Page 94 - Maxwell House

74 Chapter 2

the equilibrium is not too strong, i.e. x(t) is miniscule. Then the restoring force moving the
attached to spring electron mass toward the equilibrium is () = −() where
() is the electron displacment. From the course of physics we know that, = and,
2

0
in the case of no damping ( = 0), such free electrion bouncing around the equilibrium
might continue forever. Here is called the resonance frequency of harmonical self-
0
swaying and for most materials such resonances are located in optical part of the spectrum.
All three forces simultaneously acting should be considered as a governing force () =
() where () is the expected accelration vector. Then the motion equation for single

nd
electron based on Newton’s 2 law yeilds
() = () = Δ() − ()( ⁄ ) − () (2.76)

Here () = () and () = () . According to (1.87) from Chapter 1 the time
2
2
⁄
⁄
has the effect of multiplying its complex
differentiation of harmonic vector () = ()
amplitude () by the factor . Therefore, for the electron shifting by the harmonic E-
2
field () = () : () = () and = − . Substituting these equities in
(2.76) and grouping the terms, we obtain the frequency dependable displacement value
Δ/
() = (ω) (2.77)
2
2
( 0 ) −() +
It means that that the external field (ω) converts electrically neutral atom into an electric
dipole by slightly shifting the center of negatively charged electron “cloud” relative to
positively charged ion (look back at Figure 2.2.1b). Thereby, the virtual nonzero dipole
moments () = Δ() appears the same way as in ordinary dielectrics with electronic
polarization (!). Thereby, the difference between metals and ordinary dielectrics becomes
blurred. Continuing this analogy we can define the average polarization moment (ω) of N
atoms (check (2.8)) and then the displacement vector (ω) like (2.63) as
(ω) = (ω) + (ω) = (ω)(ω) (2.78)

Here (ω) = 1 + (ω) is the complex and frequency dependable value ( = Δ for electron)

2
� �
′′
′
() = 1 + ( 0 ) 2 = () − () ⎫

−() 2
+()
(2.79)
2
⎬
= � = 56√

0 ⎭
Here N [m ] is the number of conductive electrons per volume unit and is called the angular
-3

plasma resonance frequency. This frequency is 2620 THz for cupper, 2180 THz for silver, 2185
THz for gold, and 3852 THz for nickel. Note that frequencies are much above, when ≫
, () → 1, loss vanishes, and metals become transparent.

The theoretical and experimental data demonstrate that generally |( ) − () | ≪ for
2
2
0
metals of high conductivity at frequencies bellow 50 - 100 GHz. As such, the dielectric constant
of metal and its conductivity can be found from (2.79) as
2
� �
() ≅ 1 + = 1 + (2.80)

() 0