14
P. R. Willmott
λ
λ
1
λ
2
D
Δθ
R
(b)
l
(a)
c
(l)
l c
(t)
Fig. 1.9 Beam coherence. a The temporal, or longitudinal, coherence length is determined by the
monochromaticity of the source, while b the transverse, or spatial, coherence length depends on the
photon beam emittance. Reproduced from [3] with permission (Copyright 2019, John Wiley and
Sons)
1.3.1 Bending Magnets and Wigglers
As mentioned above, the electrons are maintained on a closed path via magnetic, or
Lorentz, forces. The Lorentz force, F L , is proportional to the cross-product of the
magnetic field strength, B, and the charged particle’s velocity, v, that is
F L = eB × v ,
(1.18)
where e = 1.6022 × 10
−19 C is the elementary charge. It acts perpendicular to the
plane defined by B and v. For the sake of simplicity, we only consider those cases
where F L , B, and v are mutually orthogonal, and drop the bold face implying their
vectorial nature. Now,
F L = eBv .
(1.19)
We equate this with a centripetal force mv
2
/ρ, where m = γ m e is the relativistic mass
of the electron travelling at a speed v ≈ c. The bending radius of the centripetal force,
ρ, is equal to the bending magnet radius. Therefore, to a high degree of accuracy,
eBc ≈
γ m e c
2
ρ
(1.20)
⇒ ρ =
E
ceB
.
(1.21)
In practical units, we obtain
ρ = 3.3
E[GeV]
B[T]
.
(1.22)
P. R. Willmott
λ
λ
1
λ
2
D
Δθ
R
(b)
l
(a)
c
(l)
l c
(t)
Fig. 1.9 Beam coherence. a The temporal, or longitudinal, coherence length is determined by the
monochromaticity of the source, while b the transverse, or spatial, coherence length depends on the
photon beam emittance. Reproduced from [3] with permission (Copyright 2019, John Wiley and
Sons)
1.3.1 Bending Magnets and Wigglers
As mentioned above, the electrons are maintained on a closed path via magnetic, or
Lorentz, forces. The Lorentz force, F L , is proportional to the cross-product of the
magnetic field strength, B, and the charged particle’s velocity, v, that is
F L = eB × v ,
(1.18)
where e = 1.6022 × 10
−19 C is the elementary charge. It acts perpendicular to the
plane defined by B and v. For the sake of simplicity, we only consider those cases
where F L , B, and v are mutually orthogonal, and drop the bold face implying their
vectorial nature. Now,
F L = eBv .
(1.19)
We equate this with a centripetal force mv
2
/ρ, where m = γ m e is the relativistic mass
of the electron travelling at a speed v ≈ c. The bending radius of the centripetal force,
ρ, is equal to the bending magnet radius. Therefore, to a high degree of accuracy,
eBc ≈
γ m e c
2
ρ
(1.20)
⇒ ρ =
E
ceB
.
(1.21)
In practical units, we obtain
ρ = 3.3
E[GeV]
B[T]
.
(1.22)