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The Derivation of the Lorentz Line Shape

In the late 19th century the Dutch physicist Hendrik Lorentz, one of the greatest physicists of all time, formulated a model of the interactions of radiation and molecules. This led to a differential equation whose solution led to the line shape later named after him. Later quantum mechanics was used to derive the same functional form. Thus the functional form is robust, being derived from alternate formulations.

One such formulation is in terms of a charged particles in an electric field oscillating at a frequency of ω. The charged particles may be molecules or electrons associated with the molecules, The conditions assumed for the model are:

The last assumption can be replaced with the assumption that the momentum of a charged particle is redirected at intervals of time τ.

The Dynamics of the Oscillating Charge

Let x be the deviation of a particle of mass m and charge e from its equilibrium position. The dynamic equation for a particle is then

m(d²x/dt²) = −kx −j(dx/dt) + eE

where k and j are constants characterizing the restoring and damping forces and E is the intensity of the electric field.

The analysis of the model is made simpler if the restoring force constant k and the damping force constant j are expressed as 4π²ω0² and 2πmζ, respectively.

The dynamics of a particle are given by the equation

m(d²x/dt²) + 2πmζ(dx/dt) + 4π²ω0² = eE

If the local electric field intensity has the form E0exp(−2πiωt) then the displacement x(t) will, after some transit motion, also have that form; i.e., x(t)=Xexp(−2πiωt), where the amplitude factor X may be a complex number to allow for a phase lag between E(t) and x(t). This form for x(t) means that

d²x/dt² = −4π²ω²Xexp(−2πiωt).

The amplitude X must then satisfy the equation

4π²m(ω0² − ω² −iζω)X = eE0
or, equivalently,
X = (e/m)(1/4π²)[E0/(ω0²−ω²−iζω)]

The fact that X is complex is not a problem. A complex X can be expressed in polar form as |X|exp(iθ), where θ is a phase difference (time lag) between E(t) and x(t).

The Electromagnetic Radiation Generated by the Oscillation

The oscillating charge radiates an electromagnetic wave which has a power proportional to

ω4/[( ω0²−ω²)² + γ²ω²]

where ω0 can now be characterized as the natural frequency of oscillation of the charged particle and γ is a parameter proportional to ω0² and to the parameter ζ. This above expression will be called the power factor.

Let δω be ω0−ω, which is very much smaller than ω. Then

ω0²−ω² = (ω+δω)²−ω²
= 2ωδω + (δω)²
which, for small δω,
is approximately equal to

When this value is substituted into the power factor and the numerator and denominator divided by 4ω² the result reduces to

(ω²/4)/[(ω0−ω)² + (γ/2)²]

Since ω and ω0 are approximately equal the ω in the numerator in the above fraction can be set equal to ω0. Since the parameter γ is proportional to ω0² the numerator is then proportional to γ

Thus the power factor can be expressed as being proportional to

(γ/4)/[(ω0−ω)² + (γ/2)²]

This is the Lorentz line shape function.

(To be continued.)

The phenomenon of line broadening can most easily be justified by the Doppler effect but the Doppler effect gives rise to a line shape somewhat different from the Lorentz line shape.

For information on the Doppler line shape and other line shape see Line Shapes

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