Lorentz curve

This article deals with the Lorentz curve in physics, for its occurrence in stochastics see Cauchy distribution . For the Loren z curve in economics see there .

The Lorentz curve , according to Hendrik Antoon Lorentz , or Breit-Wigner function , according to Gregory Breit and Eugene Wigner , is a curve that occurs in physics when describing resonances .

Two Lorentz curves with different widths. The curves are normalized so that the area under the curves is identical.

Mathematical definition and approximation

Two parameters go into the Breit-Wigner function. The parameter determines the position of the maximum , the parameter is called the width of the curve. From a physical point of view, the curve can only be interpreted for granted, since a circular frequency is usually associated with it and negative frequencies are physically nonsensical. The functional rule is: ${\displaystyle \omega _{0}}$ ${\displaystyle \gamma }$ ${\displaystyle \omega \geq 0}$ ${\displaystyle \omega }$

${\displaystyle f(\omega )={\frac {1}{(\omega ^{2}-\omega _{0}^{2})^{2}+\gamma ^{2}\omega _{0}^{2}}}}$

Another form of the curve can be obtained by reparameterization by using the following set of parameters instead of parameters and : ${\displaystyle \omega _{0}}$ ${\displaystyle \gamma }$

${\displaystyle {\omega _{0}'}^{2}=\omega _{0}^{2}{\sqrt {1+{\frac {\gamma ^{2}}{\omega _{0}^{2}}}}},\qquad {\gamma '}^{2}=2\omega _{0}^{2}\left({\sqrt {1+{\frac {\gamma ^{2}}{\omega _{0}^{2}}}}}-1\right)}$

Then

${\displaystyle f(\omega )={\frac {1}{(\omega ^{2}-{\omega _{0}'}^{2})^{2}+{\gamma '}^{2}\omega ^{2}}}}$;

in particular, it applies to that the deleted and uncoated parameters become almost identical. The first form is usually preferred in particle physics , the second form in classical physics , since they result in the corresponding forms in their respective fields from physics. The relationships are used for back conversion ${\displaystyle \gamma ^{2}/\omega _{0}^{2}\ll 1}$

${\displaystyle \omega _{0}^{2}={\omega '_{0}}^{2}-{\frac {{\gamma '}^{2}}{2}}\qquad \gamma ^{2}={\frac {{\gamma '}^{2}}{2}}{\frac {4{\omega '_{0}}^{2}-{\gamma '}^{2}}{2{\omega '_{0}}^{2}-{\gamma '}^{2}}}}$

Contrary to some belief, neither is nor the full width at half height (FWHM) of the curve. This one is instead ${\displaystyle \gamma }$ ${\displaystyle \gamma '}$

${\displaystyle {\text{FWHM}}={\sqrt {\omega _{0}^{2}+\gamma \omega _{0}}}-{\sqrt {\omega _{0}^{2}-\gamma \omega _{0}}}}$

and arises for only about . ${\displaystyle \gamma ^{2}/\omega _{0}^{2}\ll 1}$ ${\displaystyle \gamma }$

For and the Lorentz curve can go through ${\displaystyle \omega \approx \omega _{0}}$ ${\displaystyle \gamma \ll \omega _{0}}$

${\displaystyle f(\omega )={\frac {1}{4\omega _{0}^{2}}}{\frac {1}{(\omega -\omega _{0})^{2}+\gamma ^{2}/4}}}$

are approximated. With the exception of a normalization factor, it is then identical to the probability density known as the Cauchy distribution in mathematical probability theory . When the Lorentz curve is mentioned, the approximated version is also sometimes meant.

Physical meaning

Classical physics

The differential equation for the damped harmonic oscillator

${\displaystyle \left({\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}+\gamma {\frac {\mathrm {d} }{\mathrm {d} t}}+\omega _{0}^{2}\right)x(t)=F(t)}$

can be converted into the algebraic equation by Fourier transform

${\displaystyle \left(-\omega ^{2}+\mathrm {i} \gamma \omega +\omega _{0}^{2}\right){\tilde {x}}(\omega )={\tilde {F}}(\omega )}$

be convicted. The quantities appearing in these equations are:

• the damping constant ${\displaystyle \gamma }$
• the resonance frequency of the undamped harmonic oscillator ${\displaystyle \omega _{0}}$
• a stimulating function ${\displaystyle F(t)}$

The equation can now be solved elementarily, its solution is

${\displaystyle {\tilde {x}}(\omega )={\frac {{\tilde {F}}(\omega )}{-\omega ^{2}+\mathrm {i} \gamma \omega +\omega _{0}^{2}}}}$

and its square of the amount

${\displaystyle f(\omega )={\tilde {x}}(\omega ){\tilde {x}}^{*}(\omega )={\frac {{\tilde {F}}{\tilde {F}}^{*}}{(\omega ^{2}-\omega _{0}^{2})^{2}+\gamma ^{2}\omega ^{2}}}}$

the Lorentz curve in the second parameterization.

Particle physics

In particle physics, the propagators are the inverse functions of the equations of motion for the particles. These have a pole in the mass of these particles. To get around this, a so-called complex mass is introduced, which takes into account the decay width of the respective particle. Then the propagator for a certain four-pulse is proportional to ${\displaystyle m}$ ${\displaystyle \Gamma }$ ${\displaystyle k}$

${\displaystyle P(k^{2})\sim {\frac {1}{k^{2}-m^{2}+\mathrm {i} \Gamma m}}}$

and its square is the Lorentz curve in the first parameterization:

${\displaystyle P(k^{2})P^{*}(k^{2})\sim {\frac {1}{(k^{2}-m^{2}c^{4})^{2}+\Gamma ^{2}m^{2}c^{4}}}}$

example

Z ⁰ boson

The Breit-Wigner formula results especially for the decay of the Z ⁰ boson

${\displaystyle \sigma _{i\rightarrow f}(s)=12\pi (\hbar c)^{2}\cdot {\frac {\Gamma _{i}\cdot \Gamma _{f}}{(s-M_{Z}^{2}c^{4})^{2}+M_{Z}^{2}c^{4}\Gamma _{\text{tot}}^{2}}}.}$

Here is

• ${\displaystyle \Gamma _{i}}$the partial width of the input channel (i.e. for the decay Z ⁰ → e ⁺ e ^- )
• ${\displaystyle \Gamma _{f}}$ the partial width of the output channel
• ${\displaystyle \Gamma _{\text{tot}}}$the sum of the partial widths for all possible decays into fermion - anti fermion pairs
• ${\displaystyle s}$the energy in the center of gravity
• ${\displaystyle \hbar }$the reduced Planck quantum of action
• ${\displaystyle c}$the speed of light .

literature

• G. Breit, E. Wigner: Capture of Slow Neutrons . In: Phys. Rev. Band 49 , April 1, 1936, pp. 512–531 , doi : 10.1103 / physrev.49.519 (English, smu.edu [PDF; 1.1 MB ]).