[B._Beckhoff,_et_al.]_Handbook_of_Practical_X-Ray_(b-ok.org).pdf

Radiated by a relativistic monoenergetic electron on

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radiated by a relativistic monoenergetic electron on a circular orbit (CGS units): P = I ( λ, Ψ)d λ dΨ = 2 3 e 2 c R 2 E m 0 c 2 4 , (2.7) where e is the electron charge, c the speed of light, E the electron energy, m 0 c 2 its rest mass energy, R the orbit radius, λ the radiated photon wave- length (cm), Ψ the azimuthal angle (vertical – away from the orbital plane) and I is: I ( λ, Ψ) = 27 32 π 3 e 2 c R 3 λ c λ 4 γ 8 1 + (Ψ γ ) 2 2 K 2 2 / 3 ( ξ ) + γ ) 2 1 + (Ψ γ ) 2 K 2 1 / 3 ( ξ ) . (2.8) Here γ is the relativistic factor E/m 0 c 2 and K 1 / 2 and K 2 / 3 are the modified Bessel function of the second kind. λ c is the “cut-off” wavelength, given by: λ c = 4 3 πRγ 3 ; and ξ = λ c 2 λ 1 + ( Ψγ ) 2 3 / 2 . (2.9) Note the E 4 (2.7) dependence of the total radiated power, which explains why storage rings optimized for high electron energies must compensate this by a rather large ring radius R in order to keep the power P in the required range. From (2.7) it can be seen that owing to the rest mass energy denominator, electrons radiate about (2000) 4 as much power as protons. For practical vac- uum properties of the storage rings, positrons are sometimes used instead of electrons as they are less sensitive to recombination with trapped ions, which contributes to the longer lifetime of a stored beam. Other practical expressions routinely used in SR are: E c (keV) = 0 . 665 E 2 (GeV) × B ( T ) , (2.10) which defines the critical energy, and which in most cases represents the me- dian energy in the spectrum for the power distribution scheme as a function of the electron energy E and of the peak magnetic field B in the orbit. Also, the total radiated power is:
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2 X-Ray Sources 67 P tot (kW) = 26 . 6 E 3 (GeV) × B ( T ) × I ( A ) , (2.11) as a function of E , B, and the storage ring current I . The emission angle describing the cone into which all the photons are gen- erated by a relativistic electron is equal to 1 (rad) around the critical energy E c . For relativistic energies this is in the microradian range and represents one of the main advantages of SR. The great directionality and wide spectral range of the SR combine to yield a high spectral brilliance over a large range, which is the parameter of interest in the calculation of the heatload absorbed by the various beam components, such as monochromators. One of the most interesting aspects of SR is its linear polarization, which is exactly 100% in the orbit plane, with the electric vector parallel to the plane. Above and below the orbit plane, the radiation is elliptically polarized to a degree depending on the angle of observation Ψ . For “real” beams of nonzero emittance, the linear polarization in the orbit plane is slightly decreased. Another particular advantage of SR is its fast time structure. In a storage ring, electrons travel in bunches, alternatively losing their energy by emitting SR radiation, then replenishing it in the radio-frequency cavities. The length of bunches is in the ps to ns and is highly reproducible as a function of the bunch structure. Different operation modes, detailed in Sect. 2.4.2, take advantage
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