1 X-Ray Sources at Large-Scale Facilities
5
We know γ is of the order of several thousand, hence 1/γ
2 is a very small number, of
the order of 10
−8 . We use the approximation (1 − x)
n
= 1 − nx for x 1 to obtain
c − v
c
≈
1
2γ 2 .
(1.4)
In other words, the difference between c and v is very small, of the order of a few
m s
−1 .
Lastly, the mass of the electron from the perspective of a stationary observer is
equal to γ m e . We will return to these findings later.
1.2.3 Dipole Radiation and Synchrotron Radiation
Why do electrons emit EM radiation at all? First, it should be stressed that electrons, or indeed any charged particles, only emit EM radiation when accelerated.
‘Accelerated’ can include the conventional meaning of an increase in speed but no
change in direction; its opposite, that is, a deceleration; or a change in the electrons’
direction, such as in centripetal acceleration. The second case corresponds to the socalled ‘Bremsstrahlung’ (German expression for braking radiation), while the third
is associated with SR.
Given this, why then does the action of accelerating electrons cause them to emit
light? Consider Fig. 1.3. EM radiation is a form of transverse wave in which the
oscillations (of the electric and magnetic fields) are at right angles to the direction of
motion. For the sake of simplicity, we consider here only the electric field component
of the EM radiation. The electric field lines of the electrostatic field of a stationary
and isolated electron emanate out radially from the electron. Any observer looking at
the electron, therefore, sees no transverse component of the field, which thus implies
she sees no radiation [Fig. 1.3a].
If, however, the electron is made to execute oscillatory motion, the electric field
lines, which are anchored to the electron and emanate out from it at the speed of
light, will reflect this motion and thus also oscillate accordingly [Fig. 1.3b]. In all
directions except that exactly along the axis of acceleration, our observer will ‘see’
a transverse component to the electric field and therefore perceive that light is being
emitted. The amplitude of the radiation is proportional to cos χ , where χ is the polar
angle between the axis of acceleration and the observer’s direction. The intensity
distribution, shown in Fig. 1.3b, is proportional to cos
2
χ . This so-called ‘dipole
radiation’ is the reason why mirrors reflect visible light, radio antennae capture or
emit radio waves and undulators produce X-radiation.
SR is highly collimated, with natural divergences of the order of 0.1 mrad (mrad;
1 mrad is approximately equal to 0.06
◦ ). A detailed derivation of the spatial distribution of SR lies beyond the brief of this introductory overview (see, for example,
[3]). Suffice it to say, it differs substantially from the dipole distribution shown in
Fig. 1.3b; this is due to relativistic Doppler shifting. The angular power distribution
5
We know γ is of the order of several thousand, hence 1/γ
2 is a very small number, of
the order of 10
−8 . We use the approximation (1 − x)
n
= 1 − nx for x 1 to obtain
c − v
c
≈
1
2γ 2 .
(1.4)
In other words, the difference between c and v is very small, of the order of a few
m s
−1 .
Lastly, the mass of the electron from the perspective of a stationary observer is
equal to γ m e . We will return to these findings later.
1.2.3 Dipole Radiation and Synchrotron Radiation
Why do electrons emit EM radiation at all? First, it should be stressed that electrons, or indeed any charged particles, only emit EM radiation when accelerated.
‘Accelerated’ can include the conventional meaning of an increase in speed but no
change in direction; its opposite, that is, a deceleration; or a change in the electrons’
direction, such as in centripetal acceleration. The second case corresponds to the socalled ‘Bremsstrahlung’ (German expression for braking radiation), while the third
is associated with SR.
Given this, why then does the action of accelerating electrons cause them to emit
light? Consider Fig. 1.3. EM radiation is a form of transverse wave in which the
oscillations (of the electric and magnetic fields) are at right angles to the direction of
motion. For the sake of simplicity, we consider here only the electric field component
of the EM radiation. The electric field lines of the electrostatic field of a stationary
and isolated electron emanate out radially from the electron. Any observer looking at
the electron, therefore, sees no transverse component of the field, which thus implies
she sees no radiation [Fig. 1.3a].
If, however, the electron is made to execute oscillatory motion, the electric field
lines, which are anchored to the electron and emanate out from it at the speed of
light, will reflect this motion and thus also oscillate accordingly [Fig. 1.3b]. In all
directions except that exactly along the axis of acceleration, our observer will ‘see’
a transverse component to the electric field and therefore perceive that light is being
emitted. The amplitude of the radiation is proportional to cos χ , where χ is the polar
angle between the axis of acceleration and the observer’s direction. The intensity
distribution, shown in Fig. 1.3b, is proportional to cos
2
χ . This so-called ‘dipole
radiation’ is the reason why mirrors reflect visible light, radio antennae capture or
emit radio waves and undulators produce X-radiation.
SR is highly collimated, with natural divergences of the order of 0.1 mrad (mrad;
1 mrad is approximately equal to 0.06
◦ ). A detailed derivation of the spatial distribution of SR lies beyond the brief of this introductory overview (see, for example,
[3]). Suffice it to say, it differs substantially from the dipole distribution shown in
Fig. 1.3b; this is due to relativistic Doppler shifting. The angular power distribution