8
P. R. Willmott
where the approximation on the right is valid for relativistic velocities. Both the
exact and approximate expressions are shown in Fig. 1.5 up to β = 0.9999. Increases
in power of the order of 10
16 compared to electrons undergoing stationary dipole
oscillations can thus be expected—synchrotrons truly do deliver powerful beams!
1.2.4 Spectral Flux, Emittance, and Brilliance
Flux and brilliance are figures of merit of the quality of a synchrotron facility. The
spectral flux is defined as the number of photons per second per unit bandwidth
(BW), normally given as 0.1%, and is the appropriate measure for experiments that
use the entire, unfocussed X-ray beam. Brilliance essentially states how tightly the
spectral flux is collimated and how small the source size is. It is defined as
Brilliance =
photons/second
(mrad)
2
(mm 2 source area) (0.1% BW)
,
(1.9)
and is therefore equal to the flux per unit source cross-sectional area and unit solid
angle. Note that the flux (as against the spectral flux) is simply measured in photons
per second. Doubling the transmitted BW from a broadband source thus doubles the
flux, but leaves the spectral flux unchanged. From (1.9), it is seen that the brilliance is
inversely proportional to both the source size and the beam divergence. The product
of the linear source size σ and the beam divergence σ
in the same plane is known
as , the emittance in that plane, that is
x = σ x σ
x ,
(1.10)
y = σ y σ
y .
(1.11)
The goal of the machine physicist designing a synchrotron magnet lattice is to provide
as low an emittance as possible, in other words, a source with an exceedingly small
cross section emitting X-rays that are highly collimated. For a given synchrotron
storage ring, the emittance in each transverse direction (x, in the orbital plane, or y,
perpendicular to the orbital plane) is, according to Liouville’s theorem, a constant.
The emittance will be different for different facilities, in each case being determined
primarily by the degree of sophistication and perfection of the magnet lattice.
Importantly, the total emittance in a given plane is a convolution of the contribution
from the electron beam and that from the emitted photons. It follows that the total
source size σ x,y and divergence σ
x,y in the x- and y-planes perpendicular to the
direction of propagation of a given storage ring are also convolutions of contributions
from the electron and photon beams, that is
σ x,y =
(σ
e
x,y )
2
+ (σ
p
)
2
1/2 ,
(1.12)
σ
x,y =
(σ
e
x,y )
2
+ (σ
p
)
2
1/2 .
(1.13)
P. R. Willmott
where the approximation on the right is valid for relativistic velocities. Both the
exact and approximate expressions are shown in Fig. 1.5 up to β = 0.9999. Increases
in power of the order of 10
16 compared to electrons undergoing stationary dipole
oscillations can thus be expected—synchrotrons truly do deliver powerful beams!
1.2.4 Spectral Flux, Emittance, and Brilliance
Flux and brilliance are figures of merit of the quality of a synchrotron facility. The
spectral flux is defined as the number of photons per second per unit bandwidth
(BW), normally given as 0.1%, and is the appropriate measure for experiments that
use the entire, unfocussed X-ray beam. Brilliance essentially states how tightly the
spectral flux is collimated and how small the source size is. It is defined as
Brilliance =
photons/second
(mrad)
2
(mm 2 source area) (0.1% BW)
,
(1.9)
and is therefore equal to the flux per unit source cross-sectional area and unit solid
angle. Note that the flux (as against the spectral flux) is simply measured in photons
per second. Doubling the transmitted BW from a broadband source thus doubles the
flux, but leaves the spectral flux unchanged. From (1.9), it is seen that the brilliance is
inversely proportional to both the source size and the beam divergence. The product
of the linear source size σ and the beam divergence σ
in the same plane is known
as , the emittance in that plane, that is
x = σ x σ
x ,
(1.10)
y = σ y σ
y .
(1.11)
The goal of the machine physicist designing a synchrotron magnet lattice is to provide
as low an emittance as possible, in other words, a source with an exceedingly small
cross section emitting X-rays that are highly collimated. For a given synchrotron
storage ring, the emittance in each transverse direction (x, in the orbital plane, or y,
perpendicular to the orbital plane) is, according to Liouville’s theorem, a constant.
The emittance will be different for different facilities, in each case being determined
primarily by the degree of sophistication and perfection of the magnet lattice.
Importantly, the total emittance in a given plane is a convolution of the contribution
from the electron beam and that from the emitted photons. It follows that the total
source size σ x,y and divergence σ
x,y in the x- and y-planes perpendicular to the
direction of propagation of a given storage ring are also convolutions of contributions
from the electron and photon beams, that is
σ x,y =
(σ
e
x,y )
2
+ (σ
p
)
2
1/2 ,
(1.12)
σ
x,y =
(σ
e
x,y )
2
+ (σ
p
)
2
1/2 .
(1.13)