Flight time broadening

Schematic representation of when time-of-flight broadening occurs

The flight time broadening (engl. Time-of-flight broadening or transit time broadening ) is a physical effect in laser spectroscopy , the undesirable widening of the linewidth of the light emitted by moving atoms frequency entails. The effect is similar to the energy-time uncertainty relation .

overview

The effect occurs with moving particles ( atoms , ions or molecules ) that cross the laser beam but do not move parallel to it. As a result, the atoms can only interact with the laser light for a finite time . At the average speeds typical in the laboratory at room temperature ( ) and laser beams with diameters of the order of magnitude , interaction times of the order of microseconds are obtained for directions of flight perpendicular to the laser beam. The mean lifetimes of the energy levels of many particles, which are given by the spontaneous emission , are, however, in the range of milliseconds, i.e. they are about 1000 times longer. However, in order to obtain the best possible measurement results, the interaction times would have to be at least as long as the mean service life of the energy levels. ${\ displaystyle T \ approx 300 \, \ mathrm {K}}$ ${\ displaystyle {\ bar {v}} \ approx 5 \ cdot 10 ^ {4} \, \ mathrm {\ tfrac {cm} {s}}}$ ${\ displaystyle d \ approx 0 {,} 2 \, \ mathrm {cm}}$

Mathematical description

In order to make a quantitative statement about the broadening, it is assumed that the particles fly perpendicularly through the laser beam. The electric field of the laser is described by a Gaussian function that is only switched on for the time : ${\ displaystyle \ tau}$

{\ displaystyle E (t) = E_ {0} \ exp \ left (- (2r / d) ^ {2} \ right) \ cos (\ omega _ {0} t) \ Theta (\ tau) \ Theta ( t- \ tau)}

It is the Heaviside function that determines when the function is non-zero (in the time in which the particles in the beam is located), the laser frequency and the diameter of the laser beam between the points where the intensity has dropped. The variable to be determined is the frequency, which is why the Fourier transform is formed, the limits of which are reduced from up to to due to the Heaviside functions : ${\ displaystyle \ Theta (t)}$ ${\ displaystyle \ omega _ {0}}$ ${\ displaystyle d}$ ${\ displaystyle {\ tfrac {1} {e}}}$ ${\ displaystyle - \ infty}$ ${\ displaystyle \ infty}$ ${\ displaystyle 0}$ ${\ displaystyle \ tau}$

{\ displaystyle A (\ omega) = \ int _ {0} ^ {\ tau} E (t) \ exp (-i \ omega t) dt}

The square of the amount gives the frequency-dependent intensity distribution: ${\ displaystyle A ^ {*} (\ omega) A (\ omega)}$

{\ displaystyle I (\ omega) \ propto \ exp \ left [- \ left ({\ frac {d (\ omega - \ omega _ {0})} {2v {\ sqrt {2}}}} \ right) ^ {2} \ right]}

The line width is now obtained by determining the half- width of . With a Gaussian laser beam and the interaction time , the line width is minimal . The factor is specific for Gaussian laser beams, with other intensity profiles slightly different values result. B. in a "rectangular" electric field the factor . ${\ displaystyle I (\ omega)}$ ${\ displaystyle \ tau}$ ${\ displaystyle \ Gamma \ approx {\ tfrac {0 {,} 44} {\ tau}}}$ ${\ displaystyle 0 {,} 44}$ ${\ displaystyle 0 {,} 9}$

Countermeasures

This effect can be prevented, among other things, by using ion traps that hold the particles to be measured in one place, or by cooling the gas to be measured strongly, which slows down the particles and thus increases the interaction time.

Further information

Wolfgang Demtröder: Laser Spectroscopy . 5th edition. Springer, Berlin / Heidelberg / New York 2007, ISBN 978-3-540-33792-8 .