FermiGoldenRule - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


Fermi's Golden Rule

Copyright (c) RDMCHEM LLC 2023

 

 

Overview

Derivation

Fluorescence

References

Overview

The transition rate of a molecular process between a discrete state and a continuum of states can be estimated from Fermi's Golden Rule.  The rule was popularized by Enrico Fermi in a book entitled Nuclear Physics published in 1950, but it was first derived by Paul Dirac in 1927.  Fermi's Golden Rule predicts the transition rate between a discrete state l and a continuum of states m as follows:

dPmdt= 2 πℏρE__mVml2 

where ρE__mis the energy density and Vml is the absolute value of the transition element of the perturbation matrix.

Derivation

To derive Fermi's Golden Rule, we begin with the transition rate from first-order perturbation theory where the perturbation is sinusoidal with a frequency ω 

dPmdt= 2ℏ2ωmlsinωmlωtVml2 .

Switching from angular frequency to energy, multiplying by the density of final states ρE__m and integrating yields

dPmdt= 2ℏ+ρE__msinEmElℏωtℏEmElVml2 ⅆEm .

Assuming that the energy density is a constant, we have

dPmdt= 2ℏρE__mVml2+sinEmElℏωtℏEmEl ⅆEm .

But integral, whose function in the integrand is known as the sinc function, can be evaluated to a constant.  Consider the sinc function

sinxtx;

sinxtx

(2.1)

Use the Explore function to plot the sinc function as a function of time t:

Exploreplotsinxtx, t=0.1..10;

t

 

Observe that the area under the curve appears independent of t.  We can confirm this hunch by taking the integral:

Integratesinxtx, x=infinity..infinity=integratesinxtx, x=infinity..infinity assuming t>0;

Intsinxtx,x=∞..∞=π

(2.2)

Therefore,

+sinEmElℏωtEmEl ⅆEm = π

whose substitution into the transition rate equation yields Fermi's Golden Rule

dPmdt= 2 πρE__mVml2 .

Note that the predicted transition rate is independent of time t.

Fluorescence

To illustrate the Golden Rule, we consider the fluorescence decay of a molecule in front of a mirror, following the work of K. H. Drexhage, H. Kuhn, and F. P. Schäfer in Ref. [3].  The molecule's density of states changes significantly as its distance h from the mirror changes.  By Fermi's Golden Rule, we would expect the emission rate to vary in proportion to the changes in the density of states.    

G  x,y  I4HankelH10,2Pisqrtx2+y2; emission  absGx, y  Gx + 2*h, y^2;

Gx,yI4HankelH10,2πy2+x2

emissionHankelH10,2πx2+y2HankelH10,2πy2+x+2h2216

(3.1)

We can make an animation of the emission pattern as a function of the distance h from the mirror.

plotter  procexpr local p1,p2; p1  plots:-densityplotexpr1100,_rest; p2  plottools:-polygonexpr2, color=silver; return plots:-displayp1,p2; end: mirror  5, 5.1, h, 5.1, h, 5.1, 5, 5.1: emission_and_mirror  emission,mirror:plots:-animateplotter,emission_and_mirror, x= h..7, y= 5.. 5,style=patchnogrid,colorscheme=Viridis, axes=boxed, labels=h,Emission, labeldirections=horizontal,vertical, labelfont=Helvetica,16, h=3..0.0000001,frames=80,paraminfo=false;

In agreement with Fermi's Golden Rule, as the density of states increases, the molecule glows more brightly.

 

References

1. 

P. A. M. Dirac, "The Quantum Theory of Emission and Absorption of Radiation," Proceedings of the Royal Society A 114, 243–265 (1927).  Refer to equations (24) and (32).

2. 

E. Fermi, Nuclear Physics (University of Chicago Press, Chicago, 1950). Refer to formula VIII.2.

3. 

K. H. Drexhage, H. Kuhn, F. P. Schäfer, "Variation of the Fluorescence Decay Time of a Molecule in Front of a Mirror," Berichte der Bunsengesellschaft für physikalische Chemie 72, 329 (1968).