666
M. Dohlus et al.
Radiation Power and Its Angular Distribution
The radiation power density is described by the Poynting vector
− →
S =
− →
E ×
− →
B
/μ 0 = ε 0 c
− →
E rad
2 − → n , where
− →
E rad is the far zone radiation field
given in observer time t according to Eq. (11.10). In the present section, we want
to refer the radiation power P in far zone to the motion of the electron (i.e. to an
emission time interval dt’), so we need to consider
dP
t
dd
= R
2
− →
S · − → n
dt
dt = R
2
− →
S · − → n
1 − − → n ·
− →
β
.
(11.11)
where dΩ is the solid angle element into which the power is emitted. Thus,
dP
t
dd
=
q 2
(4π)
2 ε 0 c
− → n ×
− → n −
− →
β
×
˙
− →
β
2
1 − − → n ·
− →
β
5
.
(11.12)
This is the most general expression for the angular dependence of the energy loss
into radiation of an accelerated point charge in far-field approximation.
There are essentially two different acceleration mechanisms which need to be
distinguished: Acceleration ˙
v ⊥ perpendicular to the direction of motion, usually
provided by a magnetic field B ext , and acceleration ˙
v by an electric field E ext
parallel to the momentarily velocity.
The total radiation power due to ˙
v ⊥ can be shown to be
P rad,⊥ =
q 2
6πε 0 c 3 γ
4
˙
v
2
⊥ =
q 2 c
6πε 0
β
4 γ 4
ρ 2 .
(11.13)
Here, =
− → p
/ (qB ext ) is the bending radius of the particle with momentum − → p
and charge q in presence of a magnetic field B ext directed perpendicular to − → p .
The total radiation power due to ˙
v is, on the other hand,
P rad, =
q 2
6πε 0 c 3
γ
3
˙
v
2 =
q 2 c
6πε 0
dγ
ds
2
,
(11.14)
with dγ /ds the acceleration due to the longitudinal electric field. In almost all
practical cases this radiation power is much smaller than the one generated in a
magnetic field of 1 T.
Thus, for the remainder of this paper, we restrict ourselves to acceleration due to
magnetic fields.
For circular accelerators the total energy loss U rad during one turn of the charged
particle due to synchrotron radiation is an important quantity. From Eq. (11.13) one
M. Dohlus et al.
Radiation Power and Its Angular Distribution
The radiation power density is described by the Poynting vector
− →
S =
− →
E ×
− →
B
/μ 0 = ε 0 c
− →
E rad
2 − → n , where
− →
E rad is the far zone radiation field
given in observer time t according to Eq. (11.10). In the present section, we want
to refer the radiation power P in far zone to the motion of the electron (i.e. to an
emission time interval dt’), so we need to consider
dP
t
dd
= R
2
− →
S · − → n
dt
dt = R
2
− →
S · − → n
1 − − → n ·
− →
β
.
(11.11)
where dΩ is the solid angle element into which the power is emitted. Thus,
dP
t
dd
=
q 2
(4π)
2 ε 0 c
− → n ×
− → n −
− →
β
×
˙
− →
β
2
1 − − → n ·
− →
β
5
.
(11.12)
This is the most general expression for the angular dependence of the energy loss
into radiation of an accelerated point charge in far-field approximation.
There are essentially two different acceleration mechanisms which need to be
distinguished: Acceleration ˙
v ⊥ perpendicular to the direction of motion, usually
provided by a magnetic field B ext , and acceleration ˙
v by an electric field E ext
parallel to the momentarily velocity.
The total radiation power due to ˙
v ⊥ can be shown to be
P rad,⊥ =
q 2
6πε 0 c 3 γ
4
˙
v
2
⊥ =
q 2 c
6πε 0
β
4 γ 4
ρ 2 .
(11.13)
Here, =
− → p
/ (qB ext ) is the bending radius of the particle with momentum − → p
and charge q in presence of a magnetic field B ext directed perpendicular to − → p .
The total radiation power due to ˙
v is, on the other hand,
P rad, =
q 2
6πε 0 c 3
γ
3
˙
v
2 =
q 2 c
6πε 0
dγ
ds
2
,
(11.14)
with dγ /ds the acceleration due to the longitudinal electric field. In almost all
practical cases this radiation power is much smaller than the one generated in a
magnetic field of 1 T.
Thus, for the remainder of this paper, we restrict ourselves to acceleration due to
magnetic fields.
For circular accelerators the total energy loss U rad during one turn of the charged
particle due to synchrotron radiation is an important quantity. From Eq. (11.13) one