electric force
orce
restoring force
g rate c)
K natural
The color of solids:
frequency
m The
Drude and Lorentz model for light-matter interaction
0
amount r(t) gives rise to an electric dipole moment equal
to
p(t) = −er(t).
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Figure 1: Simple image of the Lorentz model. Expect for the
driving electric field the electron is affected by two additional
forces: A damping force, which is the same as in the Drude
model, and a restoring spring force, which ensures that the
electron stays bound to the atom.
rm
r
t
j
I.
INTRODUCTION
The reason that gold is yellow and silver is white can be
understood by looking at the electromagnetic response of
the material. The aim of this project is to investigate how
the Drude and the related Lorentz model can account for
the optical properties of matter.
2 II. THE MODEL
The Drude
0 model, works pretty well to describe certain
aspects of metals. A generalisation to materials where
m
r
qE
there are no free conduction electrons, i.e. insulators (or
dielectric) can be obtained by adding a spring force. This
will force the electron to stay close to its initial position,
Fourier transform
i.e. the electron can be viewed as bound to a specific
atom within the solid. The Lorentz model can be written
as
2
m
(1)
mr̈(t) 0
+ ṙ(t) + ω 2 r(t) = −eE(t).
τ
r
m
r
qE
where ω0 is a constant that is related to the binding ensee Fig. 1. A displacement of the electron by an
(2)
See also the suggested references [1–3]. To get the polarisation (i.e. dipole moment per unit volume) of the
material we sum over the dipole moments of each of the
atom and divide by the volume, i.e.
1 X
pi (t) = np(t),
(3)
V i
where n is the number of atoms per unit volume and we
assume that, on average, the response is the same for
all the atoms. By solving Eq. (1) and plugging the result
into (3) we find how the relation between the electric field
and the polarisation. From this knowledge it is possible
to determine all the relevant optical properties of the
material, such as the refractive index and the absorption.
P(t) =
III.
RESEARCH QUESTIONS
This project is quite open and you can investigate anything you find interesting. However, some specific questions which may get you going are the following:
• How does the Drude and Lorentz model account for
the light-matter interaction?
• Explain absorption and refraction and how these
are related.
• Related to the above: Investigate and explain the
Kramer-Kronig relations.
• Can all phenomena be explained? How can one
generalise the model to account for band structure?
• Make a simple simulation of the Drude/Lorentz response (might be tricky due to the random collision
frequency).
ergy,
Simplify
m
2
https://www.youtube.com/watch?v=MBH5-oHqzp4
0[1]
[2] http://kik.creol.ucf.edu/documents/ose5312notes.
r
qE
[3] http://teda.nankai.edu.cn/aps/english/apslecture/
Peter/drude.pdf
pdf
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