Spectral line shape
[1] Actual line shapes are determined principally by Doppler, collision and proximity broadening.
For each system the half-width of the shape function varies with temperature, pressure (or concentration) and phase.
A knowledge of shape function is needed for spectroscopic curve fitting and deconvolution.
A spectral line can result from an electron transition in an atom, molecule or ion, which is associated with a specific amount of energy, E. When this energy is measured by means of some spectroscopic technique, the line is not infinitely sharp, but has a particular shape.
Numerous factors can contribute to the broadening of spectral lines.
Broadening can only be mitigated by the use of specialized techniques, such as Lamb dip spectroscopy.
Thus, a combination of Doppler and pressure broadening effects yields a Voigt profile.
In Nuclear Magnetic Resonance it is possible to measure spectra in a phase sensitive manner.
The full Lorentzian lineshape is a result from the Fourier Transform of a Free Induction Decay [4] and takes the following form:
This can be expanded into the real and imaginary part by quadratic expansion of the denominator:
Taking only the real part of this expression yields the less general, but more common form of the Lorentz lineshape.
The third line shape that has a theoretical basis is the Voigt function, a convolution of a Gaussian and a Lorentzian, where σ and γ are half-widths.
The computation of a Voigt function and its derivatives are more complicated than a Gaussian or Lorentzian.
[7][note 3] For atoms in the gas phase the principal effects are Doppler and pressure broadening.
Lines are relatively sharp on the scale of measurement so that applications such as atomic absorption spectroscopy (AAS) and Inductively coupled plasma atomic emission spectroscopy (ICP) are used for elemental analysis.
The lines are relatively sharp because the inner electron energies are not very sensitive to the atom's environment.
For molecules in the gas phase, the principal effects are Doppler and pressure broadening.
Because there are many sources of broadening, the lines have a stable distribution, tending towards a Gaussian shape.
The shape of lines in a nuclear magnetic resonance (NMR) spectrum is determined by the process of free induction decay.
This decay is approximately exponential, so the line shape is Lorentzian.
In NMR spectroscopy the lifetime of the excited states is relatively long, so the lines are very sharp, producing high-resolution spectra.
Gadolinium-based pharmaceuticals alter the relaxation time, and hence spectral line shape, of those protons that are in water molecules that are transiently attached to the paramagnetic atoms, resulting contrast enhancement of the MRI image.
For example, when Beer's law applies, the total absorbance, A, at wavelength λ, is a linear combination of the absorbance due to the individual components, k, at concentration, ck.
[19] When the data points in a curve are equidistant from each other the Savitzky–Golay convolution method may be used.
[20] The best convolution function to use depends primarily on the signal-to-noise ratio of the data.
) of all single line shapes is zero at the position of maximum height.
Whereas the smaller component produces a shoulder in the spectrum, it appears as a separate peak in the 2nd.
In the co-domain (time) of the spectroscopic domain (frequency) convolution becomes multiplication.
A suitable choice of exponential results in a reduction of the half-width of a line in the frequency domain.
[24] A similar process has been applied for resolution enhancement of other types of spectra, with the disadvantage that the spectrum must be first Fourier transformed and then transformed back after the deconvoluting function has been applied in the spectrum's co-domain.
