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What is the maximum value of the Gaussian lineshape function relative to the Lorentzian lineshape?
What is the maximum value of the Gaussian lineshape function relative to the Lorentzian lineshape?
- The Gaussian peak value is exactly twice the Lorentzian peak value
- The Gaussian peak value is less than the Lorentzian peak value by almost 50%
- The Gaussian peak value exceeds the Lorentzian peak value by almost 50% (correct)
- The Gaussian peak value is equal to the Lorentzian peak value
How does the sharpness of the Gaussian lineshape compare to the Lorentzian lineshape?
How does the sharpness of the Gaussian lineshape compare to the Lorentzian lineshape?
- The Gaussian lineshape is much broader than the Lorentzian lineshape
- The Gaussian lineshape and Lorentzian lineshape have the same sharpness
- The sharpness of the Gaussian lineshape relative to the Lorentzian lineshape is not specified
- The Gaussian lineshape is much sharper than the Lorentzian lineshape (correct)
What is the result of convolving two Lorentzian lines with widths $_1$ and $_2$?
What is the result of convolving two Lorentzian lines with widths $_1$ and $_2$?
- A Gaussian line with width $ = \sqrt{_1^2 + _2^2}$
- A Lorentzian line with width $ = \sqrt{_1^2 + _2^2}$
- A Gaussian line with width $ = _1 + _2$
- A Lorentzian line with width $ = _1 + _2$ (correct)
What is the result of convolving two Gaussian lines with widths $_1$ and $_2$?
What is the result of convolving two Gaussian lines with widths $_1$ and $_2$?
What is the Voigt profile?
What is the Voigt profile?
What type of broadening is described by the Lorentzian lineshape?
What type of broadening is described by the Lorentzian lineshape?
What type of broadening is described by the Gaussian lineshape?
What type of broadening is described by the Gaussian lineshape?
Which statement about the convolution of a Lorentzian line and a Gaussian line is true?
Which statement about the convolution of a Lorentzian line and a Gaussian line is true?
What is the mathematical form of the Gaussian lineshape function?
What is the mathematical form of the Gaussian lineshape function?
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Study Notes
Width and Shape of Spectral Lines
- Emission line is described by plotting spontaneous emission radiation intensity as a function of frequency (or wavelength) for a specific lasing transition.
- The width of a spectral line (laser radiation) is described by the Full Width at Half Maximum (FWHM).
Füchtbauer-Ladenburg Equation
- An important relation that links the total area under the curve, Einstein coefficients, and population of states causing the absorption around v0.
Homogeneous and Inhomogeneous Broadening
- Homogeneous broadening occurs when all two-level atomic systems have the same lineshape function and the same atomic resonance frequency ν0.
- Inhomogeneous broadening occurs when each two-level atomic system has its own resonance frequency.
- Broadening is homogeneous when it affects all "atoms" equally and inhomogeneous when it splits them into sub-groups.
Line Width
- When both levels involving the transition are broadened, the line width is given by Dn21 = Dn1 + Dn2.
- The life time of an excited state is the inverse of the spontaneous emission probability (A21), given by t = 1/A21.
Natural Broadening
- Caused by the finite lifetime of an excited state, resulting in a damping rate g and a resonance angular frequency wo.
- The line shape function g(n) is a Lorentzian function.
- The width of spectral line due to natural broadening is given by Dn = g/2p.
- A narrower Dn is associated with low damping or longer lifetime, while a broader Dn is associated with high damping or shorter lifetime.
Calculating Width of Spectral Line
- Example: For Sodium, Ï„sp = 16ns, for Ruby, Ï„sp = 3x10-3sec, and for Semiconductor, Ï„sp = 10-9sec.
- Example: Calculate the FWHM of the natural broadening for the He-Ne laser red line (λ= 6328 Ao) obtained from the transition between the two energy levels 3S2 (τ2 = 19.6 × 10 -9 s) and 2P4 (τ1 = 18.7 × 10 -9 s).
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