σ r 0 ¼
ffiffiffi
λ
L
r
and σ r ¼
ffiffiffiffiffi
λL
p = 2π
ð Þ
ð2:16Þ
To summarize, even with infinitesimal electron beam size or divergence, the
photon beam will have a natural divergence and apparent size that depend on the
length of the source and the wavelength of the photons being used. Almost from the
first use of storage rings for synchrotron radiation, the dream has been to build a
“diffraction limited storage ring” or “DLSR” that would achieve the ultimate
possible source brightness. A final word of warning: there are many conflicting
definitions of the limiting photon properties and emittance, all of which are approximate because the true photon distribution is not Gaussian.
2.3.8 Putting Energy Back in: Storage Ring RF Cavities
A storage ring also needs one or more rf cavities to pump microwave energy into the
electron beam. The goal is primarily to restore the energy lost by synchrotron
radiation (although in some cases, the particle energy is also raised after injection).
Thus, a storage ring can be viewed as a massive microwave ! X-ray transducer.
In contrast with linac rf waveguides, which often involve traveling electromagnetic waves, the storage ring rf cavity contains a standing electromagnetic wave. In
the simplest case of a short (L < 2.03R) “pillbox” cavity with length L and radius R,
the electric and magnetic fields of the simplest TM 010 mode in cylindrical
coodinates, respectively, E z , E θ , E r , H z , H θ , and H r , are expressed in terms of Bessel
functions J 0 and J 1 (Appendix D):
E z r, t
ð Þ ¼ E 0 J 0 kr
ð Þ Á e
iωt and
ð2:17Þ
H θ r, t
ð Þ $ ÀiE 0 J 1 kr
ð Þ Á e
iωt
ð2:18Þ
where the remaining field components are zero, and the resonant frequency ω 0 is
determined by the boundary condition that the electric field is zero at the the cavity
wall (Fig. 2.11):
ω c ¼ kc ¼
2:405c
R
ð2:19Þ
In order for one bunch of particles to remain in a stable orbit, it must arrive at the
same phase of the rf cavity cycle each time. Thus, the rf cavity frequency f rf must be a
multiple h of the particle orbit frequency f particle , so that the cavity goes through an
integral number of cycles during particle orbits, and the particles always arrive at the
right time to be accelerated. For practical purposes, the bunch velocity is the speed of
light, so the period T for one orbit is the ring circumference L (actually the orbit
length L) divided by c. An acceptable rf frequency is thus:
24
2 The Storage Ring Complex
ffiffiffi
λ
L
r
and σ r ¼
ffiffiffiffiffi
λL
p = 2π
ð Þ
ð2:16Þ
To summarize, even with infinitesimal electron beam size or divergence, the
photon beam will have a natural divergence and apparent size that depend on the
length of the source and the wavelength of the photons being used. Almost from the
first use of storage rings for synchrotron radiation, the dream has been to build a
“diffraction limited storage ring” or “DLSR” that would achieve the ultimate
possible source brightness. A final word of warning: there are many conflicting
definitions of the limiting photon properties and emittance, all of which are approximate because the true photon distribution is not Gaussian.
2.3.8 Putting Energy Back in: Storage Ring RF Cavities
A storage ring also needs one or more rf cavities to pump microwave energy into the
electron beam. The goal is primarily to restore the energy lost by synchrotron
radiation (although in some cases, the particle energy is also raised after injection).
Thus, a storage ring can be viewed as a massive microwave ! X-ray transducer.
In contrast with linac rf waveguides, which often involve traveling electromagnetic waves, the storage ring rf cavity contains a standing electromagnetic wave. In
the simplest case of a short (L < 2.03R) “pillbox” cavity with length L and radius R,
the electric and magnetic fields of the simplest TM 010 mode in cylindrical
coodinates, respectively, E z , E θ , E r , H z , H θ , and H r , are expressed in terms of Bessel
functions J 0 and J 1 (Appendix D):
E z r, t
ð Þ ¼ E 0 J 0 kr
ð Þ Á e
iωt and
ð2:17Þ
H θ r, t
ð Þ $ ÀiE 0 J 1 kr
ð Þ Á e
iωt
ð2:18Þ
where the remaining field components are zero, and the resonant frequency ω 0 is
determined by the boundary condition that the electric field is zero at the the cavity
wall (Fig. 2.11):
ω c ¼ kc ¼
2:405c
R
ð2:19Þ
In order for one bunch of particles to remain in a stable orbit, it must arrive at the
same phase of the rf cavity cycle each time. Thus, the rf cavity frequency f rf must be a
multiple h of the particle orbit frequency f particle , so that the cavity goes through an
integral number of cycles during particle orbits, and the particles always arrive at the
right time to be accelerated. For practical purposes, the bunch velocity is the speed of
light, so the period T for one orbit is the ring circumference L (actually the orbit
length L) divided by c. An acceptable rf frequency is thus:
24
2 The Storage Ring Complex