5 Interactions of Beams with Surroundings
199
is characterised by the critical energy
E c =
3
2
¯
hc
γ 3
ρ
.
(5.8)
The number of photons radiated in a bending magnet of length L and bending radius
ρ is (α = e 2 / 4ππ 0 ¯
hc is the fine-structure constant and γ the Lorentz factor of the
particle) and the energy loss U 0 per beam particle and turn are
N γ ,L =
5αγ L
2
√
3ρ
,
U 0 =
4πα ¯
hc γ 4
3ρ
(5.9)
The photon energy increases with γ 3 and the energy loss in synchrotron radiation
over one turn as γ 4 . A practical limit was reached with electrons at LEP at beam
energies around 100 GeV, corresponding to a Lorentz factor γ ≈ 2 × 10 5 when 3%
of the particle energy was lost on a single turn, while for the 1836 times heavier
protons synchrotron radiation only becomes noticeable at TeV energies. Numerical
values for the main synchrotron radiation parameters for several e + e − colliders, the
LHC with protons at 7 TeV and the proposed FCC colliders[23, 24] are shown in
Table 5.5.
The normalised quadrupole power spectrum (for flat beams as typical for e + , e −
colliders) is
s d (k) =
9
√
3
8π
k
∞
k
K 5/3 (s) ds , k =
E
E cr
.
Table 5.5 Synchrotron radiation parameters; p is the beam-momentum, ρ the bending radius of
the main dipole magnets
Machine
p
γ
ρ
E c
N γ
U 0
N tot
P tot
(GeV/c)
(m)
Dane
0.51
998
97.7
457 eV
66 9.3 keV
2.1 × 10 13 98 kW
PEP-II HER 9.0
17,613 163
9.9 keV
1166 3.57 MeV 4.5 × 10 13 3.4 MW
PETRA
23.4
45,792 192
148 keV
3031 138 MeV 1.0 × 10 12 3.0 MW
TRISTAN
32.
62,622 244
298 keV
4144 380 MeV 8.8 × 10 11 5.3 MW
LEP1
45.6
89,237 3026 70 keV
5906 126 MeV 1.9 × 10 12 0.4 MW
LEP2
94.5
184,932 3026 619 keV 12,239 2.3 GeV 3.2 × 10 12 13 MW
FCC-ee,Z
45.6
89,237 10,190 22 keV
5973 36 MeV 2.8 × 10 15 50 MW
FCC-ee,tt
182.5
357,144 10,190 1323 keV 23,636 9.19 GeV 1.1 × 10 13 50 MW
LHC
7000
7460.5 2784 44.1 eV
494 6.7 keV
6.5 × 10 14 7.8 kW
FCC-hh
50,000 53,289 10,572 4.24 keV
3733 4.6 MeV 1.1 × 10 15 2.5 MW
N γ , U 0 are the number of photons radiated and the energy loss per beam particle and turn. N tot , P tot
are the number of beam particles and the power radiated in the accelerator
199
is characterised by the critical energy
E c =
3
2
¯
hc
γ 3
ρ
.
(5.8)
The number of photons radiated in a bending magnet of length L and bending radius
ρ is (α = e 2 / 4ππ 0 ¯
hc is the fine-structure constant and γ the Lorentz factor of the
particle) and the energy loss U 0 per beam particle and turn are
N γ ,L =
5αγ L
2
√
3ρ
,
U 0 =
4πα ¯
hc γ 4
3ρ
(5.9)
The photon energy increases with γ 3 and the energy loss in synchrotron radiation
over one turn as γ 4 . A practical limit was reached with electrons at LEP at beam
energies around 100 GeV, corresponding to a Lorentz factor γ ≈ 2 × 10 5 when 3%
of the particle energy was lost on a single turn, while for the 1836 times heavier
protons synchrotron radiation only becomes noticeable at TeV energies. Numerical
values for the main synchrotron radiation parameters for several e + e − colliders, the
LHC with protons at 7 TeV and the proposed FCC colliders[23, 24] are shown in
Table 5.5.
The normalised quadrupole power spectrum (for flat beams as typical for e + , e −
colliders) is
s d (k) =
9
√
3
8π
k
∞
k
K 5/3 (s) ds , k =
E
E cr
.
Table 5.5 Synchrotron radiation parameters; p is the beam-momentum, ρ the bending radius of
the main dipole magnets
Machine
p
γ
ρ
E c
N γ
U 0
N tot
P tot
(GeV/c)
(m)
Dane
0.51
998
97.7
457 eV
66 9.3 keV
2.1 × 10 13 98 kW
PEP-II HER 9.0
17,613 163
9.9 keV
1166 3.57 MeV 4.5 × 10 13 3.4 MW
PETRA
23.4
45,792 192
148 keV
3031 138 MeV 1.0 × 10 12 3.0 MW
TRISTAN
32.
62,622 244
298 keV
4144 380 MeV 8.8 × 10 11 5.3 MW
LEP1
45.6
89,237 3026 70 keV
5906 126 MeV 1.9 × 10 12 0.4 MW
LEP2
94.5
184,932 3026 619 keV 12,239 2.3 GeV 3.2 × 10 12 13 MW
FCC-ee,Z
45.6
89,237 10,190 22 keV
5973 36 MeV 2.8 × 10 15 50 MW
FCC-ee,tt
182.5
357,144 10,190 1323 keV 23,636 9.19 GeV 1.1 × 10 13 50 MW
LHC
7000
7460.5 2784 44.1 eV
494 6.7 keV
6.5 × 10 14 7.8 kW
FCC-hh
50,000 53,289 10,572 4.24 keV
3733 4.6 MeV 1.1 × 10 15 2.5 MW
N γ , U 0 are the number of photons radiated and the energy loss per beam particle and turn. N tot , P tot
are the number of beam particles and the power radiated in the accelerator