Yoav Ben

author. Professor Tim J Cole
email: tim.cole@ich.ucl.ac.uk

Malcolm D C Donaldson

Yoav Ben-Shlomo

SITAR -a useful

Keywords: Height, puberty, Turner syndrome, growth curve, random effects Height, puberty, Turner syndrome, growth curve, random effects

Introduction A child's growth curve contains considerable information about their genetic make-up and environmental exposure. [START_REF] Tanner | Foetus into man: physical growth from conception to maturity[END_REF] It can be useful to characterise the nature of the curve and relate it both to previous or current exposures and putative later outcomes. [START_REF] De Stavola | Statistical issues in life course epidemiology[END_REF] Thus the characterisation of the growth curve is an important first stage in the life course study of early growth and later outcome.

By its nature the growth curve consists of a series of highly correlated measurements, and it is important to reduce the dimensionality of the data to simplify comparisons between individual children. The traditional approach is to identify a suitable parametric model for the given measurement and age range, for example the Jenss-Bayley curve [START_REF] Jenss | A mathematical method for studying growth in children[END_REF] for weight in early life or the Preece-Baines curve [START_REF] Preece | A new family of mathematical models describing the human growth curve[END_REF] for height in puberty. The growth curve is then entirely characterised by the fitted parameters of the model, which can be used instead of the raw data to test for associations with exposure and/or outcome. This paper focuses on height in puberty, where the Preece-Baines curve with its five estimated parameters fits individual growth curves extremely well, with a residual standard deviation of 6-7 mm. [START_REF] Preece | A new family of mathematical models describing the human growth curve[END_REF][START_REF] Ledford | Mathematical models of growth in stature throughout childhood[END_REF] However as a summary of the underlying curve its five parameters are relatively high-dimensional for many purposes.

The ideal approach would be to fit a form of curve to all subjects simultaneously, and to estimate the subject-specific parameter values as subject random effects. Such an approach could in principle be applied to the Preece-Baines curve. However Beath 6 introduced a considerable simplification to the analysis of growth curves by describing a shape invariant model of infant weight. This consists of a single growth The aim here is to fit the model described by Beath [START_REF] Beath | Infant growth modelling using a shape invariant model with random effects[END_REF] and show how effectively it summarises height growth around the time of puberty. The second aim is to show how the estimated subject-specific parameters can be related to earlier exposures and later outcomes.

Methods

Data sets

Two data sets are used as examples, both of height in puberty. The first, the Christ's Hospital School (CHS) cohort, consisted of a sample of 3245 boys, who attended the School at some stage between 1939 and 1968 and were measured twice a term between the ages of 9 and 19 years (median 42 measurements per boy, interquartile range 35 to 47, range 1 to 63), a total of 129 508 heights. These were linked to followup data on 1520 adults, some 50 years later, when inter alia height and IGF-1 levels were measured. The cohort has been described in detail previously, [START_REF] Cole | Nonlinear growth generates age changes in the moments of the frequency distribution: the example of height in puberty[END_REF][START_REF] Sandhu | The impact of childhood body mass index on timing of puberty, adult stature and obesity: a followup study based on adolescent anthropometry recorded at Christ's Hospital (1936-1964)[END_REF][START_REF] Sandhu | Timing of puberty determines serum insulin-like growth factor-1 in late adulthood[END_REF] and an inverse association between age at peak height velocity (APHV) and later IGF-1 has been reported. [START_REF] Sandhu | Timing of puberty determines serum insulin-like growth factor-1 in late adulthood[END_REF] This was based on estimates of APHV for each child derived from spline curves fitted to each child's data.

The second data set came from a randomised clinical trial of oxandrolone to increase final height in girls with Turner Syndrome (TS). This is a chromosomal disorder where the second X chromosome is missing or malformed, leading to short stature and primary ovarian failure. One hundred and six TS girls already on a standard dose of growth hormone were randomised to receive either oxandrolone or placebo from One girl dropped out immediately and 13 more later on, while 92 girls remained in the study and 82 had reached final height before the time of analysis. A total of 1321 heights were included in the analysis (median 12 per child, IQR 9 to 17, range 1 to 23). The trial, which also included a second randomisation to early or late oestrogen for induction of puberty, is reported in detail elsewhere. [START_REF] Gault | Syndrome: randomised, double-blind, placebo-controlled trial[END_REF] The SITAR model

The method used to summarise the individual growth curves followed that of Beath,[START_REF] Beath | Infant growth modelling using a shape invariant model with random effects[END_REF] and was also an extension of the model developed independently by the first author and colleagues. [START_REF] Cole | Nonlinear growth generates age changes in the moments of the frequency distribution: the example of height in puberty[END_REF] This is a shape invariant model [START_REF] Stützle | Shapeinvariant modelling of human growth[END_REF][START_REF] Gasser | A method for determining the dynamics and intensity of average growth[END_REF] which involves fitting the following random effects model to the set of height growth curves:

y it = α i + h t -β i exp(-γ i )       (1)
where y it is height for subject i at age t, h(t) is a natural cubic spline curve of height versus age, and α i , β i and γ i are subject-specific random effects. The aim is to choose the values of α i , β i and γ i to make the individual growth curves as similar as possible, i.e. the mean spline curve h(t). Their interpretations are as follows:

• α i is a random height intercept that adjusts for differences in mean height -here it is termed size. Geometrically it can be thought of as a subject-specific shift up or down or translation in the spline curve, with α i smaller for shorter children. Note that its effect is to make growth curves steeper or shallower, so it effectively rotates each curve (this terminology makes clear its similarity to a random slope effect, though strictly it is a scale change rather than a rotation). The model was fitted using the nlme function [START_REF] Pinheiro | nlme: linear and nonlinear mixed effects models[END_REF] in R, [START_REF]R: a language and environment for statistical computing[END_REF] with the age term centred to mean zero, the three parameters fitted as random effects and size and velocity also as fixed effects, and h(t) a natural cubic spline curve with 6 degrees of freedom (d.f.) fitted as fixed effects. Height and age were fitted both untransformed and log transformed, to identify the optimal underlying scales.

•

Growth related to exposure and outcome

In the CHS cohort, the relationships between the SITAR parameters and the outcome measures of height and IGF-1 at follow-up 50 years later were investigated using correlation and regression.

The analysis of the TS clinical trial data involved comparing the SITAR parameters in individuals in the two trial arms. This was done in three ways:

1. comparing the mean values of each growth parameter in the two arms using t-tests;

2. extending the SITAR model to include fixed effects for each parameter by trial arm;

3. fitting the SITAR model to the data for each trial arm separately, to demonstrate differences in the two mean curves.

F o r R e v i e w O n l y 8
The first two approaches test the effect of the oxandrolone intervention on the parameters, while the third shows its effect on the mean growth curve.

Results

The SITAR model

The SITAR model fitted the CHS data best on the scales of height and log age. Table 1 shows the deviances and residual standard deviations (RSD) for the alternative models in the CHS and TS datasets, confirming that for CHS log age provided a far better fit than age (deviance 1460 units smaller), and height was better than log height. Despite this the effect on the RSD was small. For the TS data height and age were slightly better than height and log age by 19 units of deviance.

Table 1 here

In general a shift on the log scale translates to a scaling on the original scale. So a leftright shift on the log age scale corresponds to a shrinking-stretching on the age scale.

Thus for the log age model the age scale can be viewed as elastic and fixed at zero, with the scale shrunk for early puberty and stretched for later puberty.

Figure 2 shows the raw data for the CHS cohort, a series of superimposed growth curves colour-coded to identify individuals. The RSD about the fitted spline curve is 6.6 cm (not shown). Figure 3 shows the incremental effect of adjusting for each of the SITAR parameters in turn, where each subject's growth curve is plotted using the individually adjusted height and age scales. In Figure 3a Thus the estimation of, and adjustment for, the SITAR parameters reduces the RSD from 6.6 cm to 0.7 cm, a reduction in variance of 98.9%. The final RSD is similar to the median RSD for Preece-Baines curves fitted to each individual. [START_REF] Preece | A new family of mathematical models describing the human growth curve[END_REF][START_REF] Ledford | Mathematical models of growth in stature throughout childhood[END_REF] Table 2 gives the standard deviations and correlations of the growth parameter random effects for the model in Figure 3c. Size is in units of cm, tempo is fractional due to the log age scale and velocity is a fractional multiplier (the latter two can multiplied by 100 and viewed as percentages). Multiplying tempo by mean APHV (14.3 years) converts it to years, giving an SD of 0.97 years. The three growth parameters correlate with each other at around +0.3, indicating that boys entering puberty later (positive tempo β) tend to be taller (larger size α) and have a larger peak velocity (γ).

Table 2 here On this basis it can be argued that the three SITAR parameters explain all the differences in growth between individuals, so that they constitute a parsimonious summary of the individual growth curves.

Table 3 summarises the growth parameter random effects for the model in Figure 4.

Size is in units of cm and tempo in years, while again velocity is dimensionless. The tempo-velocity correlation of -0.46 differs from the +0.26 for the Christ's Hospital data, and shows that girls entering puberty early tend to have the largest height velocity. (Endocrinological note -most of the Turner girls did not enter puberty spontaneously, it was induced artificially, so tempo here relates more to the later fall in velocity than the earlier rise.)

Table 3 here

Growth related to exposure and outcome

Table 4 relates the CHS pubertal height parameters to the clinical outcomes 50 years later. It shows summary statistics and correlations with the three SITAR parameters of age, height and IGF-1 as measured at mean age 63 years. [START_REF] Sandhu | Timing of puberty determines serum insulin-like growth factor-1 in late adulthood[END_REF] In addition it gives age at peak height velocity (APHV) as derived individually for each child. [START_REF] Sandhu | Timing of puberty determines serum insulin-like growth factor-1 in late adulthood[END_REF] Table 4 here Size and height are very strongly correlated (r = 0.86), though tempo and velocity also correlate with height to some extent. To clarify their relative contributions, Table 5 shows the multiple regression of later height on the three parameters, where the standardised coefficients indicate the effect on height of a 1 SD increase in each parameter. This confirms that size is by far the strongest predictor (+5.5 cm per SD), Table 5 here Table 4 also shows a strong correlation between tempo and APHV (0.83). So tempo is closely related to APHV though their SDs differ -1.6 cm for APHV versus 0.97 cm for scaled tempo.

The next strongest correlations (0.20-0.22) are for age with tempo, APHV and IGF-1.

These represent secular trends over the period of data collection (1939-68), with puberty becoming earlier and IGF-1 higher. Finally there are smaller correlations of tempo and APHV with IGF-1 (-0.12 and -0.13 respectively, P < 0.0002). Thus the tempo of puberty (but not size or velocity) is strongly correlated with IGF-1 50 years later, and tempo and APHV are equally predictive.

As tempo and IGF-1 are both correlated with age, the tempo versus IGF-1 association could be due to the secular trend. Adjusting for age confirms this, the correlation of IGF-1 with tempo or APHV falling to -0.08 (P = 0.01, N = 1009).

Turning to the TS cohort, Figure 5 compares the individual growth parameter values by trial arm. There are small oxandrolone effects for size (+2.6 cm, 95% CI 0.4 to 4.7, P = 0.02) and tempo (-0.3 years, -0.7 to +0.1, P = 0.1), but velocity is dramatically larger in the oxandrolone arm (+23%, 16% to 30%, P = 10 -8 ). Refitting the SITAR model with fixed parameter effects for differences between trial arms confirms this pattern (not shown). Figure 6 shows the summary spline curves, each with 4 d.f., obtained for the two trial arms separately, and they demonstrate the steeper velocity in

Discussion

The findings show that Beath's shape invariant model is useful for analysing the pubertal height growth curve, both in boys and girls. As well as providing the mean growth curve it summarises the departure of each individual curve from the mean in terms of three biologically meaningful parameters: size, tempo and velocity.

Furthermore the model explains effectively all the heterogeneity in growth between individuals, as the residual variability about the fitted curves is no greater than that obtained by fitting individual Preece-Baines curves. [START_REF] Preece | A new family of mathematical models describing the human growth curve[END_REF][START_REF] Ledford | Mathematical models of growth in stature throughout childhood[END_REF] The name SITAR is proposed for the model as it SuperImposes by Translation And Rotation.

The SITAR model has some similarities with the random intercept-random slope model, in that size is a random intercept and velocity can be viewed as a random slope. However where the two differ is that SITAR expects a nonlinear mean curve, and this allows the tempo random effect to be estimated -if the mean curve were linear the size and tempo effects would be indistinguishable. Thus the most innovative aspect of SITAR is its random shift on the age scale.

The SITAR model should also be useful for endocrinologists in that it clarifies the nature of growth in puberty. The substantial population heterogeneity in pubertal height velocity shows itself as variability in peak height velocity (PHV) and APHV.

However these two quantities are effectively the same as the SITAR parameters velocity and tempo, and the present analysis makes clear that these parameters explain all the differences in growth velocity in both the CHS and TS cohorts. Note that the The correlations between the three growth parameters indicate that CHS boys entering puberty later tend to be taller and have a larger peak velocity (Table 2). TS girls show similar correlations except that tempo and velocity are inversely related, with early puberty associated with greater velocity (Table 3). This accords with the fact that only a minority of Turner girls enter puberty spontaneously -the others have puberty induced artificially with oestrogen therapy, and their growth spurt is much smaller.

The positive correlation between size and tempo appears to contradict the previously reported negative correlation between height at school entry and APHV. [START_REF] Sandhu | Timing of puberty determines serum insulin-like growth factor-1 in late adulthood[END_REF] Clearly the correlations of the growth parameters with height at any particular age depend on the values of the other parameters, as seen in Table 5 for later height. Repeating this analysis for height at different ages in puberty shows that for size, a 1 SD increase predicts a ~6 cm increase in height at all ages (which fits with the SDs of ~6 cm for size in Tables 2 and3). For tempo, a 1 SD increase is associated consistently with a reduction in predicted height, peaking at -7 cm at age 14. Thus around APHV the tempo effect eclipses the size effect. For velocity the effect is smaller, ranging from -2 to +1 cm. So conditional on the other parameters, tempo is consistently inversely associated with height at any particular age, which accords with the earlier finding. [START_REF] Sandhu | Timing of puberty determines serum insulin-like growth factor-1 in late adulthood[END_REF] The SITAR model is useful for relating individual growth parameters to previous exposures and/or later outcomes. The TS example shows how SITAR provides a parsimonious analysis of the growth clinical trial, where the oxandrolone intervention

F o r R e v i e w O n l y
14 is not only highly effective, but also is seen to operate primarily by increasing growth velocity (γ) rather than tempo (β) or size (α). The strength of the oxandrolone effect (P = 10 -8 ) is remarkable given the modest number of growth curves involved (N = 105). Its effect on the size parameter of +2.6 cm is less than on final height (+4.5 cm) as it reflects mean height throughout the study. In fact the Turner study design was more complex than described here, with two separate randomisations in a factorial design, and the net effect of oxandrolone on final height exceeded 7 cm. [START_REF] Gault | Syndrome: randomised, double-blind, placebo-controlled trial[END_REF] The CHS analysis relates the growth parameters to IGF-1 50 years later, and again finds just one of the three parameters to be important, this time tempo. So earlier puberty is associated with higher IGF-1 after a lag of 50 years. The association is exaggerated by secular trends in tempo and IGF-1, but it remains after adjusting for age.

APHV measured from individual growth curves gives the same results as tempo. The high correlation of 0.83 between the two suggests that they are measuring essentially the same quantity, albeit on different measurement scales. However tempo is much easier to estimate, being a random effect from a single analysis whereas APHV involves analysing each individual growth curve.

Discussion of findings

The fact that the model fits well both in boys and girls, even growth disordered girls with Turner Syndrome, highlights its generality. The fit in the CHS boys was far better with a log age scale, where a left-right shift on the log age scale corresponds to a shrinking-stretching on the age scale anchored at birth. This elasticity of the growth curve affects the curve's slope (i.e. velocity), so that it is steeper for individuals with This provides a statistical explanation for the clinical observation that PHV (velocity) is inversely correlated with APHV (tempo). [START_REF] Tanner | Growth at adolescence[END_REF] Thus log age provides a better fit for CHS because it models this correlation. Indeed it over-compensates, as the negative correlation of -0.19 between tempo and velocity on the age scale (result not shown) switches to +0.28 on the log age scale (Table 2).

The log age scale did not provide a better fit for the TS girls, which may be due to their mean velocity curve lacking a clear peak, due to most of them failing to enter puberty spontaneously. Thus there was less opportunity for an association between PHV and APHV.

Strengths and weaknesses

The strengths of the study are the two substantial and contrasting datasets that provide a useful test of the generality of the SITAR approach. In addition they include examples of both exposure-growth and growth-outcome associations, which highlight respectively the velocity and tempo parameters of the model.

A weakness of the study is that the results apply only to height in puberty. Their relevance to height at other ages, e.g. pre-school, or to other measurements, e.g. weight, is not known. That said, Beath's paper describing the method 6 applied it to weight from 0 to 2 years, where it appeared to fit well, though the goodness of fit of the model was not examined in the same way as here.

There 3. The effect on Figure 2 of adjusting in turn for size α (shifting curves vertically, Figure 3a), tempo β (shifting curves horizontally, Figure 3b), velocity γ (shrinking-stretching age scales, Figure 3c), and extreme outliers (Figure 3d). See text for details.

  be applied to all subjects by applying just three subject-specific translations and rotations of the curve to fit the individual subject growth curves.

  9 years (or the age at recruitment if later) until final height was reached.

Figure 1

 1 Figure1illustrates the meanings of the three parameters geometrically, i.e. the three

  tempo parameter β), which shifts individual curves left-right and reduces the RSD to 1.2 cm. Finally in Figure3cthe age scaling is added (the velocity parameter γ), which shrinks-stretches individual curves and reduces the RSD to 0.8 cm. The three graphs show increasingly obvious spiky outliers above and below the main body of data, corresponding to height recording errors at the time of measurement. To adjust for this, Figure3dtrims residuals exceeding 2.4 cm in absolute value (n = 952 or 0.74%) so as to remove the spikes but retain the main body of data. This reduces the RSD to 0.7 cm.

Figure 4

 4 Figure4shows the individual raw and SITAR-adjusted growth curves for the TS

  and velocity are also predictive in opposite directions (-0.6 and +0.5 cm per SD respectively).

  . Analysis of the final height data confirms that oxandrolone increases final height by 4.5 cm (1.9 to 7.2, P = 0.001).[START_REF] Gault | Syndrome: randomised, double-blind, placebo-controlled trial[END_REF]

  which defines mean height and hence is important to set the relative position of the height distance curve, has no effect on the height velocity curve. The SITAR parameterisation shows that differences in height velocity between individuals are entirely explained by differences in their internal "growth body clock".

  and shallower for those reaching puberty later (see Fig 7 of ref 7 ).

4 .

 4 Turner Study data -Left: 105 height growth curves and 1321 heights from 9-19 years as randomised (oxandrolone red solid lines, placebo blue dashed lines). Right: the same curves after SITAR adjustment. See text for details. 5. Turner Study. Comparison of the SITAR growth parameters by trial arm. Velocity is far greater for oxandrolone (P = 10 -8 ). 6. Turner Study. The mean growth curves for the two arms, indicating the greater velocity for oxandrolone.

  β i is a random age intercept to adjust for differences in the timing of the pubertal growth spurt in individuals i.e. the age at peak height velocity, and it is here called

	tempo following Tanner's usage. Geometrically it corresponds to a subject-
	specific left-right shift or translation in the spline curve, with β i negative for early
	puberty and positive for late.
	• γ i is a random age scaling that adjusts for the duration of the growth spurt in
	individuals. Its parameterisation as exp(-γ i ) ensures that both positive and
	negative values are permissible, with zero corresponding to average velocity, and
	the minus sign means that it measures velocity rather than its inverse. (This minus
	sign is the reverse of Beath's notation.) Geometrically it corresponds to a F o r shrinking or stretching of the age scale. Thus for a child with a relatively short
	spurt and a steep growth curve, γ i is positive to stretch the age scale and reduce R the slope (i.e. peak velocity). Conversely if γ i is negative the age scale is shrunk e v and the curve slope increased. For this reason the parameter is termed velocity.
	i e
	w
	O n l
	y

Table 1 .

 1 is a need for future work to test the generality of the SITAR model, particularly in puberty where the heterogeneity of height growth, which appears to depend on the individual's time clock, may be related to the timing of changes in sex hormone Comparison of SITAR models of height on age with and without log transformation. Deviances and residual standard deviations for the two cohorts under the four models, ranked by size for CHS, with the minima shown in bold italic font.

	F o r
	R
	e v
	i e
	w
	O n l
	y

Table 2 .

 2 Christ's Hospital School data: summary statistics of SITAR pubertal growth parameter random effects for 3245 subjects and 129 508 measurements.

		SD Correlations
	Size (cm)	6.1 Size Tempo
	Tempo	0.068 0.33
	Velocity	0.15 0.36 0.28
	Residual (cm) 0.79

Table 3 .

 3 Turner Study data: summary statistics of SITAR pubertal growth parameter random effects for 105 subjects and 1321 measurements.

		SD	Correlations
	Size (cm)	5.9	Size Tempo
	Tempo (years) 1.04 0.29
	Velocity	0.23 0.28 -0.46
	Residual (cm)	0.63

Table 4 .

 4 Christ's Hospital School data: summary statistics and correlations of age at peak height velocity (APHV) and clinic measurements in later life[START_REF] Sandhu | Timing of puberty determines serum insulin-like growth factor-1 in late adulthood[END_REF] with SITAR pubertal growth parameter random effects.

	Correlations

Table 5 .

 5 Christ's Hospital School data: multiple regression of height (cm) in later life on SITAR random effects, adjusted for age of measurement. Illustration of the SITAR model for height in puberty. The solid line is the mean growth curve; the short dashed lines indicate a vertical or height shift in the curve corresponding to size (α); the long dashed lines indicates a horizontal or age shift corresponding to tempo (β), and the dot-dashed lines represent a shrinkingstretching of the age scale reflecting velocity (γ). See text for details. 2. Christ's Hospital School data -3245 height growth curves and 129 508 heights from 9-19 years.

		Standardised
			Standard error	t	95% Confidence interval
		regression coefficient
	(Intercept)	178.2	0.10	1860	178.0 to 178.4
	Size	5.5	0.10	53.1	5.3 to 5.7
	Tempo	-0.59	0.10	-5.7	-0.79 to -0.39
	Velocity	0.54	0.10	5.3	0.34 to 0.74
	Residual SD 3.0 cm, R 2 0.77

Acknowledgements

We thank the reviewer for comments on the paper.

Contributorship

All authors have contributed to the drafting of the paper, and have approved the final version. TJC is guarantor for the paper. TJC also confirms that the references have been checked for accuracy and completeness, and that the manuscript has been carefully edited by someone with an excellent mastery of the English language. The paper has not previously been published (or submitted for publication) in a substantively similar form.

Funding

This work was supported by the Medical Research Council [grant number G0700961

to TJC].

Conflict of interest

None declared.


S Briefi

U Fantz

Simulation of the A -X and B -X transition emission spectra of the InBr molecule for diagnostics in low pressure plasmas

Keywords: numbers: 52.70.Kz, 33.20.Lg, 33.20.Vq, 33.70.Ca, 33.70.Fd

Inductively coupled low pressure discharges containing InBr have been investigated spectroscopically. In order to obtain plasma parameters such as the vibrational and rotational temperature of the InBr molecule, the emission spectra of the A 3 Π 0 + → X 1 Σ + 0 and the B 3 Π 1 → X 1 Σ + 0 transitions have been simulated. The program is based on the molecular constants and takes into account vibrational states up to v = 24. The required Franck-Condon factors and vibrationally resolved transition probabilities have been computed solving the Schrödinger equation using the Born-Oppenheimer approximation. The ground state density of the InBr molecule in the plasma has been determined from absorption spectra using effective transition probabilities for the A -X and B -X transition according to the vibrational population. The obtained densities agree well with densities derived from an Arrhenius type vapour pressure equation.

Introduction

Metal halides such as InBr are discussed to be an efficient molecular radiator in low pressure plasmas. Up to now, rare gas discharges containing mercury in the high and low pressure range are commonly used for lighting purposes. Several properties of mercury are hereby utilized such as a high vapour pressure and the strong and efficient radiation in the visible and especially the ultraviolet spectral range. Recent research both in low and high pressure lamps ( [1,2,3] and [4,5]) focuses on finding efficient substitutes for mercury due to its toxicity. Molecular radiation -especially from metal halides -is under discussion to be a good alternative to mercury [6]. Some of the suggested molecules are already used as additives in common high pressure lamps to achieve a higher colour rendering index [START_REF] Waymouth | Electric Discharge Lamps[END_REF]. The performance of low pressure discharges containing indium halides (InCl, InBr and InI) has been investigated recently [START_REF] Körber | XXVIII Int. Conf. on Phenomena in Ionized Gases[END_REF][START_REF] Hayashi | [END_REF]. The most intense emission bands of their molecular spectrum are in the near UV range (330 -420 nm) which makes the conversion into visible light by a phosphor more efficient compared to mercury in fluorescent lamps due to the lower Stokes shift. These emission bands consist of transitions from the first two excited states to the ground state: A 3 Π 0 + -X 1 Σ + 0 and B 3 Π 1 -X 1 Σ + 0 . The minima of the potential curves of the InBr molecule are at 3.30 eV for the A state and at 3.39 eV for the B state [10]. The resulting band spectrum is in the wavelength range between 350 and 400 nm. It is still under discussion if the third electronic state, the C 1 Π 1 state, is bound or repulsive [11]. The C -X band emission is discussed to be around 285 nm [12]. A plasma which contains InBr (ionization energy E Ion (InBr) = 9.41 eV [13]) also emits intense radiation from indium in the blue spectral range at 410.18 nm and 451.13 nm due to the dissociation of the molecule (dissociation energy of InBr D e = 3.99 eV [10]).

InBr has a very low vapour pressure at ambient temperature [START_REF] Körber | XXVIII Int. Conf. on Phenomena in Ionized Gases[END_REF]. To obtain a significant partial pressure of InBr in a plasma, the discharge vessel has to be heated up to several hundred degrees centigrade. The amount of InBr in the plasma which strongly influences the intensity of the molecular emission is determined by the temperature of the coldest spot of the vessel wall.

For lighting purposes, the aim is to maximize the molecular near UV radiation of a rare gas plasma containing InBr at a fixed discharge power. Therefore it is important to investigate the population mechanisms of the involved electronic states in the molecule. The vibrational and rotational population of the electronic states determine the shape of the molecular band spectrum and, as the single vibrational levels have different radiative lifetimes, also the intensity of the emission. A well-known example of these characteristics are nitrogen bands [14]. Spectroscopic diagnostics are common techniques to determine the influence of external parameters such as pressure, power and heating temperature on the population densities and on the emission respectively. For InBr, however, the vibrational and rotational population cannot be obtained from an analysis of individual lines from the measured spectrum because of the manifold of overlapping ro-vibrational lines of the two electronic transitions.

The aim of this paper is to determine the rotational and vibrational population of the molecule from emission spectra by comparing the measured relative intensity of the band with simulated spectra. With the knowledge of these populations the ground state density of InBr can be obtained from absorption spectroscopy measurements. First, the simulation procedure will be described. The required Franck-Condon factors and vibrational resolved transition probabilities of InBr have not been available in literature and were determined numerically. Examples of simulated spectra with different spectral resolution of the optical system and with varying vibrational and rotational temperatures are given in section 2.4. Section 3 describes the application to low pressure plasmas and presents the results together with a discussion on the accuracy of the determination of the vibrational and rotational temperature. In addition, the ground state density of InBr obtained from absorption measurements is compared with the density calculated from an Arrhenius type vapour pressure equation.

Simulation of InBr emission spectra

The band spectrum of InBr between 350 and 400 nm arises from overlapping emission of the electronic A -X and B -X transition and can be seen easily in the measurements presented in section 3. The C -X band spectrum could not be observed. However, it does not contribute to the near UV radiation and, therefore, the focus is put on the A -X and B -X transition. Due to the large mass of InBr (atomic weight 194.722 u) the molecular constants and hence the energy differences between vibrational and rotational states characterized by their quantum numbers v and J are very small (∆E v ≈ 0.03 eV, ∆E J ≈ 3 • 10 -5 eV). States with high quantum numbers v and J can be populated with rather small amounts of energy and many emission lines occur in a narrow wavelength range. Due to the Doppler broadening of emission lines it is only possible to resolve the structure of the sequences (transitions with fixed ∆v = v ′ -v ′′ ) of the band spectrum even with a high resolution spectrometer (∆λ FWHM ≈ 25 pm). This is demonstrated in Figure 1 which already shows simulated spectra with a FWHM of the apparatus profile of 25 pm. The simulation was performed using the procedure described in the following section.

Computing method

As the energy difference between the individual states is very low, it is necessary to take into account states with high vibrational and rotational quantum numbers for the X, A and B state. First investigations showed that it is sufficient for a good match of simulated and measured spectra to limit the calculation to the energetically lowest 25 vibrational states and 300 rotational states in each electronic state. Significant deviations have been observed by reducing the vibrational quantum number below v = 20. The energy of the rovibrational states of the X, A and B state is calculated using the molecular constants taken from [10]. The wavelength of a transition from the upper electronic state i (A or B) with vibrational and rotational quantum numbers v ′ and J ′ to the lower state k (the ground state X) with v ′′ and J ′′ is derived from the energy difference of the states.

The relative vibrational population of the ground state is calculated assuming a Boltzmann distribution with the vibrational temperature T Vib . This assumption is reasonable due to the small energy difference between the vibrational states (about 0.03 eV). For the excited states, the relative vibrational population n v ′ i is derived from the population of the ground state applying the Franck-Condon principle [15]. To obtain the relative population n J ′ i,v ′ of the rotational states with the vibrational quantum number v ′ of the excited electronic states, again a Boltzmann distribution characterized by the rotational temperature T Rot is used. The relative population of a single state with v ′ 0.0 0.5 and J ′ , n v ′ ,J ′ i , is then given by

1.0 v = -2 v = -1 v = 0 v = +2 v = +1 v = -3 v = -2 v = -1 v = 0 v = +1 v = +2
n v ′ ,J ′ i ∝ n v ′ i (T Vib ) • n J ′ i,v ′ (T Rot ). (1)
The relative populations are normalized that the population of the whole electronic state is equal to unity:

v ′ n v ′ i = 1 and J ′ n J ′ i,v ′ = n v ′ i . (2)
According to [START_REF] Herzberg | Molecular Spectra and Molecular Structure: I. Spectra of Diatomic Molecules[END_REF] the relative intensity

I ik,v ′ v ′′ ,J ′ J ′′ of the transition (i, v ′ , J ′ ) → (k, v ′′ , J ′′ ) is determined by I ik,v ′ v ′′ ,J ′ J ′′ ∝ ν d i n v ′ ,J ′ i A v ′ v ′′ ik S P,Q,R (J ′ ), (3)
where ν denotes the frequency of the emitted photon, d i the degeneracy, n v ′ ,J ′ i the population density in the state i and A v ′ v ′′ ik the vibrationally resolved transition probability. S P,Q,R (J ′ ) is the Hönl-London factor for the P,Q or R branch, which describes the intensity of the emission line depending on the rotational quantum numbers of the states involved [START_REF] Herzberg | Molecular Spectra and Molecular Structure: I. Spectra of Diatomic Molecules[END_REF]. The Hönl-London factors depend on the Hund's coupling cases of the electronic states which are involved in the transition. As InBr is a very heavy molecule Hund's case (c) is valid for both the A -X and the B -X transition [10]. Furthermore, both transitions are singlettriplet transitions which means spinorbit interactions may lead to perturbations in the intensity distribution of rotational lines [START_REF] Kovács | Rotational Structure in the Spectra of Diatomic Molecules[END_REF]. The exact formulae for the Hönl-London factor for the A -X and B -X transition can be found in [START_REF] Kovács | Rotational Structure in the Spectra of Diatomic Molecules[END_REF]. They contain constants which describe the magnitude of perturbation due to the spin-orbit interaction. In the case of InBr, these constants are not available in literature. Therefore the Hönl-London factors of Hund's case (a) are used in the simulation as an approximation since the resultant angular momentum from the coupling of the single electronic and rotational angular momenta for the electronic state is the same as in case (c) [START_REF] Herzberg | Molecular Spectra and Molecular Structure: I. Spectra of Diatomic Molecules[END_REF]. Thus, the spin-orbit interactions are neglected. The comparison of simulated spectra to measurements has to demonstrate the validity of this approximation.

For each transition line a Gaussian line profile is calculated with the peak of the profile at the wavelength of the transition and folded with the relative intensity of the line emission. The FWHM of the line profile is an input parameter of the simulation. As only low pressure plasmas are investigated the dominating line broadening mechanism is the Doppler broadening. Due to the large mass of the InBr molecule the FWHM of the Doppler profile is still much smaller than the apparatus profile of the optical system. Therefore the FWHM used for the simulation is given by the experimental setup. It is determined by utilizing the nearby indium lines. To obtain a spectrum, the intensities of all computed lines are added up. The relative intensities of the two electronic transitions can be adjusted separately to consider the possibility of a different electronic population of the A and the B state.

The InBr molecule basically consists of two relevant isotopic species, 115 In 79 Br with natural abundance of 48.51% and 115 In 81 Br with 47,19%. Therefore the emission is simulated separately for both species and then added, taking into account the natural abundances to obtain the final spectrum. The isotopes 113 In 79 Br (2,18%) and 113 In 81 Br (2,12%) are neglected due to their low abundance.

Franck-Condon factors and transition probabilities

The simulation requires vibrationally resolved Franck-Condon factors (F CF ) and transition probabilities A v ′ v ′′ ik as input parameters. F CF of the InBr molecule can be found in literature only for vibrational quantum numbers up to v = 13 [START_REF] Singh | [END_REF] and radiative lifetimes can be found only for selected vibrational levels of the A or B state [START_REF] Körber | XXVIII Int. Conf. on Phenomena in Ionized Gases[END_REF]. Therefore F CF and A v ′ v ′′ ik were calculated using the program TraDiMo [19]. This program derives the eigenvalues and vibrational wave functions via numerically solving the Schrödinger equation. The calculations are based on the Born-Oppenheimer potential curves, the electronic dipole transition moments and the reduced mass of the molecule as input parameters. The overlap integral of two vibrational wave functions in different electronic states yields the F CF . The A v ′ v ′′ ik are calculated from the overlap of the vibrational wave functions with the electronic dipole transition moments. A more detailed description of TraDiMo can be found in [19].

There are three complete sets of potential curves for the X, A and B state of InBr available in literature: RKR potential curves [START_REF] Singh | [END_REF], Hulburt-Hirschfelder potential curves [20] and potential curves from MRDCI calculations [11]. The RKR potential curves are only given for vibrational quantum numbers up to v = 12. As already mentioned, for a good match of the simulated spectrum and the measurement vibrational states up to v = 24 have to be taken into account. Hence, Morse potential curves were calculated using the molecular constants of InBr [10] which reproduced the RKR potential curves quite well. Morse potential curves are a very good approximation close to the minimum of the potential curve and therefore for low vibrational energies. The energy of the state with vibrational quantum number v = 24 of InBr is still rather low due to the large mass of the molecule. TraDiMo calculations were carried out with all three sets of potential curves: with the Morse potential curves as extended fit to the RKR curves, with the Hulburt-Hirschfelder curves and with those from the MRDCI calculations. The electronic dipole transition moments D AX (r) and D BX (r) for the A -X and B -X transition are taken from [START_REF] Körber | XXVIII Int. Conf. on Phenomena in Ionized Gases[END_REF]. They are empirically determined and given as:

D AX (r) = D AX (r eX ) 1 + 4 • r r eX -1 D BX (r) = D BX (r eX ) (4)
where r eX is the equilibrium nuclear distance of the ground state and with D AX (r eX ) = 0.0705 a.u. and D BX (r eX ) = 0.053 a.u.. As the different potential curves deviate from each other the obtained sets of F CF and A v ′ v ′′ ik also show deviations. Simulations were carried out with all sets of data and compared to measured emission spectra. The best results by far regarding the match of the relative intensity of simulated and measured spectra were achieved with the data derived from the Morse potential curves.

Uncertainties of the F CF or transition probabilities are determined by the accuracy of the input parameters (potential curves and electronic dipole transition moments). If the potential curves vary slightly, different absolute values of the vibrational eigenvalues are computed [19]. Small changes of the eigenvalues lead to deviations in the wave functions and hence to differences in the F CF . The deviation depends on the absolute value of the F CF and transition probabilities (the smaller the absolute value, the larger the deviation) which means the precision is rather determined by the digit than a percentaged value [19].

Since the Morse potential curves depend on molecular constants, slightly different curves are obtained for the isotopic species. Calculations of the F CF and A v ′ v ′′ ik have been carried out on the one hand with the mass of both relevant isotopes ( 115 In 79 Br and 115 In 81 Br) and on the other hand with the averaged mass. The simulation of the band spectrum using either the data gained considering the isotopic species or the averaged mass only differ in the range of a few percent. So, for simplicity Franck-Condon factors and transition probabilities derived with the average mass are used for the simulation of the relative intensity. These F CF and A v ′ v ′′ ik can be found in the tables A1 to A8 in the Appendix. Considering the uncertainty discussion above, all calculated Franck-Condon factors that are smaller than 1 • 10 -5 and all transition probabilities smaller than 1 s -1 are replaced by zero.

In [START_REF] Singh | [END_REF] F CF up to v = 13 of the A -X and B -X transition of InBr are calculated using the RKR potential curves published in the same paper. Table 1 compares the F CF calculated via TraDiMo to the data taken from [START_REF] Singh | [END_REF], which is denoted in brackets, up to a vibrational quantum number of v ′′ = v ′ = 3 for the A -X transition. Even though the Morse potential curves used for the TraDiMo calculations reproduce the RKR curves well the potential curves diverge slightly. As discussed above, this leads to differences in the calculated F CF . However, it can be seen that for the relevant Franck-Condon factors (high absolute values) the difference is only a few percent. The same consideration is valid for the F CF of the B -X transition.

Table 1. Comparison of the derived F CF to the data taken from [START_REF] Singh | [END_REF] (in brackets) for the first four vibrational states in the A -X transition.

v ′ → v ′′ v ′′ = 0 v ′′ = 1 v ′′ = 2 v ′′ = 3 v ′ = 0 0.
i v ′ = v ′′ A v ′ v ′′ ik -1
with those available in literature [START_REF] Körber | XXVIII Int. Conf. on Phenomena in Ionized Gases[END_REF]. The lifetimes differ slightly, but unfortunately in [START_REF] Körber | XXVIII Int. Conf. on Phenomena in Ionized Gases[END_REF] no specification of the input data of the calculation and no details about the calculation itself is given.

Table 2. Calculated radiative lifetimes τ i v ′ of selected vibrational levels for two electronic states. For comparison, the values taken from [START_REF] Körber | XXVIII Int. Conf. on Phenomena in Ionized Gases[END_REF] are denoted in brackets.

State

A B v ′ = 0 6.81 µs (6.4 µs) 9.24 µs (8.7 µs) v ′ = 10 4.55 µs (4.5 µs) 9.20 µs (9.3 µs)

Effective transition probabilities

The determination of the population density of the electronic states of InBr with spectroscopic measurements requires the knowledge of the effective transition probability A eff ik , where i denotes the A or B state and k the X state. It can be obtained by weighting the inverse lifetimes of the single vibrational levels (

τ i v ′ ) -1 = v ′′ A v ′ v ′′ ik
with a vibrational population according to Boltzmann:

A eff ik (T Vib ) = v ′ v ′′ A v ′ v ′′ ik • n v ′ i (T Vib ) with v ′ n v ′ i (T Vib ) = 1. (5)
The vibrational temperature is taken from the fit of the simulated spectrum to the measurement. Table 3 shows effective transition probabilities calculated with different T Vib . It can be seen that the A eff ik for the A -X transition (which has a electronic dipole transition moment varying over the internuclear distance [START_REF] Körber | XXVIII Int. Conf. on Phenomena in Ionized Gases[END_REF]) changes with increasing vibrational temperature whereas for the B -X transition (which has a constant electronic dipole transition moment [START_REF] Körber | XXVIII Int. Conf. on Phenomena in Ionized Gases[END_REF]) the A eff ik is almost constant.

Influence of input parameters on simulated spectra

With the simulation program it is now possible to determine the influence of the vibrational and rotational temperature on the structure of the emission band. Furthermore, spectra simulated with a rather wide FWHM of the line profile (1 nm, low resolution, typical for a broad band spectrometer) can be compared to those calculated with the same parameters but a narrow FWHM of 0.025 nm (high resolution spectrometer). Figure 2 shows simulated spectra with fixed T Vib and two different T Rot for low and high spectral resolution. Rotational temperatures of 500 K and 1000 K are used as this temperature range is typical for low pressure plasmas. In case of the low resolution spectra the rotational temperature hardly changes the structure of the band.

Even for low T Rot the sequences are overlapping strongly. In the case of high resolution the spectrum simulated with higher rotational temperature shows a distinct broadening of the single sequences which overlap only slightly.

Varying the vibrational temperature with low spectral resolution leads to a strong change of the relative intensity of the sequences and to a slight broadening of the sequences. This can be seen in figure 3. The change in the relative intensity can also be seen with high resolution. However, the broadening can now be identified to be caused by the population of states with higher vibrational quantum number in the case of increased vibrational temperature. Transitions from higher vibrational quantum number are located in the shoulder of the single sequences. This leads to a broadening of the whole sequence in spectra calculated with low resolution as these transitions cannot be resolved. With high resolution, these transitions occurring with high vibrational temperature can be seen at 360 and 370 nm, for example.

Thus, a measurement with a high resolution is advantageous for the determination of the vibrational and the rotational temperature of the InBr molecule.

All measurements presented in this paper are carried out with a high resolution spectrometer with a FWHM of the apparatus profile of 25 pm.

Applications to low pressure plasmas

Experimental setup

Rare gas discharges containing InBr have been generated via inductively RF-coupling at a frequency of 13.56 MHz and a generator power of 100 W in sealed cylindrical quartz cells. These cells are filled with a few mg InBr salt and Helium or Argon at a pressure of a few mbar. As InBr has a very low vapour pressure at ambient temperature, the whole cell was heated up to several hundred degrees centigrade using hot air flowing through a heat container. The amount of InBr in gas phase and in the plasma is determined by the coldest spot of the wall of the cell which is not welldefined. The temperature of the hot air inside the heat container is measured with three thermocouples which are placed on the left, in the middle and on the right part of the heat container. The heating temperature T H of the vessel is defined as the averaged measured temperatures of the three thermocouples. Figure 4 shows a schematic view of the experimental setup. The line-of-sight (LOS) for optical emission and absorption spectroscopy is axial (radial centered). A high resolution spectrometer with a focal length of 750 mm, a 1800 lines/mm grating and a CCD detector resulting in a FWHM for the apparatus profile of 25 pm is used. For white light absorption measurements a stabilized high-pressure Xe-lamp (450 W) is applied.

Comparison of measurements with simulation

Measurements have been carried out with Argon and Helium as background gas. The emission spectrum of these plasmas in the actively heated case is dominated by the InBr band (A -X and B -X transition) and by atomic indium lines arising from the dissociation of InBr whereas the emission from the background gas only plays a minor role. Figure 5 shows exemplary two measured spectra of InBr with different external parameters (different heating temperature, type and pressure of background gas) and the simulation which gives the best fit to the measurement. It can be seen that the structure of the band spectrum, and therefore T Vib and T Rot , varies strongly. The simulation Figure 6 shows the ∆v = 0 sequence of the A -X transition of the measurement in Argon and the appendant simulation in detail. The double peak structure of the sequence is well reproduced. In literature [START_REF] Körber | XXVIII Int. Conf. on Phenomena in Ionized Gases[END_REF], however, it was discussed if this "asymmetric shape" is caused by population transfer out of metastable states into certain vibrational levels. Since the simulation presented in this paper is only based on molecular constants, such transfer mechanism seems not to be the cause for the double peak structure.

Determination of the InBr density

An important parameter of InBr discharges which strongly influences the intensity of the band emission is the ground state density of the molecule. It is determined by the coldest spot of the wall of the quartz cell which is not well-defined in this experimental setup. However, the ground state density of InBr can be derived from absorption spectroscopy measurements using the effective transition probabilities A eff ik of the A -X or the B -X transition. As shown in section 2.3 the calculation of the A eff ik needs the vibrational temperature of InBr as input parameter which is gained from adjusting the simulation to the measured band spectrum. If the amount of InBr is very high, reabsorption of the intense sequences in the emission spectrum occurs. This can distort the shape of the spectrum drastically as shown in figure 7 and make a determination of the vibrational and rotational temperature with the simulation program impossible. In this case, however, the ground state density of InBr can still be obtained via absorption spectroscopy by using only the B -X transition since the effective transition probability A eff BX is almost constant for all vibrational temperatures. For low densities of InBr, which means negligible reabsorption effects, the A and the B state can both be analyzed using the individual effective transition probabilities. The resulting densities differ only by a few percent. Therefore it is reasonable to use only the B -X transition for the determination of the ground state density of InBr via absorption spectroscopy. To compare the population density of the ground state determined via absorption spectroscopy to the density calculated from an Arrhenius type vapour pressure equation [START_REF] Körber | XXVIII Int. Conf. on Phenomena in Ionized Gases[END_REF], the lowest temperature measured by the three thermocouples T low was used as an approximation for the coldest spot temperature (see table 4). Figure 8 shows measured densities in a discharge with Argon as background gas as a function of the heating temperature. A very good agreement is observed at high T H . For low heating temperatures the difference between the values obtained by the two methods increases. At T H = 300 • C the deviation is about a factor 4 which indicates that the approximation of using the heating temperature as coldest spot temperature is only valid for high heating temperature values. The distance between the thermocouple and the discharge cell is a few centimeter. Therefore the measured heating temperature can underestimate the cold spot temperature due to temperature gradients occurring especially for low heating temperatures. Figure 8. Ground state density of InBr with increasing heating temperature T H of the cell determined on the one hand with absorption spectroscopy (using only the B -X transition) and on the other hand with the vapour pressure equation [START_REF] Körber | XXVIII Int. Conf. on Phenomena in Ionized Gases[END_REF] using the lowest temperature measured by the three thermocouples T low as approximation for the coldest spot of the cell wall.

Conclusion

Recent research in lighting technology focuses on finding efficient substitutes for mercury in low and high pressure lamps due to the toxicity of Hg. Metal halides such as InBr are discussed as an efficient molecular radiator in low pressure plasmas. Thus, low pressure rare gas discharges containing InBr have been investigated spectroscopically with focus on the near UV band emission arising from the A 3 Π 0 + → X 1 Σ + 0 and the B 3 Π 1 → X 1 Σ + 0 transitions. The absolute intensity of this emission strongly depends on the amount of InBr in the plasma whereas the relative intensity is determined by the vibrational and rotational population of the electronic states. To determine these populations, the relative band emission has been simulated using Boltzmann distributions for the vibrational and rotational states. The simulation requires a complete set of Franck-Condon factors and vibrationally resolved transition probabilities which was not available in literature so far. Therefore this data has been derived by applying the Schrödinger equation to Born-Oppenheimer potential curves of the X, A and B states of InBr. The simulation matches the measured spectra very well; the vibrational and rotational temperature of the molecule are used as fitting parameters. The dependencies of the spectra on vibrational and rotational temperature have been demonstrated. Using high resolution measurements, these temperatures can be determined with high accuracy. It has been pointed out that the simulation is not applicable if reabsorption of the emission distorts the shape of the band spectrum. Knowing the vibrational temperature, an effective transition probability for the electronic A -X or B -X transition can be calculated. For the A state, the transition probability changes with varying T Vib whereas the transition probability of the B state is nearly independent of the vibrational temperature. Using these transition probabilities together with white light absorption spectroscopy measurements the ground state density of InBr in the plasma has been determined. Even if the emission spectra are influenced by reabsorption effects, the B -X transition can still be used to determine the InBr ground state density. The results are in good agreement with the densities calculated from an Arrhenius type vapour pressure equation and are independent on the knowledge of any experimental parameters such as the temperature of the coldest spot of the discharge cell wall. In summary, it has been demonstrated that the simulation of the relative intensity of the emission spectrum of InBr allows the determination of the vibrational and rotational temperature of the molecule in optically thin low pressure plasmas and the ground state density of InBr via absorption spectroscopy with high sensitivity.

Table A5. Transition probabilities

A v ′ v ′′ ik (s -1 ) for the A -X transition, part 1 (from v ′′ = 0 to v ′′ = 12). v ′ v ′′ = 0 v ′′ = 1 v ′′ = 2 v ′′ = 3 v ′′ = 4 v ′′ = 5 v ′′ = 6 v ′′ = 7 v ′′ = 8 v ′′ = 9 v ′′ = 10 v ′′ = 11 v ′′ =
0 0 0 0 0 0 0 0 1 13 21 20 0 0 0 0 0 0 0 0 0 0 0 0 27 21 0 0 0 0 0 0 0 0 0 0 0 0 0 22 0 0 0 0 0 0 0 0 0 0 0 0 1 23 0 0 0 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 Table A6. Transition probabilities A v ′ v ′′ ik (s -1 ) for the A -X transition, part 2 (from v ′′ = 13 to v ′′ = 24). v ′ v ′′ = 13 v ′′ = 14 v ′′ = 15 v ′′ = 16 v ′′ = 17 v ′′ = 18 v ′′ = 19 v ′′ = 20 v ′′ = 21 v ′′ = 22 v ′′ = 23 v ′′ =

Figure 1 .

 1 Figure 1. Simulated emission spectra of InBr. Upper part: only A -X transition; center: only B -X; bottom part: both transitions. The most intense vibrational sequences of each transition are annotated.

Figure 2 .

 2 Figure 2. Simulated emission spectra with varying T Rot for low (upper part) and for high spectral resolution (bottom part). T Vib is fixed at 1000 K.

Figure 3 .

 3 Figure 3. Simulated emission spectra with varying T Vib for low (upper part) and for high spectral resolution (bottom part). T Rot is fixed at 500 K.

Figure 4 .Figure 5 .

 45 Figure 4. Sketch of the experimental setup.

Figure 6 .

 6 Figure 6. Simulation and measurement of the ∆v = +1 sequence of the A -X transition.

Figure 7 .

 7 Figure 7. Examples of the emission band structure of InBr without (upper part) and with reabsorption caused by high InBr ground state densities at high heating temperatures (bottom part).

ik (s - 1 )

 1 for the B -X transition, part 1 (from v ′′ = 0 to v ′′ = 12).v ′ v ′′ = 0 v ′′ = 1 v ′′ = 2 v ′′ = 3 v ′′ = 4 v ′′ = 5 v ′′ = 6 v ′′ = 7 v ′′ = 8 v ′′ = 9 v ′′ = 10 v ′′ = 11 v ′′ = 12 0 679

Table 2

 2

shows a comparison of the calculated radiative lifetimes of a vibrational level τ

Table 3 .

 3 Effective transition probabilities A eff ik calculated for different vibrational temperatures.

	T Vib (K) A eff AX (10 5 s -1 ) A eff BX (10 5 s -1 )
	400	1.560	1.084
	600	1.603	1.084
	800	1.647	1.085
	1000	1.692	1.085
	1200	1.736	1.084
	1400	1.776	1.083

Table 4 .

 4 Heating temperature and the lowest measured temperature inside the heat container which is used as an approximation for the coldest spot temperature.

	T H (K) T low (K)
	299	269
	324	288
	346	298
	362	319
	377	337
	390	356

Table A7 .

 A7 Transition probabilities A v ′ v ′′

	8	2980	922	218	36	4	0	0	0	0	0	0	0
	9	6992	3227	1053	264	44	5	0	0	0	0	0	0
	10	8518	6941	3389	1170	303	53	7	0	0	0	0	0
	11	3563	7675	6807	3487	1258	337	61	8	0	0	0	0
	12	2410	3118	6893	6624	3524	1318	366	66	9	0	0	0
	13	2277	3309	2932	6157	6440	3492	1364	378	73	8	0	0
	14	678 57	766	4409	2991	5465	6257	3419	1378	387	75	8	0
	15 1387 30	801 62	23	5823	3324	4786	6108	3299	1371	388	75	8
	16	3264 1324 10	939 84	429	7699	4001	4093	6017	3136	1346	381	71
	17	122 42	345 1208 00 1083 50	2368	102 30	5152	3357	6021	2917	1318	359
	18	484	158 22	638 1037 50 1217 80	6147	136 23	6962	2571	6146	2651	1280
	19	796	78	184 48	5308	820 18 1323 60	118 71	180 93	9706	1745	6447	2325
	20	3	1153	78	192 40	148 82	575 04 1378 40	192 94	237 82	137 23	941	6993
	21	43	7	1427	916	175 22	287 94	332 87 1361 30	277 02	306 78	194 10	285
	22	1	57	77	1458	2940	132 68	450 68	133 20 1258 50	358 16	384 61	271 37
	23	1	5	57	263	1162	6178	7445	603 65	1687 1069 40	419 43	464 04
	24	0	1	16	39	583	597	101 11	2114	704 84	1364	812 57	443 05

Acknowledgments

The authors would like to thank Dirk Wünderlich from the Max-Planck-Institut für Plasma Physik in Garching, Germany, for his kind assistance in using the TraDiMo package.

Table A1. Franck-Condon factors for the A -X transition, part 1 (from v ′′ = 0 to v ′′ = 12).

v ′′ = 10 v ′′ = 11 v ′′ = 12 0 0.581 82 0.293 43 0.094 45 0.024 06 0.005 09 0.000 98 0.000 16 0.000 02 0 0 0 0 0 1 0.343 17 0.136 01 0.279 79 0.161 50 0.058 67 0.016 24 0.003 75 0.000 73 0.000 13 0.000 02 0 0 0 2 0.069 79 0.407 45 0.011 62 0.193 27 0.185 00 0.090 80 0.031 16 0.008 56 0.001 90 0.000 37 0.000 06 0.000 01 0 3 0.005 10 0.148 56 0.371 48 0.003 81 0.112 48 0.178 04 0.113 60 0.047 24 0.014 91 0.003 80 0.000 81 0.000 14

Franck-Condon factors for the A -X transition, part 2 (from v ′′ = 13 to v ′′ = 24).

0.000 04 0 0 0 0 0 0 0 0 0 0 0 5 0.000 49 0.000 08 0.000 01 0 0 0 0 0 0 0 0 0 6 0.003 47 0.000 76 0.000 14 0.000 02 0 0 0 0 0 0 0 0 7 0.015 95 0.004 71 0.001 09 0.000 21 0.000 03 0 0 0 0 0 0 0 8 0.049 01 0.019 25 0.005 98 0.001 46 0.000 29 0.000 04 0 0 0 0 0 0 9 0.093 80 0.053 38 0.022 20 0.007 28 0.001 86 0.000 38 0.000 05 0 0 0 0 0 10 0.095 37 0.093 69 0.056 61 0.024 84 0.008 49 0.002 26 0.000 48 0.000 07 0 0 0 0 11 0.029 95 0.087 29 0.092 56 0.058 88 0.027 03 0.009 60 0.002 64 0.000 58 0.000 08 0 0 0 12 0.013 51 0.025 30 0.080 08 0.090 79 0.060 27 0.028 72 0.010 56 0.002 98 0.000 67 0.000 10 0 0 13 0.039 62 0.017 03 0.022 57 0.073 71 0.088 83 0.060 80 0.030 03 0.011 27 0.003 30 0.000 73 0.000 11 0 14 0.203 12 0.028 14 0.020 54 0.021 48 0.068 16 0.086 76 0.060 71 0.030 83 0.011 82 0.003 52 0.000 79 0.000 12 15

0.000 02 0 0 0 0 0 0 0 0 0 0 0 5 0.000 26 0.000 04 0 0 0 0 0 0 0 0 0 0 6 0.002 05 0.000 40 0.000 06 0.000 01 0 0 0 0 0 0 0 0 7 0.010 20 0.002 69 0.000 56 0.000 09 0.000 01 0 0 0 0 0 0 0 8 0.034 74 0.012 09 0.003 37 0.000 72 0.000 13 0.000 01 0 0 0 0 0 0 9 0.077 80 0.037 65 0.013 80 0.003 98 0.000 89 0.000 16 0.000 02 0 0 0 0 0 10 0.100 97 0.078 70 0.039 78 0.015 17 0.004 55 0.001 03 0.000 19 0.000 02 0 0 0 0 11 0.068 45 0.094 87 0.078 80 0.041 05 0.016 26 0.004 99 0.001 17 0.000 21 0.000 02 0 0 0 12 0.000 91 0.066 61 0.088 68 0.078 37 0.041 56 0.017 00 0.005 31 0.001 26 0.000 23 0.000 02 0 0 13 0.001 67 0.002 63 0.067 75 0.082 33 0.077 72 0.041 34 0.017 41 0.005 49 0.001 33 0.000 23 0.000 02 0 14 0.446 64 0.000 04 0.005 28 0.072 09 0.075 64 0.077 12 0.040 42 0.017 54 0.005 52 0.001 35 0.000 23 0.000 02 15 0.171 19 0.472 99 0.003 37 0.009 05 0. (s -1 ) for the B -X transition, part 2 (from v ′′ = 13 to v ′′ = 24).