Page 46 - Electrical Properties of Materials

Some properties of waves 29

needed. The focusing is possible because the electron is a charged particle,
and the great resolution is possible because it is a wave of extremely short
wavelength. It must be admitted that the
resolving power of the electron
2.3 Some properties of waves microscope is not as large as
would follow from the available
You are by now familiar with all sorts of waves, and you know that a wave of wavelengths of the electrons. The
frequency, ω, and wave number, k, may be described by the formula limitation in practice is caused by
lens aberrations.
u = a exp iϕ; ϕ =–(ωt – kz), (2.10)
where the positive z-axis is chosen as the direction of propagation.
The phase velocity may be defined as

ω
∂z
v p = = = f λ. (2.11)
∂t k
ϕ=constant
This is the velocity with which any part of the wave moves along. For a single
frequency wave this is fairly obvious. One can easily imagine how the crest
moves. But what happens when several waves are superimposed? The resultant
wave is given by

u = a n exp{–i(ω n t – k n z)}, (2.12)
n
where to each value of k n belongs an a n and an ω n . Going over to the continuum
case, when the number of components within an interval k tends to infinity,
a(k)
we get
∞

u = a(k)exp{–i(ωt – kz)} dk, (2.13)
–∞
Δk
where a(k) and ω are functions of k. 1
We shall return to the general case later; let us take for the time being,
t = 0, then

∞ k k

u(z)= a(k)exp(ikz)dk (2.14) 0
–∞
Fig. 2.4
and investigate the relationship between a(k) and u(z). We shall be interested in
The amplitude of the waves as a
the case when the wavenumber and frequency of the waves do not spread out function of wavenumber, described by
too far, that is a(k) is zero everywhere with the exception of a narrow interval eqn (2.15).
k. The simplest possible case is shown in Fig. 2.4 where
k k
a(k)=1 for k 0 – < k < k 0 + (2.15)
2 2
and

a(k)=0
outside this interval. The integral (2.14) reduces then to
k 0 + k/2
u(z)= exp (ikz)dk, (2.16)
k 0 – k/2