P M Weaver
email: paul.weaver@npl.co.uk

M G Cain

M Stewart

Temperature dependence of high field electro-mechanical coupling in ferroelectric ceramics

Keywords: numbers: 77.80.-e, 77.65.-j, 85.50.-n, 85.80.Jm

A study of the temperature dependence of the electromechanical response of ferroelectric Lead Zirconate Titanate (PZT) ceramics at high electric fields (up to 1.3 kV /mm) is reported. Simultaneous measurements were performed of strain, electric field and polarisation to form a complete response map from room temperature up to 200C. An electrostrictive model is shown to provide an accurate description of the electromechanical response to high levels of induced polarisation and electric field. This provides a method for decoupling strain contributions from thermal expansion and polarisation changes. Direct measurements of electrostriction and thermal expansion, above and below the Curie temperature are reported. Electrostriction coefficients are shown to be temperature dependent in these ceramic materials, with different values above and below the Curie temperature.

Introduction

Recent technological interest in piezoelectric materials has been fuelled by new application areas as a component in magneto-electric composites [1] for high density electronic memory [2], highly sensitive magnetic field sensors [2,3], and for electrical control of magnetization for a range of spintronics applications [4]. Piezoelectric materials are also widely used as actuators, where they continue to find new and extended applications such as injector valves for diesel engines [5], ultra-low power electronic locks and valves [START_REF] Weaver | Proceedings of Actuator[END_REF] for industrial and domestic fluid controls. 'Soft' PZT (Pb Zr x Ti 1-x O 3 ) ceramics are the materials of choice in many of these applications due to their high piezoelectric coefficients. However, these materials are sensitive to temperature changes with strong coupling between thermal, mechanical and electrically induced polarisation changes.

For data storage applications the ability to write electrically and read magnetically would improve speed and data retention [START_REF] Eerenstein | [END_REF]2]. In a composite magneto-electric material the coupling is achieved through strain. Piezoelectric strain in response to an applied electric field is indistinguishable from thermally induced strain. This could lead to thermally induced writing and data loss. In sensor applications, it may be possible to partially cancel a.c. thermal noise due to pyroelectric currents [8], but strain-coupled d.c. drift could not be so easily compensated. Careful expansion matching or compensation for the thermal strain in the piezoelectric material would be required to minimize this effect. In actuator applications thermal drift can severely limit the operating range of the electromechanical device. For instance, in a valve, the position of the sealing member relative to the valve seat is a critical parameter in controlling the flow rate. In bender actuators the ceramic is often bonded to a metal substrate, so careful matching to the substrate expansion is required to achieve stable mechanical output across a wide temperature range. In all these applications, a strain reference is fixed either at the bonding stage in the manufacturing process, or with reference to a fixed mechanical point such as the seat of a valve. It is therefore essential to consider the strain output relative to a datum at a particular temperature -a factor that is not usually taken into account in measurements of piezoelectric properties.

In most of these applications large electric fields are needed to maximise energy density. In this high field regime, linear coupling coefficients are of limited value, and the temperature dependency of the non-linear high field electromechanical coupling dominates performance. Many of the fundamental characteristics of ferroelectric materials are defined by the temperature dependency of their properties. Changes in permittivity and spontaneous polarisation around the Curie temperature are central to the characterisation and understanding of these materials and have been well studied [START_REF] Lines | Principles and Applications of Ferroelectrics and Related Materials[END_REF]. The effects of temperature on the electromechanical properties are less well understood. Data are available on the temperature variation of low field properties such as piezoelectric coefficients obtained from resonance measurements [10]. These are not sufficient to provide information on the high field properties which are dominated by non-linear effects, and tell us nothing about thermal drift in the electromechanical performance. Where measurements have been made of large field ferroelectric strain loops [11], these are not related to a datum, so provide no information on the absolute strain. Thermal expansion measurements have been made in PZT ceramics [12,13], reporting very low values for the thermal expansion coefficient of depoled ceramic (0-2 ×10 -6 • C -1 ) and negative values for poled ceramics [12,13]. Whilst these measurements provide very useful information on phase transitions, they were performed under short circuit conditions, so provide no information on the finite field electromechanical coupling. The need to obtain finite field piezoelectric measurements as a function of temperature relative to a datum has been recognised more recently with regard to the application in diesel injector valves [14]. Direct strain measurements were not reported and the strain relative to the reference was calculated from separate measurements of non-referenced loops, remanent strain, and thermal expansion. The relationship to the polarisation and behaviour above the Curie temperature were not reported.

Thermal expansion in a ferroelectric material is complicated by coupling between the temperature dependent remanent polarisation and the strain. This coupling is influenced by the polarisation induced by an applied electric field, leading to a complicated interdependency between temperature, electric field and strain. This coupling can be understood by studying the electrostrictive properties of the material. It is common to neglect electrostriction when dealing with small field piezoelectric effects on a poled ceramic [15]. In this case the total remanent polarization is not changed significantly by relatively small changes in the induced polarization. High electric fields induce polarisation changes which may be comparable to the remanent polarization. This is particularly significant at high temperatures, but also at more moderate temperatures for soft materials. In these cases the electrostrictive coupling becomes important. The low Curie temperature of soft PZT means that large changes in spontaneous polarization occur over a temperature range encountered in many practical situations, and this electrostrictive coupling to the temperature dependent polarization dominates the thermal expansion and high field electromechanical response.

It can be shown [16,[START_REF] Shirane | Ferroelectric Crystals[END_REF] that the linear piezoelectric effect can be described in the low field limit as an electrostrictive strain with the polarisation equal to the remanent polarisation. This relationship can be used to estimate piezoelectric coefficients from measurements of remanent polarisation [18] assuming the electrostriction coefficients do not change with temperature. Room temperature measurements of the electrostrictive coefficients have long been available for some ferroelectric materials from resonance techniques, particularly for Barium Titanate [19,16]. It is often asserted that the electrostrictive coefficients are independent of temperature, but experimental evidence for this is scant. This is particularly true for PZT. This assertion is based on values obtained indirectly by means of a number of assumptions, or measurements over a restricted temperature range or for single crystals or for materials other than PZT [20].

The objective of the work reported here was to obtain for the first time, a complete mapping of the strain -polarisation -electric field relationship from room temperature to above the Curie temperature for soft PZT ceramics. It was a key objective to relate all the strain measurements to a value at a reference temperature so that drift in the actuation position from zero to high electric field could be measured. Measurements were performed for both unipolar and bipolar (switching) cycles. We also provide for the first time electrostriction coefficients obtained from direct measurements of the strain and polarisation, and show that these are temperature dependent above 100 • C.

Experimental Details

A vertical pushrod dilatometer was used to measure electrically and thermally induced strain. Corrections were made for the pushrod expansion by calibration against a sample of known expansivity. Samples used for the experiments were 25 mm x 5 mm x 0.3 mm sheets of a commercial soft PZT (Fuji Ceramics C91), with approximately 2µm of Ni electrode on both 25mmx5mm faces. Strain was measured in the 25 mm length for an electric field applied across the 0.3mm thickness (this was also the direction of poling). This is referred to in this paper as transverse expansion or contraction. Strain in the poling direction is referred to as longitudinal. A positive strain refers to an expansion. Note that results in this paper are mostly plotted as negative strain (noted on the graph labels) to present the familiar direction of the strain loops.

For polarisation measurement a capacitor (C = 17.2µF) was placed in series with the piezoelectric element (approximately 17nF at room temperature). The voltage across the capacitor (V ) was measured by means of a high input impedance unity gain instrumentation amplifier. The capacitor was discharged at the start of the experiment. Changes in polarisation from this point were computed from the relation:

∆P = CV A ( 1
)
where A is the ceramic electrode area. The small amount of drift due to input bias currents was measured and compensated for in the results. Electric field was applied using a high voltage amplifier under computer control. Simultaneous measurements were made of the applied voltage, charge, extension and temperature. Separate experiments were performed for bipolar and unipolar electrical cycles. For the bipolar experiment a loop sequence of linear ramps 0 V to +400 V to -400 V to 0 V was applied over a period of 40 s. Two loops were taken for each temperature point (approximately every 10 • C) for a heating/cooling rate of 20 • C per hour. The unipolar loop sequence was 0 V to 400 V to 0 V over 80 s. The heating/cooling rate for the unipolar experiments was 10 • C per hour.

Results and Discussion

Electrical loops recorded at different points across the temperature range are shown in Figure 1. m -2 at room temperature, typical of this type of material [START_REF] Jaffe | Piezoelectric ceramics[END_REF]. At 186 • C the hysteresis has disappeared and the dielectric response becomes paralectric, albeit with a high and non-linear permittivity ( r ≈ 15000). The transition from ferroelectric to paraelectric behaviour occurs approximately between 140 • C and 150 • C. This will be referred to as the transition temperature. Figure 1(b) shows the strain -field loops. The familiar 'butterfly' loop is observed at the lower temperatures. Above the transition temperature the strain hysteresis also disappears. However, there is still a strong eletromechanical response well above the transition temperature. This is shown in Figure 2 which plots the loop maxima, minima, and the loop end point at 0V(the remanent strain). The maxima and minima of the strain loops show no discontinuity or other anomaly on passing through the transition temperature. The remanent strain shows a step change in strain as well as a change in slope at the transition temperature. The remanent strain closely follows the remanent polarisation (see Figure 4) except for a gradual strain drift with increasing temperature corresponding to an underlying thermal expansion. It is not possible to specify a unique expansion coefficient for these loops because it depends on the point in the loop that is measured i.e. the thermal expansion depends on the electrical conditions. For example, the loop maxima and remanent strain show an expansion of approximately 7.5-8.0×10 -6 • C -1 while the loop minima give a value of 2.9×10 -6 • C -1 . Figure 1(c) shows the strain -polarisation loops. Above the transition temperature, these loops become very closely parabolic. This is shown more closely in Figure 3(a) where the experimental data are compared with a parabolic line obtained by a least squares fit (see below for line fitting). The close parabolic relationship is only seen in the strain as a function of polarisation and is characteristic of electrostrictive behaviour [START_REF] Shirane | Ferroelectric Crystals[END_REF]. The strain shows a much more complicated dependency on electric field (Figure 1(b)).

Considering only the transverse strain (orthogonal to the poling axis) with the ∞m point group symmetry of a poled ceramic [START_REF] Damjanovic | [END_REF], the electrostriction can be expressed as [START_REF] Shirane | Ferroelectric Crystals[END_REF]:

S 1 = Q 12 P 2 3 (2)
where S 1 is the strain , and P 3 is the polarisation in the direction of the poling axis. Q 12 is the longitudinal electrostriction coefficient. The experiment measures changes in polarisation, so the determination of zero polarisation requires some care. Figure 4 shows the values of the polarisation at different points in the loop, and their variation with temperature. The maximum and minimum polarisation diminished gradually with increasing temperature, and showed no anomaly at the transition temperature, approximately tracking the strain maximum. The remanent polarisation decreased somewhat more rapidly, and underwent a step change at the transition temperature. The polarisation averaged over each loop remained very nearly constant over the temperature range. The average of the positive and negative remanent polarisation was equal to the same value, and also did not vary with temperature. Above the transition temperature, the remanent polarisation approached the same value asymptotically from both directions. The correlation of the different measures of the mean polarisation, and the fact that the remanent polarisation approached the same value above the transition temperature, allows the identification of a single unique value for the polarisation zero. This was set as zero in Figure 4 and all subsequent figures in this paper.

In these experiments the strain varied both with temperature and polarisation and was measured relative to a reference value. Equation 2 must therefore be re-stated more generally as:

S 1 (T, P ) = S 0 (T ) + Q 12 (T )(P 3 ) 2 (3)
where S 1 is the strain measured from the fixed reference and S 0 is the strain at zero polarisation at a particular temperature, measured from the same reference. S 0 (T ) therefore represents the thermal expansion of the lattice which is additional to the strain related to polarisation changes. Note that Q 12 is usually negative reflecting the fact that the striction is negative in the transverse direction, so S 1 < S 0 (negative strain -S is the quantity plotted in Figure 1 -Figure 3). S 0 and Q 12 can therefore be found from a plot of S 1 against P 2 3 by linear regression. An excellent fit to the experimental data is obtained above the transition temperature as shown in Figure 3(a). Below the transition temperature, the points in the cycle where the electric field was decreasing from its peak value to zero (open circles in Figure 3(b)) were found to provide an excellent fit to Equation 3. The electrostriction coefficient, and S 0 thus obtained are plotted as functions of temperature in Figure 5.

The electrostriction coefficient (Figure 5(a)) is approximately constant up to 100 • C with a value between -0.0100 C -2 m 4 and -0.0105 C -2 m 4 . This is slightly lower than the value of -0.0158 C -2 m 4 measured by Haun [20] for undoped PZT with x(Ti)=0.52. Above 100 • C the coefficient drops more rapidly up to the transition temperature. Above the transition temperature the electrostriction coefficient (approximately 0.0075 C -2 m 4 ) reduces linearly with increasing temperature. This could be explained by an increasing contribution from 180 • domain wall motion with increasing temperature. Zhang [18] reports that under weak external driving fields the major part of the domain-wall response is generally thought to be from non-180 • walls and 180 • walls are almost idle. The reason given is that 180 • wall width is much narrower (probably on the order of one unit cell) in comparison with non-180 • wall which implies that there is a much deeper potential well for 180 • walls. Whilst Zhang's results suggest that there is a larger response from 180 • domain walls in soft PZT, it is still reasonable to expect that 180 • domain walls will become more active as the temperature increases.

Because the piezoelectric activity in soft PZT is strongly related to non-180 • domain wall movements, but a large part of the polarisation response is due to 180 • domain wall movement, the electrostriction coefficient might be expected to vary with temperature in these materials. Increased 180 • domain wall mobility may also explain why the electrostriction coefficient values reported here for soft PZT are somewhat lower than previously reported values for harder compositions. It is interesting to note that this process continues even above the transition temperature suggesting that domain processes are still contributing significantly to the dielectric and electromechanical response even well above the transition temperature. The value of S 0 thus obtained can be interpreted by consideration of the free energy expansion in the presence of an applied field. The free energy expansion as a function of electric displacement, D, is given by [START_REF] Lines | Principles and Applications of Ferroelectrics and Related Materials[END_REF]:

G 1 = α 2 D 2 + γ 4 D 4 + δ 6 D 6 (4)
E = ∂G 1 ∂D T = αD + γD 3 + δD 6 (5)
where α varies with temperature, being positive above the Curie temperature, T C , and negative below it. The coefficients γ and δ are assumed to be constant and positive corresponding to a second order phase transition. These assumptions are unlikely to be valid for the temperature range of interest here, but this does not affect the qualitative conclusions of this discussion. In the following discussion we use the approximation D ≈ P and treat the two terms interchangeably. From Equations 4 and 5 we can evaluate the free energy in the presence of an electric field:

G = G 1 -ED (6)
At temperatures T > T C there are no states of lower free energy for any value of the electric field (function not plotted). The states described by Equation 6 are therefore stable for all values of the electric field. The free energy function (Equation 6) for T < T C is plotted in Figure 6(I). The stable states (a to c) are identified in the same way as for T > T C , but now there are metastable states (c to d) where a lower free energy exists for the same electric field. An energy barrier separates the metastable from the stable states. The states d to e are unstable with no energy barrier to the transition. The corresponding points on the PE loop are shown in Figure 6(II). The transition does not occur at zero polarisation, but switches from one polar state to another. The unstable states d to e include the state of zero polarisation i.e. the unstable centro-symmetric crystal. Using the electrostrictive relationship (Equation 2) with Equation 6provides the strain as a function of electric field. The familiar butterfly curve is obtained as shown in Figure 6(III) for T < T C . The unstable centro-symmetric state corresponds to the strain minimum, S 0 . If the polar switching was the only factor, then S 0 would not vary with temperature. However, anharmonicity in the vibration of the relatively fixed ionic structure causes an underlying thermal expansion and therefore a change in S 0 . Extrapolation of the electrostrictive curve therefore provides a means of separating temperature dependent polarisation-induced strain from the underlying lattice expansion. The actual response is the sum of this lattice expansion and the electrostriction calculated through equation 3.

Up to 100 • C a linear expansion in S 0 is observed with an effective expansion coefficient of 1.33×10 -6 • C -1 . Just below the transition temperature the expansion coefficient reduces to a very low value. A contraction is observed in the region of the transition temperature, indicative of a first order crsytallographic phase change. Above the transition, the expansion proceeds linearly with a coefficient of 1.77×10 -6 • C -1 . These figures are in line with previous measurements for unpoled ceramics [12,13].

For the transverse expansion measurements reported here, changes in the remanent polarisation couple electrostrictively to the strain causing an expansion on heating. This reinforces the underlying thermal expansion of the lattice. If measured in the direction of polarisation, the changes in remanent polarisation would lead to a contraction on heating, which would oppose the underlying lattice expansion. For unpolarised samples, the electrostrictive contraction would be approximately zero, so a weak expansion would be observed. For poled samples, the electrostrictive contraction would dominate leading to negative expansion. Above the Curie temperature there is no remanent polarisation, so zero field strain in both directions reverts to a normal expansion on heating. The transverse measurements reported here therefore help to explain previous reports of negative thermal expansion in poled PZT ceramics [12,13].

The strain -polarisation curves shown in Figure 7 have been adjusted by applying a temperature dependent offset of 0.8×10 -6 • C -1 . This is the value of the total expansion across the temperature range shown in Figure 5(b). The value is so low due to the the contraction at the transition temperature. It is used here to compensate the curves to remove the average overall expansion (this figure is arbitrary as discussed below). We observe that all the curves nearly lie on the same parabolic curve, indicating that there is an approximate underlying electrostriction that applies across the temperature range. Very similar results were obtained for both unipolar and bipolar (stable region) cycles. However, the family of curves can not be made to lie on the same curve for any single pair of values of expansion and electrostriction coefficients. This is a reflection of the temperature dependence of these coefficients shown in Figure 5.

These results confirm that the electrostrictive model provides an accurate description of the electro-mechanical response at high levels of induced polarisation and electric field, as long as the system is kept within the stable region of the free energy diagram. This is true even when there is significant non-linearity and hysteresis present in the strain -field response. This implies that the electrostrictive description of the piezoelectric behaviour applies not just to intrinsic crystallographic distortions, but also to the changes in strain and polarisation due to extrinsic domain wall motion. The fact that the relationship holds across the temperature range, and the electrostriction coefficient reduces with increasing temperature is a strong indication that both 180 • and non-180 • domain wall processes are accommodated within this model. However, the model does not apply in the region of metastability where the field is applied against the direction of the remanent polarisation.

Conclusions

Measurements are reported of the temperature dependent transverse actuation response of PZT ceramics relative to a reference length for large field bipolar and unipolar electrical cycles. Simultaneous measurements were made of the strain, electric field and polarisation to form a complete Strain -Polarisation -Electric Field map from room temperature up to 200 • C. Remanent strain and polarisation show a transition in the region of the Curie temperature, although significant actuation activity is observed well above this transition temperature. The temperature dependent movement range is offset by thermal expansion.

An electrostrictive model provides an accurate description of the device response to high levels of induced polarisation and electric field below, as well as above, the transition temperature as long as the system is kept within the stable region of the free energy diagram. This is true even though there is significant non-linearity and hysteresis in the strain -field response.

Direct measurements of electrostriction coefficients and thermal expansion, above and below the transition temperature are reported. Electrostriction coefficients are shown to be temperature dependent in these ceramic materials, with different values

Figure 1 .Figure 2 .

 12 Figure 1. Composite plots showing bipolar a) polarisation-field, b) strain-field and c) strain-polarisation loops at different temperatures. The x axis values for each loop (electric field or polarisation) were offset to centre each loop on the temperature at which it was measured.

Figure 3 .

 3 Figure 3. SP loops at a) 186 • C (T > T C ) and b) 19 • C (T < T C ). Dashed lines show the quadratic fit to the points in the stable state region (see discussion), marked as red circles.

Figure 4 .

 4 Figure 4. Variation of bipolar PE loop maximum and minimum (+), mean (•) and remanent ( ) polarisation with temperature

Figure 5 .

 5 Figure 5. Electrostriction coefficient, Q 12 , and strain zero S 0 as functions of temperature (bipolar cycles). In (b) the dashed lines represent straight line fits to the approximately linear portions of the curves.

Figure 6 .

 6 Figure 6. Stability from free energy expansion for T < T c . (I) free energy G 1 -ED, (II) polarisation, and (III) strain as functions of electric field. a to c is stable, c to d is metastable.

Figure 7 .

 7 Figure 7. Strain -v-Polarisation compensated for a thermal expansion of 0.8×10 -6 • C -1 . The left half of the plot shows the stable regions of the bipolar cycles (negative half only shown). The right half of the plot shows unipolar cycles.

Acknowledgements

The authors would like to acknowledge the support of the Technology Strategy Board and the UK National Measurement Office. We would also like to acknowledge the support of the APAHOE project partners Servocell Ltd., Diameter Ltd., and University of Southampton.

above and below the Curie temperature.


G Dilecce

F Ambrico

S De Benedictis

On the collision quenching of N + 2 (B 2 Σ + u , v = 0) by N 2 and O 2 and its influence on the measurement of E/N by intensity ratio of nitrogen spectral bands

Keywords: numbers: 39.30.+w, 52.70.kz, 52.80.-s Spectroscopic Techniques, Plasma Diagnostics, LIF

Laser induced fluorescence measurements of the rate coefficients of N + 2 (B 2 Σ + u , v = 0) collision quenching by N 2 and O 2 are presented. The values of (8.84 ± 0.37) × 10 -10 cm 3 s -1 and (10.45 ± 0.45) × 10 -10 cm 3 s -1 have been found for N 2 and O 2 respectively. The present results agree well with literature data obtained by selective methods for ion B state excitation. The data are discussed in the frame of the spectroscopic evaluation of the reduced electric field in electrical discharges at high pressure, that makes use of the Second Positive System (SPS) and First Negative System (FNS) emissions of nitrogen.

Introduction

The determination of the electric field strength in a discharge by spectroscopic methods is an old issue [1] that is nowadays receiving renewed attention, in both low and high pressure discharges [2], [3]. In air or, more generally, in nitrogen containing discharges, this is achieved by looking at emissions belonging to the Second Positive System (SPS) of N 2 : C 3 Π u → B 3 Π g and to the First Negative System (FNS) of N + 2 : B 2 Σ + u → X 2 Σ + g . The intensity ratio of these two emissions is very sensitive to the mean electron energy, and then to the reduced electric field E/N, due to the large difference of excitation thresholds of the two excited states, 11.03 eV for N 2 (C 3 Π u , v = 0) and 18.7 eV for N + 2 (B 2 Σ + u , v = 0) (bands belonging to v=0 vibrational quantum number of both electronic states are usually employed). The emission ratio is proportional to the ratio of the populations of the emitting states. Populations are calculated at the stationary state as the ratio of excitation and quenching rates. Excitation rates are calculated by solving the Boltzmann equation with the proper cross sections, with the reduced electric field -E/N -as a parameter, in the hypothesis of local field approximation -LFA (for expample with a Boltzmann solver like BOLSIG+ [4]). Another working hypothesis is that both N 2 (C 3 Π u ) and N + 2 (B 2 Σ + u ) are excited by electron impact with ground state nitrogen molecules:

e + N 2 (X 1 Σ + g ) → e + N 2 (C 3 Π u ) (1) e + N 2 (X 1 Σ + g ) → e + N + 2 (B 2 Σ + u ) (2)
E/N is deduced from the best match of calculated and measured SPS and FNS emission ratio. This method has been recently reviewed and analysed thoroughly in [5], in a wide pressure range, from 300 to 10 5 Pa. The relevant cross sections and collision quenching data typically adopted can be found there. As pressure increases the collision quenching,

N + 2 (B 2 Σ + u ) + M → N + 2 (X 2 Σ + g ) + M (3) N + 2 (C 3 Π u ) + M → N + 2 (X 2 Σ + g ) + M (4)
overcomes the radiative quenching and becomes dominant at atmospheric pressure for both emitting states. The accurate knowledge of collision quenching rate constants is then of fundamental importance. In [5] the data of [6] and [7] were adopted for the collision quenching of N 2 (C 3 Π u ) and N + 2 (B 2 Σ + u ) respectively. Later in [8] it was pointed out that the associative ion conversion

N + 2 (B 2 Σ + u , v = 0) + N 2 + M → N + 4 + M (5)
with its 5 × 10 -29 cm 6 s -1 rate constants, becomes dominant at high pressure, and has to be taken into account in the calculation of the total quenching of the FNS emitter. In order to choose the collision quenching rate constants values, one has to deal with quite a large amount of literature. A good convergence of data is found about the quenching of N 2 (C 3 Π u , v = 0, 1) by N 2 (see [9]), and by O 2 (see [6]), and the data of [6] are a good choice. A refinement in the calculation of C state population can be achieved by taking into account also vibrational relaxation, that has been measured in [9], as it was done in [10]. Large discrepancies exist, instead, between the many measurements of rate constants for N + 2 (B 2 Σ + u , v = 0) by N 2 and O 2 . We have reported in table I these literature data: rate constant values differ by about a factor four for nitrogen quencher and by about a factor two for oxygen quencher. In the table the data are listed according to the method used for the excitation of the electronic state of the nitrogen ion, and a further grouping is made according to the method being selective or non-selective. It is clearly seen that selective excitation methods, i.e. those based on the absorption of resonant radiation, give the largest values. Among the nonselective methods, i.e. those that excite the whole manifold of electronic states, pulsed discharge based methods give the lowest values, while continuous electron/proton beam and soft X-ray methods give intermediate values. It is quite evident that the rate constants values are dependent on the measurement method, and that the choice of the right value is a matter that deserves a thorough discussion. For instance at 1 atm of nitrogen, using the value of [7] leads to a total collision quenching, processes (3) plus ( 5), of about 3.5 × 10 10 s -1 , that, with the data of [11] instead amounts to 5 × 10 10 s -1 . The model calculation of the SPS and FNS emission ratio can then be affected by a factor 1.43 variation according to the choice of quenching rate constants. In the pulsed discharge method of [7] (and [6]) a fast rise and fall time voltage pulse, with duration of few tens of ns, was applied to produce a discharge with the fast ionization wave (FIW) mechanism. After discharge-off, the decay of the emission intensity was measured and fitted by a single exponential function to get the quenching rate. A weak point of this method is that in the FIW, as pointed out by the same authors in [6], the relaxation of the EEDF in the energy range for inelastic processes proceeds for a time comparable to that of depopulation of the investigated levels. In other words the excitation of electronic states is not totally switched off during the emission decay time, and this might lead to a possible slowing down of the depopulation itself, i.e. to an underestimation of the measured quenching rate constants. This was argued to be a minor effect in [6] on the basis of two identical quenching results obtained at two different discharge durations, i.e. at two different electron density and mean energy conditions. A strong point of the FIW excitation method was that, due to the short duration and low repetition rate of the discharge, the dissociation degree and the concentration of excited species is very low, so that it is possible to consider for the quenching the species and concentrations that constitute the initial gas feed only. Electron/proton beam based methods use a continuous beam for excitation of electronic species in a background gas. The emission intensity is then recorded as a function of the background gas. It is an indirect method since, in addition to the precise characterization of the electron beam characteristics, it requires to take into account the effect of excitation by low-energy secondary electrons coming from the ionizations produced by the principal beam. Such an effect depends on the background gas pressure. Finally resonant excitation methods [11] resonant laser (LIF) 7. 5 11 [12] resonant radiation 5.4 ± 1.0 [13] resonant radiation 3.9 ± 1.4 [14] electron beam non sel. 4.3 ± 0.27 7.2 ± 3 [15] electron beam (6.0 -7.7) ± 1.5 [16] electron beam 3.66 (at 300 K) [17] electron beam 4.15 [18] pulsed proton beam 4.53 7.36 [19] soft X-ray (1.37 -2.17) [20] pulsed discharge (invertron) 2.1 ± 0.2 5.1 ± 0.5 [7] pulsed discharge (FIW)

are the most direct and less affected by spurious effects, since they are based on the measurement of the quenching rate of an excess population of the single vibronic state that is selectively excited by absorption of resonant radiation. In [11], laser radiation was used to excite N + 2 (B 2 Σ + u , v = 0) from the ion ground state and the fluorescence pulse decay provided the quenching rate. A sufficient population of ions was produced in a continuous d.c. discharge. This is a possible weak point, since the exact gas composition in presence of gases that can be easily dissociated might not be neatly known. A possible role of quenching by electron impact was instead reasonably ruled out in [11].

In this paper we have revisited this subject by combining the cleaniness of laser induced fluorescence measurements with the advantage of production of ground state ions by means of a short duration, low repetition rate discharge, that are the same advantages as those described for the FIW excitation method. New data on N + 2 (B 2 Σ + u , v = 0) quenching by N 2 and O 2 are provided that can help in the decision on which values to adopt in the application of spectroscopic methods to the E/N determination.

Experimental

discharge

The ground state nitrogen ions necessary for LIF process (see next section) are produced in a short pulse dielectric barrier discharge. A drawing of the electrodes arrangement is shown in figure 1. The bottom electrode is made of an alumina slab, 0.7 mm thick, with a 8×10 mm rectangular surface covered by a silver loaded paint, on the side not exposed to the discharge. The top electrode is a bare, flat stainless steel disc in a ceramic holder. Both electrodes are mounted on sliding supports that allow to change the interelectrode distance under vacuum by computer controlled stepping motors. The whole apparatus is described in more detail in [21].

The high voltage supply is composed of a low voltage pulse generator (TTi mod. TG1010A), a power amplifier (Industrial Test Equipment Powertron 1000A), and a HV transformer with an operating range 1-7 KHz, 25 kV max. voltage. A negative square pulse, 10µs duration and 60 Hz repetition rate, is applied to the amplifier, resulting in a distorted waveform after the transformer, as shown in figure 2. In the same figure, as a monitor of the current, the discharge emission of the FNS (0,1) band is shown. The emission pulse is composed of a low initial part, lasting about 5 µs, followed by a more intense part that lasts about 15 µs. The LIF pulse is superimposed to the discharge emission, and its position marks the t=0 point. The gas flow is operated by MKS flow meters/controllers, and pumping is achieved by a 30 m 3 /h rotary pump, that serves both for evacuating the vessel and for maintaining the dymanic pressure under gas flow by pumping through an adjustable needle valve. N 2 and O 2 (99.995% purity) are used, at a fixed flow rate of 50 sccm, in the pressure range 0.2-4 Torr. The pressure is measured by a capacitance manometer. We have used two gas compositions only: pure N 2 and N 2 + 50% O 2 . In these conditions the interelectrode gap is fixed at 18 mm, while the peak negative voltage varies in the range 2-4 kV.

laser induced fluorescence

We have made mainly measurements of LIF pulses with large bandwidth detection, in order to measure the decay rate of the LIF pulse that carries the information on the collision quenching of the ion electronic state. Few fluorescence and emission spectra have also been recorded to explore the details of the LIF process under observation and to measure the gas temperature. The LIF excitation-detection scheme is as follows: where ν L and ν F are the frequencies of the laser and fluorescence photons respectively. The laser apparatus consists of a Nd-YAG pumped tunable pulsed dye laser, with LDS 751 dye in DMSO solution, and a BBO second harmonic generator. The pulse width and energy are about 10 ns and 1 mJ respectively, at a wavelength of 391.424 nm, and the nominal bandwidth is 0.4 cm -1 , i.e. about 0.006 nm. The laser wavelength is measured with 5 × 10 -6 relative accuracy by a High Finesse WS5 wavemeter. The excitation wavelength is chosen such as to maximize the fluorescence signal. It corresponds to a position close to the P-branch bandhead, with simultaneous excitation of N=10-16 rotational levels. A fluorescence spectrum, measured at P=2 Torr, is shown in figure 3. The spectrum is well reproduced by a simulation, calculated by LIFBASE [22], with the rotational distribution shown in the inset figure. Most of the fluorescence comes from the rotational levels N=10-16 that are directly excited by the laser, while a weak collisional relaxation towards lower N levels is also present.

N + 2 (X 2 Σ + g , v = 0)+hν L → N + 2 (B 2 Σ + u , v = 0) → N + 2 (X 2 Σ + g , v = 1)+hν F (6) Collision quenching of N + 2 (B 2 Σ + u , v = 0)
Both laser induced fluorescence and discharge emission light are collected perpendicularly to the laser beam, by a 1:1 telescope made of two 2 inches diameter, 30 cm focal length quartz lenses. Fluorescence spectra are measured by a 500 mm focal length SPEX 500M monochromator, equipped with two gratings, 600 gr./mm, 500 nm blaze and 1800 gr./mm 300 nm blaze, and an Andor DH5H7-18F-01 gated ICCD detector. 1 mm input and output slit width respectively. The trapezoidal spectral transmission function of the slits is positioned in such a way as to collect the whole P-branch rotational levels wavelengths that contribute to the LIF signal (see figure 3). With this setting there is no possibility of a spurious contribution to fluorescence decay due to some excitation that escapes the observation window by rotational relaxation. The current signal from the photomultiplier is measured by a digitizing oscilloscope (Le Croy waveRunner 6030A). The measurement output is a voltage vs. time signal. The voltage is the drop of the photomultiplier current across the 50Ω input impedance of the oscilloscope, and is then proportional to the number of photon/sec impinging on the photocathode. We will denote this quantity by E t . The 50 Ω input impedance of the oscilloscope is necessary to match the cable impedance and avoid reflections that could influence the measurement of the decay of such a fast signal. The laser firing, the ICCD gate and the oscilloscope are triggered by the discharge TTL trigger pulse.

Experimental method

The time resolved fluorescence pulses are recorded and their decay is fitted to a single exponential yielding the decay rate. In figure 4 a typical fluorescence pulse, both in linear and logarithmic scale is shown. Repeated measurements at various pressures allow contruction of Stern-Volmer plots from which the rate constant is deduced. The concentration is calculated from the pressure after the measurement of the gas temperature. This is achieved by fitting the emission spectrum of FNS (0,0) band, as shown in figure 5. N + 2 (B) state is infact excited from the ground state by electron impact, that is known to reproduce the ground state rotational distribution on the electronic excited state. The rotational distribution observed in FNS emissions is always Boltzmann-like, with a temperature of 330±10 K that is constant in all the pressure and gas composition conditions. We take this value as the gas temperature. In a few cases we have also verified that this value agrees well with that of N 2 (C) state measured from SPS emission. LIF and gas temperature measurements are made at about 8 µs after the discharge onset, as shown in figure 2. The short discharge duration and long time between two successive pulses prevents significant dissociation or formation of other species before the laser pulse. In [23] calculations experiments in a pulsed d.c. discharge show that, in an air-like N 2 -O 2 mixture, oxygen dissociation and NO formation are low at short pulse lenghts. In particular, at pressures of the order of 0.1 Torr, O and NO fractional concentrations are less than 10 -3 for a pulse lenght of 100 µs, the minimum lenght reported in that paper, with a fast rising trend with increasing pulse duration. It is then reasonable to assume that, in our case of 8 µs discharge duration before the LIF measurement, dissociation and formation of other species is very low, and that, from the point of view of electronic states quenching, the concentration of quenchers is practically the same as that prepared by flow gas mixing and initially present at the discharge onset. Figure 5. Emission spectrum of FNS (0,0) at 2 Torr of N 2 +50%O 2 . The measurement is taken with the ICCD and the 1800 gr/mm grating, with a gate width of 5 µs centered at the time of the laser shot. The simulation is calculated by LIFBASE with a Voigt instrumental profile (gaussian + 50 % Lorentzian), 0.12 nm bandwidth, T rot =330K

Collision quenching of N

+ 2 (B 2 Σ + u , v = 0)

Results and discussion

The Stern-Volmer plots of the quenching rate as a function of pressure are reported in figure 6 and figure 7 for the pure N 2 and for the N 2 + 50% O 2 mixture respectively. The corresponding rate coefficients are reported in table 1, in which the due to gas temperature uncertainty is included. The rate coefficient for quenching by O 2 (k O2 ) is calculated, after that by N 2 (k N 2 ) has been determined, according to the simple relationship

k mix = 1 2 k N 2 + 1 2 k O2 ,
where k mix is the rate coefficient of the mixture determined from the slope of the plot of figure 7. The present results agree very well with all the literature data relevant to selective excitation methods, enforcing the idea that, as noted in the introduction, the measurement methods carry some systematic error that must be discussed in order to understand which values should be taken to be closer to real ones. From the point of view of E/N estimation from SPS and FNS emission measurements, this is not an irrelevant issue. We have estimated the effect of such indetermination in the case of a pure nitrogen discharge at atmospheric pressure. In this condition the total quenching rate of N + 2 (B 2 Σ + u , v = 0) varies by about a factor 1.47 going from the lowest to the highest value of table 1 for the quenching rate coefficient. Given an experimentally determined [N + 2 (B)]/[N 2 (C)] population ratio, the ratio of electron impact rate coefficients R = r 2 /r 1 deduced from the experimental population ratio using the quenching data of this work will be about 1.47 times larger than that

Collision quenching of N + 2 (B 2 Σ + u , v = 0)
(B 2 Σ + u , v = 0) quenching rate in pure N 2 .
The linear fit gives a slope of (2.5871 ± 0.027) × 10 7 Torr -1 s -1 and an intercept of (1.5228 ± 0.055) × 10 7 s -1 . The inverse of the intercept, τ = 65.67 ± 2.37 ns is in good agreement with the 62.33 ns radiative lifetime of [24].

obtained by the data of [7], in nitrogen at one atmosphere. We have calculated by BOLSIG+ the ratio R as a function of E/N in pure nitrogen, and plotted E/N vs. R in figure 8. The error introduced by the factor 1.47 uncertainty in the quenching of the ion B state grows with growing E/N. In the figure three cases are shown graphically. Around E/N=100 Td the indetermination in E/N amounts to about 8%, and grows to about 12% around E/N=200 Td and to about 19% around 400 Td. The present measurements should be the most direct and less affected by collateral problems. The agreement with the data obtained by selective excitation is striking, for both N 2 and O 2 quenchers. As already stated in the introduction, a quick look at table 1 shows quite clearly that in each class of methods data are similar, and quite different from class to class. Each class of methods then could be affected by systematic errors ascribable to the method. Selective excitation gives the largest quenching values, therefore its possible systematic error would be an additional loss channel for the overpopulation of ion B states induced by radiation absorption. At present we are not aware of such an additional channel.

Inspection of table 1 shows also that the ratio of rate constants by O 2 and by N 2 is also class-dependent. In the pulsed discharge method it amounts to 2.43, and decreases to 1.6 for electron beam methods down to 1.18 in present work case. This might be

Collision quenching of N + 2 (B 2 Σ + u , v = 0)
(B 2 Σ + u , v = 0) quenching rate in N 2 + 50%O 2 .
The linear fit gives a slope of (2.8236 ± 0.0128) × 10 7 Torr -1 s -1 and an intercept of (1.5752 ± 0.0178) × 10 7 s -1 . The inverse of the intercept, τ = 63.48 ± 0.72 ns is in good agreement with the 62.33 ns radiative lifetime of [24].

the spy of the influence of changing conditions on the rate coefficient results in indirect, non-selective methods. In continuous charged particles beam or X-ray excitation, this may be the effect of a non exact calculation of secondary electrons excitation. In pulsed discharge methods, the discharge decay time can influence the measured quenching of the excited species, and the addition of oxygen may shorten the discharge decay, allowing the measurement of faster rate coefficients. An analogous difficulty of the pulsed discharge method in recovering fast decay times was perhaps already evidenced in the measurement of N 2 (C 3 Π u , v) of [6] compared to the same measurements by LIF of our group [9]. The two data sets agree well for v=0, 1 and 2, showing increasing values of the rate coefficients up to about 4.3 × (10 -11 cm 3 s -1 ), but for v=3 the value of [6] remains slightly lower than that of v=2, while the LIF measurement shows an increase to 6.34 × (10 -11 cm 3 s -1 ) and confirm the increasing trend with a 9.86 × (10 -11 cm 3 s -1 ) value for v=4. . The cross sections are taken from [25] and from [26] for processes (1) and ( 2) respectively.

obtained by selective methods for ion B state excitation. These data are compared to collision data obtained by various further non-selective methods. Possible systematic conceptual errors relevant to three classes of measurement methods are briefly discussed.

Selective methods appear to be at present conceptually more reliable, but, in the absence of a definitive confirmation, we must accept an indetermination in these values with a preference, we believe, for selective methods results. A precise knowledge of these collision quenching data is very important when spectroscopic methods that employ the ratio of FNS and SPS emissions are applied to the estimation of the reduced electric field in an electrical discharge at high pressure. The effect of such an indetermination has been briefly analyzed for a nitrogen discharge at atmospheric pressure.

Figure 1 .

 1 Figure 1. Sketch of the discharge hardware, with indication of the gas flow and LIF geometry (see text). In the annexed picture the discharge at 1.6 Torr is shown.

Figure 2 .

 2 Figure 2. Oscillograms of the applied voltage and FNS (0,1) band emission in N 2 at 2 Torr. The emission scale is compressed in the main figure. In the inset figure the whole emission plus LIF signals are shown in an uncompressed scale.

Figure 3 .

 3 Figure 3. Fluorescence spectrum of FNS (0,1) band after laser excitation at 391.424 nm. The measurement is taken with the ICCD and a bandwidth of about 0.4 nm. The simulation is calculated by LIFBASE and with the rotational populations shown in the inset figure. The dashed lines define the position of the trapezoidal slit spectral transmission function used for LIF pulse measurements.

Figure 4 .

 4 Figure 4. LIF pulse at P= 0.2 Torr, pure N 2 (after background subtraction). The curve shown is the average over 5000 laser shots. The same pulse is shown in logarithmic scale in the inset figure. Single exponential decay starts at about 30 ns, i.e. after about 20-25 ns from the beginning of the pulse. The red line is the exponential fit of the decay.

Figure 6 .

 6 Figure 6. Stern-Volmer plot of the N + 2(B 2 Σ + u , v = 0) quenching rate in pure N 2 .The linear fit gives a slope of (2.5871 ± 0.027) × 10 7 Torr -1 s -1 and an intercept of (1.5228 ± 0.055) × 10 7 s -1 . The inverse of the intercept, τ = 65.67 ± 2.37 ns is in good agreement with the 62.33 ns radiative lifetime of[24].

Figure 7 .

 7 Figure 7. Stern-Volmer plot of the N + 2(B 2 Σ + u , v = 0) quenching rate in N 2 + 50%O 2 .The linear fit gives a slope of (2.8236 ± 0.0128) × 10 7 Torr -1 s -1 and an intercept of (1.5752 ± 0.0178) × 10 7 s -1 . The inverse of the intercept, τ = 63.48 ± 0.72 ns is in good agreement with the 62.33 ns radiative lifetime of[24].

Figure 8 .

 8 Figure 8. E/N vs. R = r 2 /r 1 calculated by BOLSIG+ in pure N 2 . Three uncertainty cases are presented, with R ranging from 10 -3 to 1.47 × 10 -3 , and from 10 -2 to 1.47 × 10 -2 and from 8.3 × 10 -2 to 1.23 × 10 -1 . The cross sections are taken from[25] and from[26] for processes (1) and (2) respectively.

Table 1 .

 1 N + 2 (B 2 Σ + u , v = 0)quenching rate constants (10 -10 cm 3 s -1 ) by N 2 and O 2 . Present results and literature data. Data are grouped according to selective and nonselective excitation methods.

	quencher	ref.	excitation	sel./non sel.
	N 2	O 2	method
	8.84 ± 0.37	10.45 ± 0.45 this work	resonant laser (LIF)	sel.
	8.2 ± 1.2

  Fluorescence pulses are instead measured by a Hamamatsu R2949 photomultiplier after spectral selection throught a 1000 mm focal lenght McPherson monochromator with a 1200 gr./mm, 300 nm blaze grating with a choice of 0.2 mm and

	Collision quenching of N + 2 (B 2 Σ + u , v = 0) by N 2 and O 2	7
			0.30
														P branch
			0.25
	100	Rot. pop. (norm)	0.10 0.15 0.20
	I (a.u.)		0.00 0.05	0	2	4	6	8	10	12	14	16	18	20
	50								N					R branch
						exp.
						mod.
	0
	412		414		416		418		420	422	424	426	428	430	432
										wavelength (nm)

u , v = 0) by N 2 and O 2