Light absorption:

When light passes through a medium, some of the photons may interact with the matter (atoms, molecules, clusters etc.) and be absorbed (their energy being transformed in internal energy). As a result, the intensity decreases as light passes through the absorbing medium.

We find that the variation ‘dI’ of the light intensity ‘I’ crossing a very thin slice ‘dx’ of the absorbing medium is proportional to the incoming intensity and the thickness of the thin slice ‘dx’, i.e. each thin slice absorbs a given portion of the incoming light. The proportionality factor ‘k(n)’ depends both on the frequency of the incoming light and on the nature of the matter.

If we assume ‘k’ to be 0.9, 90% of the incoming light will pass through a slice of unity thickness and 10% will be absorbed.
If we assume a homogenous medium, a uniform layer for example, we can easily integrate the linear first order differential equation. The solution is of the form:

Where I₀ is the initial intensity at ‘x=0’. This relation ship is known as Lambert-Beer’s law first described by Pierre Bouguer in the early 18^th century. Check .wikipedia or britannica.com for more information on PierreBouguer
For a non-uniform absorbing layer we obtain:The optical depth or optical thickness ‘t‘ of a specific light path through a medium with absorption coefficient ‘k(n,x)’ and thickness ‘L’ is defined as the integral of the absorption coefficient over the entire light path. Despite its name, the optical depth is dimensionless. Values for ‘t‘ larger than one indicate strong absorption.When considering absorption of light in the discharge cell, the absorption is obviously due to the interaction between the photons and the different plasma species. We are mainly concerned by the interaction between the photons and the atoms or ions. The absorption coefficient depends therefore on the “atomic absorption probability” (proportional to the Einstein coefficient) and the density of the absorbing species, e.g. atoms in an electronic state able to interact with the photon. Light emitted by the plasma species in the discharge may be re-absorbed by the same species at different places in the discharge. This effect is known as self-absorption.
As the distribution of atoms and ions in the discharge cell is not homogeneous, the absorption coefficient varies within the discharge volume.
A well known example for the use of light absorption in analytical chemistry is Atomic Absorption Spectroscopy (AAS), using for example a flame or an electrothermal atomiser.

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

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

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Fourier Transform, Lorentz line shape and limited life time of a excited states.

In the classical picture, interpreting the photon emission as a damped oscillation, a Fourier analysis of the oscillation leads to a Lorentz shape. This can be found in most text books on the subject, i.e. Anne Thorne's book on spectroscopy.
Spontaneous photon emission by an atom in an excited that, can be explained by the coupling of the atom to the quantised electro-magnetic field. Treating the interaction as a first order perturbation leads to both the exponential decay behaviour and Lorentz shape in the emitted photon distribution. The coupling also introduces a shift in the energy levels, which in most spectroscopy text books is treated in the chapter on pressure broadening and pressure shift.
The following is no attempt to derive the full theory again, it only intends to demonstrate the apparent link between Fourier transform, the exponential decay of states and the Lorentzian line shape.
If we assume a particle in the state A with energy EA and we further assume the probability of finding the particle in the state A to decrease exponentially with the characteristic decay time g-1, we can write the time dependent wave function as (h=1)

We can now express this wave in terms of an orthonormal basis set for the time dependence of stationary solutions of a Hamilton operator.

Where the coefficients a(E) are projection of the wave function on the basis set functions. They are given by the scalar product.

Solving this integral using 0 and infinity as limits we obtain:

The probability density of finding the particle in a state with energy E is now:

Which is pretty close to the Lorentz shape of the spectral lines observed and the presence of the integrals of the typical Fourier form is quite obvious. This uncertainty in the energy of an unstable state also ensures the conservation of energy, when photons with a Lorentzian energy spread are emitted.
As usual a proper derivation is a little more complex, and can be found in the following references.

References:

• V. Weisskopf, E. Wigner, Z. f. Phys.; 63; 1930; 54-73
• F.Hoyt; Phys. Rev,; 36, 1930, 860-870
• W. Heitler, S.T. Ma, Proc. R.I.A. (A); 52; 1949, 109-125

Reference books:

• A.Messiah, Mécanique Quantique, tome 2, Dunod, 2003.
• Anne P. Thorne, Spectrophysics; Chapman and Hall Ltd; 2nd ed. 1988
• Robert D. Cowan, The Theory of Atomic Structure and Spectra; University of California Press; 1981

First published on the web: 15. 02. 2008.

Author: Thomas Nelis. The text is an extension of the discussion following Prof. E.B Steers lecture during the first GLADNET Training course held in Antwerp, Be, in September 2007.

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Physical Lineshape and Doppler Profile

Atomic collisions can reduce the life time of an excited state and thus enlarge this Lorentzian shape. This is called collisional or pressure broadening.

Due to the thermal movement of atomic light emitters the observed line width is altered from its natural (Lorentzian) shape. The light emitted by an atom moving towards the detectors will be seen blue shifted, light emitted by atoms moving away from the detector will appear red shifted. The shift is proportional to the speed of the atom.  The velocity distribution of the atoms is random and follows the normal Gauss distribution. The mean speed increases with the temperature of the atoms and for a given temperature decreases with the atomic mass. The Doppler shift in association with the random movement of the emitters leads therefore to a Gaussian spectral.

Doppler Profile (a minus is missing in the exponent) Doppler width

The observed Doppler profile can therefore be described as a Gauss function. The Doppler width, FWHM, is given in the lower equation, with

• Boltzmann constant: k= 1.380658 10^-23 JK^-1
• atomic mass unit: u=1.660542 10^-27 kg
• speed of light: c= 2.99792458 10⁸ ms^-1

Choosing two examples, H (mass 1u) and Ar (mass 40 u) at a wavelength of 600 nm, both atoms have emission lines in that spectral region, we can calculate the expected line broadening as function of the gas temperature.

Even for the high temperatures corresponding to an ICP plasma the broadening is less then 20 pm for the lightest atom Hydrogen. In order to observe the Gaussian profile the spectral resolution of the spectrometer needs to be significant higher then the width of the spectral feature to be observed. The spectrometer required for measuring the Doppler width must therefore provide a resolution in the low pico-meter range or even better fractions of a pico-meter. This corresponds to a resolving power of several hundred thousands.

Spectrometers operating in the visible region and providing a resolution in the "some-pico-meter range", are available, but can already be considered as high-end instruments, for higher resolving power considerable efforts must be made.

Other broadening mechanisms altering the physical lineshape are fine and hyperfinestructures, electric and magnetic fields. These effects will however, rather split the line into several components of slightly different wavelengths. A superposition of the original lineshape will therefore be observed.

As the natural line shape of one atom is Lorentzian and the population of the moving atoms generate a Gaussian lineshape the resulting physical lineshape is a convolution of the Lorentzian and Gaussian line shape.

The combination of Lorentzian and Gaussian profiles which results from random movement of atoms is called a Voigt profile.

Authors: Richard Payling and Thomas Nelis

First published on the web: 15 November 1999.

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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’:

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.

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:

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.

 

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:

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:

Integrating this equation we find

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

or

and finally

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.

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.

case 1: evolution of lineshape towards self reversal

case 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.

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Measured or instrumental Lineshape

The measured intensity must pass through the slits of the spectrometer: very narrow slits introduce diffraction broadening, while wide slits reduce the resolution. In general wide slits introduce a trapezoidall shape which is then convoluted with the natural Voigt profile to produce the lineshapes we typically observe in analytical spectrometers. The trapezoidall shape itself is linked to the convolution the two rectangular slits, entrance slit and exit slit respectively..

Instrumental lineshape FWHM ~50 pm

Different effects such as astigmatism may render the observed lineshape asymmetric.

For more details and a description of the equations used see: R Payling and P L Larkins, in R Payling, D G Jones and A Bengtson (Eds), Glow Discharge Optical Emission Spectrometry, John Wiley & Sons, Chichester (1997), pp 364-375.

Authors: Richard Payling and Thomas Nelis

First published on the web: 15 November 1999.

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