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Selfabsorbed lines

Effect of Selfabsorption on lineshapes:

Self absorption appears when light emitted by some atoms in a light source is absorbed by atoms in the source. This effect not only reduces the light emission, but also changes the line shape to differ from the typical Gaussian profile. Predicting the exact emission line shape in the presence of self absorption is quite difficult as it requires information on the spatial distribution of the emitting and absorbing species, including their velocity (or temperature). The effect can, however, be illustrated by assuming a simplified model of the real situation. If we assume the emission cell to consist of two different layers, an inner layer emitting light and an outer layer partly re-absorbing this light before it can reach the detector, the effect on the line shape can be calculated rather easily using Lambert-Beer’s law and the frequency dependence of the emission and absorption profiles.
The shape of the emission line ‘IE(n)’ before it is affected by re-absorption, will be given by a Gaussian profile with a total line intensity of ‘IL’:

Gaussian Line shape

The line shape of the absorbing medium will be described by the same function. The temperature of the atoms in the absorbing medium, however, may be different from the temperature in the emitting region. Note that the temperature influences the line width. The line width is proportional to the inverse of the square root of the temperature.
In order to calculate the transmission coefficient ‘T(n,d)’, we apply Lambert-Beer’s law.

absorption equation

The ‘T’ in the above formula describes the fraction of light transmitted while passing through the absorbing medium of thickness ‘d’. The absorption is strongest in the centre of the absorbing line and becomes smaller at frequencies away from the centre.

To calculate the transmitted light ‘IT(n)’, and its line shape, we need to multiply the emission profile by the absorption profile:

transmitted light equation


lineprofile with selfabsortpion dip

In the example shown in the above figure, we have assumed that the gas temperature in the emitting region is twice the temperature in the absorbing region. The ratio of line widths or the emitting and absorbing light, respectively, is therefore a square root of two. For this example, absorption coefficient was chosen just strong enough to generate a dip the line shape of the transmitted line profile. Depending on the strength of the absorption, the dip may be more pronounced or not observed at all, as shown in the experimental spectral profiles below.


Experimental Line profile in HDC Experimental line profile Grimm source

 


Each of these recordings from a high resolution spectrometer show the hyperfine doublet of the 324.75nm Cu I transition. The recording on the left was acquired using a hollow cathode source at moderate operation conditions, whereas the spectrum on the right was acquired using a typical Grimm type source in end-on view. The 324.75 nm line is obviously affected by severe self absorption in the Grimm source as can be seen in the (misleading) dips of each of the two hyperfine components of the electronic transition.

A different model for the distribution of the emitting and absorbing species which can be easily described is the homogeneous emitter – absorber system. Here we assume a constant distribution of emitting and absorbing species. Emitting and absorbing species have the same temperature.  To see the main difference from the situation described above with distinct emitting and absorbing areas, we calculate the effect of emission and re-absorption.

For this situation we can write the differential equation:

Differintial ligth absorption equation

Where ‘dI’ is the change in light intensity in a space interval ‘dx’. ‘El‘ is the linear emission density (emitted light intensity per length unit) and ‘ka‘ is the proportionality factor describing the portion of light intensity absorbed by a thin slice ‘dx’. The frequency dependence of ‘ka‘ is not specifically mentioned here.
By rearranging this equation we obtain:

light emission equation

Integrating this equation we find

ligh emission and absorption equation


Assuming now that ‘I(0)=0’, there is no light emitted yet at ‘x=0’. We get

light emission equation or light emission equation

and finally

Light emission from homogeneous source

For increasing thickness, the intensity approaches a constant value given by the ratio of the emission density and the absorption coefficient. Once this limit is reached in each slice, the amount of absorbed light equals the amount of emitted light. The transmitted light intensity does not vary anymore.
We can now use this formula to describe the effect on the line shape.

Flat Top

The above figure displays the effect of self-absorption by a homogenous emitter/absorber system on the line shape. The line shape is calculated for two strength or thickness. As the source gets (optically) thicker the emitted intensity reaches a limit at the emission profile centre. The profile therefore shows a flat top.
The following two graphics summarise the effect of self absorption on the emission intensity and the line shape in an emission cell. Here we assumed, for the two cases, that the density of both emitters and absorbers is continuously increased. In the first case, separated emission and absorption regions, the intensities first increase until a clear dip in the line centre is developed. In the second case, mixed emitters and absorber, again the transmitted intensity first increases and then a flat top line shape.

Lineshape with dipcase 1: evolution of lineshape towards self reversal Flat top lineshapecase 2: evolution of lineshape towards a flat top profile

When discharge sources are used as light sources both effects are likely to be present. Emitters (excited atoms or ions) are present in the same region as the absorbers (ground state atoms or ions). Their absolute and relative density however will vary over the entire discharge region observed by the spectrometer. Additionally the gas temperature will not be constant in the different discharge regions. It is therefore rather difficult to predict the exact line shape observed by a high resolution spectrometer. The effect of stimulated emission, necessary to make the transition towars black body radiation, has been neglected in this discussion.

First published on the web: 09. 01. 2008.

Author: Thomas Nelis. The text resumes and extends topics presented by Prof. E.B Steers during the first GLADNET Training course held in Antwerp, Be, in September 2007.