LCLS Conceptual Design Report - Stanford Synchrotron Radiation ...

L C L S C O N C E P T U A L D E S I G N R E P O R T
Based on composition values and atomic weights of the constituents, the effective values of
radiation length, atomic number, and material critical energy of the material are: Xo = 11.54
g/cm 2 ; Xo/ρ~1.55 cm; Z~29; εo~17.3 MeV respectively. The minimum ionization loss is
dE/dx = εo/Xo ~1.5 MeV/(g/cm 2 ). Note: these values very closely match those of copper.
For a square hit up front with no shower multiplicity (i.e.,
( ) −
Π e = 1), the normalized power
deposition is P’ ~1.35 W/cm. For the highest envisioned incident beam energy of 15 GeV,
shower maximum of the electromagnetic cascade occurs at a depth of Tmax~5.8 cm Xo ≡ 9.0 cm,
( − )
and the maximum shower multiplicity is Π max ~106 [14]. Consequently, the maximum
normalized power deposition is ~145 W/cm.
e
( e−
)
Pmax
′ = P′
Π max
First, the exposure at the undulator entrance is estimated where the assumed Gaussian
distributed beam has a predicted transverse size of σ = 32 µm. Assuming a uniform particle
distribution inside 0 < r σ, a heat source term is defined as S = CP'/Ab ~16.5 × 10 3 W/cm 3 ,
where C ~0.4, and Ab = σ 2 π ~32.2 × 10 -6 cm 2 ~<
. For a specific heat capacity of ρc = 3.43
Ws/(cm 3 °C), the temperature rise per pulse (RF-bunch) for a pulse repetition rate (PRR) = 120 Hz
is
∆T =
S
≈ 40° C/pulse. (8.31)
ρcPRR
The consequential thermal stresses are proportional to the product of the coefficient of
thermal expansion α and the modulus of elasticity E:
σ th ∝ Eα ∆ T . (8.32)
Numerically σth is ~ 2.83 × 10 7 N/m 2 (4100 psi or ~0.35 σUT) for a fully restrained body.
Since this is near a surface, actual stresses are somewhat lower. This should not present any
structural challenge to the magnetic material, even for repeated exposures. At Tmax the effective
transverse beam size increases to σeff ~220 µm (from Monte Carlo simulations for copper and
(e − )
scaling.). Using Π = 106, the heat source term is S ~37 kW/cm 3 max
, and the resulting single
pulse temperature rise is ∆T ~90°C/pulse.
Somewhat higher temperatures are actually observed short of Tmax, at a depth of ~ 3.5 to 4 Xo
for Eo = 15 GeV, since the expanding transverse shower has not yet caught up with the rapidly
( ) −
Π e is ~ 75. The resulting effective
increasing shower multiplicity. At 4 Xo, σeff is ~130 µm and
3
heat source term is S ~ 73kW/cm , for which the temperature rise per pulse is
4 Xo
∆T ∼ 175°C/pulse, (8.33)
and the consequential thermal stress rise is of the order of σth is ~ 1.24 × 10 8 N/m 2 (18,000 psi or
~1.6 σUT).
U N D U L A T O R ♦ 8-41