power law (and not an exponential as expected for a very narrow energy distribution). 34,41 Nevertheless, the internal energy distribution of the anions decaying at a given time 𝑡 is relatively sharp and peaks at the energy 𝐸 c given by 𝐾(𝐸 c ) = 1 𝑡 ⁄ where 𝐾(𝐸 c ) is the rate for electron emission from an anion of internal energy 𝐸 c . 24,34,41 In the following, the time 𝑡 stands for the time of flight from the ECR source to the VMI spectrometer (48 or 39 µs ; average value : 44 µs). The model would be reinforced if agreement was found between the experimental time of flight 𝑡 and the inverse 𝜏(𝐸) of the theoretical emission rate 𝐾(𝐸) with 𝐸 the internal energy of the anion deduced from the experimental KERD via the temperature 𝑇 ( = 667 𝐾 of the neutral.

Using the caloric curve of the neutral, 𝑇 ( = 667 𝐾 leads to 𝐸 -𝐸 + = 0.36 𝑒𝑉 and 𝐸 = 1.39 𝑒𝑉 (𝐸𝐴 = 1.03 𝑒𝑉 from Ref. 19 ). For such an internal energy, radiative cooling can be disregarded: it has been shown to compete with VAD only if 𝐸 -𝐸 + ≲ 50 -100 𝑚𝑒𝑉. 21 By integrating Eq. ( 6), one deduces the rate for electron emission from a sulphur hexafluoride anion of vibrational energy 𝐸 :

𝐾(𝐸) = 𝑚 , 𝜋 # ℏ - 𝑔 . 𝑔 $ 𝜌 . (𝐸 -𝐸 + ) 𝜌 $ (𝐸) h 𝜀𝜎(𝜀) 𝑒𝑥𝑝 P- 𝜀 𝑘 / 𝑇 ( -⋯ R 9$9 ' . 𝑑𝜀 (16)
Rate calculations are carried out to the second order in 𝜀 so as to be consistent with the 𝜅 distribution fits. As defined above, 𝑔 . is the electronic degeneracy of the neutral times the spin degeneracy of the free electron and 𝑔 $ is the electronic degeneracy of the anion. Since all molecular orbitals of SF6 are full, 11 𝑔 . and 𝑔 $ are both equal to 2. As mentioned previously the densities of state and the caloric curve are calculated in the harmonic approximation using the Whitten-Rabinovitch expression (see eq. A.3 and A.8 of Appendix II). Using the vibrational frequencies of Oh symmetry, 14 one finds for SF6:

𝜌 . (𝐸 -𝐸 + ) = 23 /𝑐𝑚 $! (17) In a first approach, Oh symmetry is assumed for SF6 -. Using vibrational frequencies from Ref. 14 , one gets: 𝜌 $ (𝐸) = 2.5 • 10 6 /𝑐𝑚 $! (18) leading to a rate 𝐾(𝐸) = 5.2 • 10 @ 𝑠 $! (with 𝐸 = 1.39 𝑒𝑉). A lifetime is defined as the inverse of the rate: : 𝜏(𝐸) ∼ 2 𝜇𝑠. It would mean that, following the emission rate of the model, the internal energy distribution of the molecular anions decaying after a delay 𝜏(𝐸) ∼ 2 𝜇𝑠 would peak at the internal energy 𝐸 = 1.39 𝑒𝑉. 24,34,41 Consequently, the inverse of the rate (∼ 2 𝜇𝑠) may be compared with the ion time of flight between the ECR source and the detection zone (∼ 44 𝜇𝑠). It is found one order of magnitude lower than the experimental delay. However, as mentioned earlier, recent calculations demonstrate a C4v distorted geometry for SF6 -, 15 which leads to a density of states one order of magnitude larger:

𝜌 $ (𝐸) = 2.2 • 10 A /𝑐𝑚 $! (19)
In this case, the rate is given by 𝐾(𝐸) = 5.8 • 10 4 𝑠 $! and the lifetime amounts to 𝜏 ∼ 17 𝜇𝑠, improving significantly the agreement with the experiment. The distortion of the anion from Oh geometry results in considerable anharmonicity, 15,19 that should significantly increase the density of states 𝜌 $ (𝐸) and the lifetime 𝜏, leading certainly to a better agreement with the experimental time of flight. However precise data and spectroscopic constants for describing the anharmonicity are not available. The simple harmonic calculation we have developed already gives a satisfactory agreement with the experiment.

VI. Conclusion

The KERD of VAD from sulfur hexafluoride anion SF6 -has been measured for the first time. It has been compared to outputs from two detailed balance models: Langevin model and a more sophisticated modeling based on a measured attachment cross section. The latter considers a relatively realistic scenario for the attachment: s-wave electron capture followed by IVR and including competition with VEX. The temperature derived from the fitting of the experimental KERD significantly depends on the considered processes. It is also necessary to take into account the second order term in the expansion of the density of states as a function of 𝜀. The temperature derived from the Langevin model is significantly lower than the ones from the more realistic model. Langevin model is not well suited for the VAD from SF6 -: it describes the attachment as a classical capture which is far from the s-wave capture observed for SF6 -and it does not deal with IVR. Besides, it should also include higher order terms in power of 𝜀.

VAD rate derived from the realistic model gives lifetime slightly lower (by a factor 2-3) than the experimental time window, which is considered as encouraging for calculations based on harmonic densities of states. It is expected that the C4v distorted geometry of SF6 -results in considerable anharmonicity, 15,19 which should increase significantly the calculated lifetime.

The electron attachment to the model molecule SF6 has been well characterized thanks to a large number of studies. 7,9,10 Thus, the influence of the different processes involved in the attachment can be determined on the VAD rate. It has been shown in this work that the temperature derived from a fit of the KERD is meaningful only if the attachment cross section used in the fit function is realistic and enough terms are included in the expansion as a function of powers of 𝜀. These calculations employ the vibrational frequencies respectively from Ref. 14 for both SF6 and SF6 -of Oh symmetry and from Ref. 15

  ℎ𝜈 D the 𝑠 vibrational quanta and 𝐸 B = point energy. The approximation relies on the correction function 𝑎(𝐸) = 1 -𝛽𝜔 which depends on 𝜔 $

Figure 1 :Figure 2 :

 12 Figure 1: measured electron KERD of the vibrational autodetachment of SF6 -(full black line), fit based on the Langevin model with the parameter 𝑇 ( = 319 𝐾 (blue dotted line) and empirical fit derived from the Langevin model with the parameters 𝛾 = 0.44 and 𝑇 ( = 346 𝐾(red full line). Inset: electron raw image

Figure 3 :

 3 Figure3: measured κ distribution (black line) compared to fits based on different assumptions concerning the attachment cross section: pure s-wave capture (blue line), s-wave capture followed by IVR (dashed red line), s-wave followed by IVR and VEX (full red line). VEX threshold (𝜅 . = 0.560) is pointed by a red arrow. All fit expressions include the second order term in 𝜀.

Figure 3

 3 Figure 3

  for SF6 -of C4v symmetry. Using the definition of the microcanonical temperature

	𝑑𝜔 𝑑𝐸	= -𝜔 # ]	5 𝐸 B	+	2.73 2}𝐸𝐸 B	^	for 𝐸 < 𝐸 B	(A.9)
	𝑑𝜔 𝑑𝐸	= -𝜔	1.056𝑙𝑛10 4	1 !/4 𝐸 -/4 𝐸 B	for 𝐸 > 𝐸 B	(A.10)
				1 𝑘 / 𝑇	=	𝑑 𝑑𝐸	𝑙𝑛𝜌(𝐸)	(A.7)
	an analytical relation between the temperature 𝑇 and the vibrational energy 𝐸 may be calculated:
			(𝑠 -1)𝑘 / 𝑇 =	𝐸 + (1 -𝛽𝜔)𝐸 B 1 -𝛽 𝑑𝜔 𝑑𝐸 𝐸 B	(A.8)
	with

Corresponding author : bruno.concina@univ-lyon1.fr

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Appendix I: Taylor expansion of 𝒍𝒏$𝝆 𝟎 (𝑬 -𝑬 𝒃 -𝜺)+ 𝑙𝑛K𝜌 . (𝐸 -𝐸 + -𝜀)L may be expanded through a Taylor expansion in powers of 𝜀 using the daughter temperature 𝑇 ( . 24 The first term leads to an Arrhenius factor. The expression of the second and the third terms depends on the heat capacity 𝐶 ( (temperature derivative of the energy) of the neutral (daughter) SF6 defined for 𝑇 = 𝑇 ( . Finally, the kinetic energy distribution reads:

Using the reduced wave number 𝜅 = √2𝛼𝜀 (atomic unit expression), the distribution becomes :

Appendix II: the Whitten-Rabinovitch approximation

The harmonic vibrational density of states of SF6 and SF6 -may be conveniently calculated in the frame of the Whitten-Rabinovitch approximation: 8,36,37 𝜌(𝐸) = (𝐸 + 𝑎(𝐸)𝐸 B ) C$! (𝑠 -1)! ∏ ℎ𝜈 D