12.3 Electron Microscopy 289
Box 12.1 Wavelength of Accelerated Electrons
Electrons have, like many other elementary particles, a two-fold nature: They
can act as particles or as waves. In electron microscopy, both natures are essential. Looking at the wave nature: The wavelength of electrons is controlled by
the acceleration voltage. According to de Broglie the wavelength λ associated to
a particle with the mass m is given by
λ =
h
mv
.
(12.13)
In Eq. (12.13), the quantity h is Planck’s constant (h = 6.63 × 10
−34 J s ) and v
the velocity of the electrons. From the energy balance
U eV
mv
v
eV
m
=
=
⇒ =
2
1
2
2
2
,
(12.14)
the velocity of the electrons may be calculated. In Eq. (12.14), e is the electrical
charge (e = 1.602 C), and m the rest mass of the electrons and V the acceleration
voltage of the system. In electron microscopy, voltages above 100 kV are applied.
At these high energies, the velocity of the electrons come into a range where
mass is increased by relativistic phenomena. Using the Lorentz transformation,
the mass m of a particle (in this case an electron) traveling with the speed v
and the mass m 0 = 9.11 × 10
−31 kg at v = 0 (“rest mass”), is calculated by
m
m
v
c
=
−
0
2
1
2
1
,
(12.15)
where c is the speed of light (c = 2.998 × 10
8 m s
−1 ). After inserting in Eq. (12.14),
one obtains
λ =
+
h
m eV
eV
m c
2
1 2
0
0
2
1
2
.
(12.16)
The relativistic increase of the electron mass reduces the wavelength of the
electrons. Figure 12.8 was calculated using this formula.
Now, one can estimate the necessary voltage for electron microscopes. If one
wants to see points at a distance of 0.5 nm apart and the numerical aperture of
bypassed. The problem of spherical aberration was solved by the development of
correction elements, which are not circularly symmetric. New electron microscopes equipped with both types of correction elements are able to depict, for
example, the position of every atom in a graphene layer at an electron energy of
only 80 keV (see Figure 12.17).
Box 12.1 Wavelength of Accelerated Electrons
Electrons have, like many other elementary particles, a two-fold nature: They
can act as particles or as waves. In electron microscopy, both natures are essential. Looking at the wave nature: The wavelength of electrons is controlled by
the acceleration voltage. According to de Broglie the wavelength λ associated to
a particle with the mass m is given by
λ =
h
mv
.
(12.13)
In Eq. (12.13), the quantity h is Planck’s constant (h = 6.63 × 10
−34 J s ) and v
the velocity of the electrons. From the energy balance
U eV
mv
v
eV
m
=
=
⇒ =
2
1
2
2
2
,
(12.14)
the velocity of the electrons may be calculated. In Eq. (12.14), e is the electrical
charge (e = 1.602 C), and m the rest mass of the electrons and V the acceleration
voltage of the system. In electron microscopy, voltages above 100 kV are applied.
At these high energies, the velocity of the electrons come into a range where
mass is increased by relativistic phenomena. Using the Lorentz transformation,
the mass m of a particle (in this case an electron) traveling with the speed v
and the mass m 0 = 9.11 × 10
−31 kg at v = 0 (“rest mass”), is calculated by
m
m
v
c
=
−
0
2
1
2
1
,
(12.15)
where c is the speed of light (c = 2.998 × 10
8 m s
−1 ). After inserting in Eq. (12.14),
one obtains
λ =
+
h
m eV
eV
m c
2
1 2
0
0
2
1
2
.
(12.16)
The relativistic increase of the electron mass reduces the wavelength of the
electrons. Figure 12.8 was calculated using this formula.
Now, one can estimate the necessary voltage for electron microscopes. If one
wants to see points at a distance of 0.5 nm apart and the numerical aperture of
bypassed. The problem of spherical aberration was solved by the development of
correction elements, which are not circularly symmetric. New electron microscopes equipped with both types of correction elements are able to depict, for
example, the position of every atom in a graphene layer at an electron energy of
only 80 keV (see Figure 12.17).