22 unifying physics of accelerators, lasers and plasma
∇ · E = 4πρ , ∇ · B = 0
1 ∂B
1
∂E
∇ × E = −
, ∇ × B =
4πJ +
(2.6)
c ∂t
c
∂t
Charge of electron in SI:
(
)
≈ 1.6 × 10 −19 C or A · s
And the Lorentz force is
v
F = q E +
(2.7)
c
× B
Charge of electron in Gaussian units: ≈ 4.8 × 10 −10 cgs The above equation highlights why the Gaussian system is
units
so useful: the electric and magnetic fields are expressed in
the same units, which stresses that these fields have a similar
nature.
Deriving a formula, instead
Throughout this text we will use both SI and Gaussian
of using quantities such as e units. We will, however, plan to construct equations or at
or h, express the end result least the end results in such a way that allows for easy convervia more natural quantities sion between different units. As a result, we will avoid hav(m e c 2 ,r e ,λ e ,α, etc.).
ing quantities such as electric charges or Planck’s constant in
our equations, and instead will use only quantities of length,
speed and energy.
Let’s write down some of the equations that will be useful
for such conversions into the natural and unit-independent
form.
The most useful one is for the classical radius of an electron:
1
e 2
e 2
SI : r e =
, Gauss : r e =
(2.8)
4πε
2
2
0 m e c
m e c
r e ≈ 2.82 · 10 −15 m
And the one for the fine structure constant:
e 2
e 2
SI : α =
, Gauss : α =
(2.9)
(4πε 0 ) nc
nc
α ≈ 1/137
The latter also gives us the reduced Compton wavelength for
the electron (¯ λ e = r e /α ≈ 3.86
ful
· 10 −13 m), which is very usewhenever the Planck’s constant needs to be hidden in an
equation.
2.2 Simplest accelerator
An example of a simple accelerator is a thermionic gun, in
which thermionic emission from the cathode generates electrons that are accelerated across a high voltage gap to the anode. A variety of materials can be used for these cathodes.
FIGURE 2.1
Pure metals are the most robust choice but require extremely
Simple electron gun.
high temperatures. On the other hand, the oxides cathodes
are covered with alkaline earth metal oxides (e.g., the BaO
impregnated tungsten cathode), and provide a higher emission at lower temperatures. However, they are more delicate
DOI: 10.1201/b18696-1
∇ · E = 4πρ , ∇ · B = 0
1 ∂B
1
∂E
∇ × E = −
, ∇ × B =
4πJ +
(2.6)
c ∂t
c
∂t
Charge of electron in SI:
(
)
≈ 1.6 × 10 −19 C or A · s
And the Lorentz force is
v
F = q E +
(2.7)
c
× B
Charge of electron in Gaussian units: ≈ 4.8 × 10 −10 cgs The above equation highlights why the Gaussian system is
units
so useful: the electric and magnetic fields are expressed in
the same units, which stresses that these fields have a similar
nature.
Deriving a formula, instead
Throughout this text we will use both SI and Gaussian
of using quantities such as e units. We will, however, plan to construct equations or at
or h, express the end result least the end results in such a way that allows for easy convervia more natural quantities sion between different units. As a result, we will avoid hav(m e c 2 ,r e ,λ e ,α, etc.).
ing quantities such as electric charges or Planck’s constant in
our equations, and instead will use only quantities of length,
speed and energy.
Let’s write down some of the equations that will be useful
for such conversions into the natural and unit-independent
form.
The most useful one is for the classical radius of an electron:
1
e 2
e 2
SI : r e =
, Gauss : r e =
(2.8)
4πε
2
2
0 m e c
m e c
r e ≈ 2.82 · 10 −15 m
And the one for the fine structure constant:
e 2
e 2
SI : α =
, Gauss : α =
(2.9)
(4πε 0 ) nc
nc
α ≈ 1/137
The latter also gives us the reduced Compton wavelength for
the electron (¯ λ e = r e /α ≈ 3.86
ful
· 10 −13 m), which is very usewhenever the Planck’s constant needs to be hidden in an
equation.
2.2 Simplest accelerator
An example of a simple accelerator is a thermionic gun, in
which thermionic emission from the cathode generates electrons that are accelerated across a high voltage gap to the anode. A variety of materials can be used for these cathodes.
FIGURE 2.1
Pure metals are the most robust choice but require extremely
Simple electron gun.
high temperatures. On the other hand, the oxides cathodes
are covered with alkaline earth metal oxides (e.g., the BaO
impregnated tungsten cathode), and provide a higher emission at lower temperatures. However, they are more delicate
DOI: 10.1201/b18696-1