Absorption lines equivalent width

FIGURE 14-8 (a) Meaning of equivalent width, W (b) Doppler and Lorentzian line-shapes for equivalent half-widths (c) transmission curves for an absorption line for a weak and strong absorber, respectively (adapted from Lenoble, 1993). 

In absorption spectroscopy it is more useful to define the equivalent width of a spectral line as the width of an idealized, totally absorbing rectangular line having the same area (Fig. 4). The equivalent width of an isolated absorption line is therefore given by... 

The equivalent width W of an isolated absorption line is independent of spectrometer broadening effects. 

Fig. 3 Upper traces, apparent peak absorptance vs AxDop/Axres, the Doppler width per unit resolution. Each trace is identified by the actual peak absorptance. Lower traces, percentage error incurred when (Ax op + Axr2es)1/2 is used to approximate Axobs for an absorption line vs A cDop/Ajcres. The curves are labeled with the appropriate equivalent width per unit Doppler width as EQW/DOPW.

Ion abundances can be obtained either from the equivalent widths of the absorption lines, or from their residual intensity at line center, r. The two parameters are basically equivalent in that they are directly related to each other, although the use of is preferable because it is directly proportional to abundance in a model-independent way when the lines are unsaturated, e.g., finite spectral resolution increases the observed residual line intensity, but not the equivalent width. In the supernova the lines are all easily resolved, and the residual intensity at the bottom of the lines is more readily measured than W, hence we use ri to obtain the abundances. 

In the optical region a detailed measurement of the line profiles is usually difficult, and for absorption lines it has therefore become usual to measure an absorption line by its equivalent width Wv defined in frequency units (Wv =... 

Hence the highest sensitivity is obtained with large A (equivalent to large H) and low temperature. At the energy corresponding to A"-band resonance, and at room temperature, the ratio is about 0.998. It may be shown that the ultimate sensitivity of a spectrometer working in the A -band at room temperature is approximately 10 spins where AH is the width of the absorption line, and t is the time... 

Absorption lines are defined relative to the continuum. In the case of a resolved line, one may describe the line in terms of its depth relative to the local continuum across the line, say R(AA) = F[(A )/FC where AA is the wavelength measured from line centre, Fc is the flux in the continuum, and F) is the flux at a wavelength within the line. The total absorption by the line obtained by integrating R(AA) over the line profile is known as the equivalent width W. When a line profile is not resolved yet unaffected by blending from neighbouring lines, the equivalent width is independent of the resolution even though the line profile is set by the instrumental profile and not by the intrinsic stellar profile. 

Development of the W — fN H relation, the CoG, beyond the weak-line limit depends on the profile of the absorption coefficient <)>(AA). An extreme form for the profile is effective at illustrating this point. Suppose (AA) = a for AA = AXD and 0 for AA > AAj> Normalization of provides the relation connecting the constant a, the width AXd, and the /-value - the derivation is left as an exercise for the student With increasing fN H, I(AX)/Iq falls within the line to its minimum value of zero. At which point, the equivalent width has saturated at W = 2AAd- Note that, unlike W in the weak-line limit, the CoG beyond the weak-line portion depends on the shape of the line absorption coefficient - here, the width AAd. This dependence means that conversion of a measured W to JNlH for realistic absorption coefficient profiles demands observational or theoretical knowledge of the absorption coefficient s profile. This requirement plus the reduced sensitivity of W to /NlH make this part of the CoG less attractive for abundance determinations. This stretch of the CoG is variously referred to as the flat, Doppler or saturated part. 

In deriving chemical abundances in QSO absorbers and high redshift galaxies, we shall make use of some of the same techniques which are applied locally to interpret the spectra of stars, cool interstellar gas and H II regions. These methods are discussed extensively in other articles in this volume, particularly those by Don Garnett, David Lambert, and Grazyna Stasiriska, and will therefore not be repeated here. The derivation of ion column densities from the profiles and equivalent widths of interstellar absorption lines is discussed in a number of standard textbooks, as well as a recent volume in this series (Bechtold 2002). 

There are many frequently measured quantities that are related to the absolute strength of an absorption line or band (Pugh and Rao, 1976). The fundamental quantity to which the others are related is the integrated cross section, <7°-, which has units of area times frequency (often cm2 cm-1 molecule-1 or, equivalently, cm/molecule). The integrated cross section is more fundamental than the peak cross section, 0), because it is insensitive to environmental factors (e.g., temperature, pressure, solvent or matrix effects), the effects of which are typically manifested in the width of the transition, Az/y. 

The pulsed NMR apparatus (12 MHz) and methods of procedure have been described previously (10, 13). The spin-spin relaxation time, T2, equivalent to the inverse absorption line width, is a time constant for the exponential decay toward internal equilibrium in the nuclear spin system. The spin-lattice relaxation time, Tu is the time constant for the exponential decay toward equilibrium between the nuclear spins and all other degrees of freedom of the system. The data are presented in Figure 1. Note that at higher temperatures T2 is much lower than Ti. 

Stellar spectra normally consist of absorption lines which occur when the intense continuum radiation from the hot interior of the star is filtered on passing through the cooler outer stellar atmosphere. The strength of the absorption line is a measure of the abundance of the element. As a measure in the determination of the number of absorbing atoms the so-called equivalent width of the spectral line is used. The equivalent width is defined as the width of a square box that covers the same surface as the actual absorption profile of the line, as illustrated in Fig.6.79. An example of an experimentally recorded stellar line is included in the figure. 

We then turn to a consideration of the absorption lines produced by the transmission of a continuous spectrum through an absorbing vapour. The concept of equivalent width is explained and we show that the equivalent width of an absorption line is determined by the product N.f..L. Details are given of the measurement of relative oscillator strengths by the absorption technique using a King furnace. 

In the atmospheres of the Sun and stars the excitation temperature also falls from the centre outwards to the boundary of the photosphere. In the outer regions this temperature gradient is responsible for the absorption line spectrum (the dark Fraunhofer lines) which is superimposed on the continuous spectrum of the Sun. By measuring the equivalent widths of these absorption lines (see section 10.4), and by solving the equation of radiative transfer, it is possible to deduce the chemical composition and physical state of the stellar atmosphere. 

If the incident radiation comes from a tungsten lamp or high-pressure xenon arc, 1( (0) will be essentially constant in the region of an absorption line. The transmitted intensity will then have a dip centred on and the spectral line is observed in absorption, as shown in Fig.10.3. The equivalent width, W, of the absorption line is defined as the width of a rectangular strip of height 1 (0) which has the same area as that of the absorption line. This equality is represented by the two shaded areas in Fig.10.3 and leads to the expression ... 

Fig.10.3. Definition of the equivalent width of an absorption line, W, in angular frequency units. The areas of the shaded sections are equal.

Referring to equation (10.7) we see that the equivalent width of an absorption line is determined by the product B ,N-L, or alternatively f.,N-L when the oscillator strength is in-troduced. Thus the measurement of equivalent widths is an important method for determining either oscillator strengths or atomic densities and further details of these techniques are given in the sections which follow. 

In the optically-thin case the f-value of an absorption line can be obtained directly from a measurement of the equivalent width provided that the density of absorbing atoms,... 

Most of these measurements involved the use of strong absorption lines and required the analysis of curves of growth. The discussion of section 10.4.2 shows that considerable simplifications are possible if the absorbing sample is optically thin. Consequently an apparatus has been developed by Peach (1969) and Blackwell and Collins (1972) in the Department of Astrophysics at Oxford for the measurement of the equivalent widths of weak absorption lines in metal vapours. The main components of this system, which is shown schematically in Fig.10,5, are discussed in the following sections. 

Calculations along these lines have been performed by Blackwell et at. (1972) using experimentally determined gf-values. When the results were compared with the measured equivalent widths of the manganese solar absorption lines they obtained a solar abundance of... 

A and its absorption oscillator strength is 0 325. Show that the column of vapour is optically thin at this wavelength and that the equivalent width of the absorption line formed by the vapour is 5 42 X 10 s". ... 

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