Particle accelerators
RadiationWhen does a charged particle emits?
Charge at rest No
Charge in constant motion in vacuum
No
More exotic situations
Charge in constant motion in a gas Yes if v>c/n Cherenkov Radiation Charge in constant motion through a foilYes ,Transition Radiation
Charge in constant motion through a hole
Yes ,Diffraction Radiation
Moving charge
The field of a point charge at rest. The electric field points
directly away from the charge.
In a reference frame where the charge is moving to the
right, the field is longitudinally contracted but the vertical component of the field is stronger. The field again point directly away from the current location of the charge.
Accelerating charge
A positively charged particle, initially traveling to the right bounces off a wall at point
B.
The particle is now at point A, but if there had been no bounce it would now be at
C.
The circle encloses the region of space where news of the bounce has already arrived Inside this circle (as at D) the electric field points directly away from A. Outside the
Field lines of an accelerating charge
Pillbox to calculate the flux of the
The flux must be zero!
On the outside (right-hand) portion there is a positive
flux, while on the inside (left-hand) portion there is a negative flux.
These two contributions to the flux do not cancel each
other, since the field is significantly stronger on the
outside than on the inside. This is because the field on the outside is that of a point charge located at
C
, while the field on the inside is that of a point charge located atA
, andC
is significantly closer thanA
. To cancel this positive flux, the remaining edges of the
pillbox must contribute a negative flux.
We refer to this component as the
transverse
field,since it points transverse (i.e., perpendicular) to the purely radial direction of the field on either side
Transverse components
Now the fluxes through ab and ef cancel.
Segments bc and de are chosen to be precisely parallel to the
field lines in their locations, so there is no flux through these portions of the surface.
In order for the total flux to be zero, therefore, the flux must be
zero through segment cd as well. This implies that the electric
Field of an uniform moving particle
When a point charge moves at constant velocity, its electric field always points directly away
from it.
This may seem strange, since no information can travel faster than the speed of light. The
particle has been traveling at constant velocity. So if you’re at a faraway place, you could have arranged for the particle to send you information about its position and velocity some time ago, so that when you receive this information you can extrapolate its motion from the past into the present and figure out where it must be by now.
Particle stops
Particle direction Acceleration takes place between t=0 and t0>0 Observation time is T>>t0
Quantitative estimation of E
t Using T=R/c
Radiation Field is proportional to 1/R
Vettore di Poynting
Il flusso del vettore di Poynting attraverso l’elemento di
superficie dA rappresenta l’energia elettromagnetica che l’onda trasporta nell’unità di tempo attraverso dA
Bibliography
J. D. Jackson, "Classical Electrodynamics", 3rd
Implosion
• This process represents a thin sphere that is converging with velocity c towards the point P while integrating over all charges that contribute to the potential V(t)
• Charges moving towards P while this integration is carried out are ‘counted’ for a longer time and contribute more to the potential V(t) • On the other hand, charges moving away from P will contribute less to
Moving frame
Assume the lights left the front end at the time t1. If the light left the rear end at t1−Δt and arrives the
front end also at the time t1, both signals (light) will then arrive at the observer at the same time t.
The retarded time for integral in the retarded potentials are different from the front end and rear end. This makes the apparent length for integration dependent of the velocity.
We have to make sure that the signals from both the front and rear ends arrive at the observation point at the same time
Shrinkage factor
During this time, the rod moved a distance
l’
−
l
Equating the right side we obtain
‘
‘
Please notice
If the velocity of an accelerated charge is small
Larmor’s formula
where d=dt/ is the proper time and p is the
Linear acceleration
In a linear accelerator the motion is 1-D. The
radiated power is
The rate of change of momentum is equal to the rate of
Power radiated
For linear motion the power radiated depends only on
the external forces which determine the rate of change of particle energy with distance, not on the actual
energy or momentum of the particle
The ratio of power radiated to power supplied by
external sources is
The radiation loss in an electron linear accelerator will
be unimportant unless the gain in energy is of the
order of 1014MeV/meter. Typically radiation losses are
completely negligible in linear accelerators since the gains are less than 50 MeV/meter
Circular motion
In circular accelerators like synchrotron or betatron the
momentum changes rapidly in direction as the particle rotates, but the change in energy per revolution is
small
Relativistic limit
In a 10 GeV electron Synchrotron (Cornell with
100m) the loss per revolution is 8.85MeV.
In LEP (CERN) with beams at 60 GeV ( =4300m)
Energy radiated
The energy per unit area per unit time measured
at an observation point at time t of radiation
emitter by charge at time t’ = t-R(t’)/c is
The energy radiated during a finite period of
Power per solid angle
Shift in frequency
Electron with velocity β emits a wave with period Temit
while the observer sees a different period Tobs because the electron was moving towards the observer
Geometry
Because the duration of the pulse is
very short, it is necessary to know the velocity and the position over only a small arc of the trajectory.
The origin of time is chosen so that at
t=0 the particle is at the origin of coordinates.
Notice that only for very small angles
there will be appreciable radiation intensity.
The trajectory lies on the plane
xy with instantaneous radius of curvature .
The unit vector n can be chosen
to lie in the xz plane, and is the angle with the x-axis.
Bessel plot
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 K2/3() K1/3()Total irradiated energy
Mainly polarized in the Horizontal plane
In the orbit plane Perpendicular to the orbit plane
Critical angle and frequency
Defined with =1/2 and =0
For frequencies much larger than the critical frequency
Different energy
No energy
dependence at long wavelength
Main features
The radiation is polarized mainly in the horizontal
plane
Half of the power is emitted below the critical
frequency and half above it
The critical frequency goes as
3
The total energy growths as
4
The frequency and the angle are closely related
The long wavelength part of the spectrum is
Wavelength shifter
Usually the central field is much larger than that of the lattice bending magnets
and has the purpose of providing very-short-wavelength radiation.
For this reason it is called a wavelength shifter.
Apart from the different total power and critical frequency, the properties of the
emitted radiation are the same as those of that emitted in the bending magnets of the ring
Only the high-frequency radiation emitted in the center part is used. Owing to its
much higher field and the geometrical separation, the radiation from other magnets gives a small contamination to the spectrum
Spectrum of wavelength shifter
A comparison between the spectral photon flux emitted by a 1.2 T
bending magnet and the 6.0 T wavelength shifter.
Properties of a Wavelength shifter
Since the magnetic field along the length of the
wavelength shifter is not constant, the critical photon energy also varies along its length
This means that although the SR produced has the
same characteristics of bending magnet radiation, the exact characteristics observed depend on which part of the electron trajectory the observer is looking at.
Furthermore, the observer may simultaneously also see
SR produced by the side poles, which may enhance the flux but also give a light source with more than one
source point.
A wavelength shifter will typically deflect the electron
Multipole Wigglers
Each wavelength shifter is independent of the
other and the electron returns back to the beam
axis after passing through each one
An observer will see an enhancement in the flux
received simply because there are now several
sources emitting radiation in his direction.
Properties of Wiggler radiation
The radiation is therefore enhanced by a factor of
2
N
, where
N
is the total number of periods of the
wiggler.
The spectrum from a wiggler has the same form as
that from a bending magnet. Therefore the
formula describing the emitted power is similar to
that for a bending magnet.
Bibliography
Peter Schmuser, Martin Dohlus, Jorg Rossbach,
“Ultraviolet and Soft X-Ray Free-Electron Lasers”,
Springer
James A. Clarke “The Science and Technology of
Potential I
B
yhas to be symmetric with respect to the plane
y
= 0 hence
c
2= 0.
Potential and field
We restrict ourselves to the symmetry plane
y
= 0
Solution and initial condition
Radiation angle
It is a general property of the radiation emitted by
relativistic electrons in a magnetic field that at large distance most of the intensity is concentrated in a narrow cone of opening angle 1
/
. The cone is centered around the instantaneous tangent
to the particle trajectory. The direction of the tangent varies along the sinusoidal orbit in the undulator
magnet, the maximum angle with respect to the axis being
If this directional variation is less than 1/
the radiationfield contributions from various sections of the trajectory overlap in space and interfere with each other
Undulator radiation
K<1 Undulator
K>>1 Wiggler
Second order analysis
zz
z
Some steps
z
Dipole radiation
This is mainly a transverse harmonic oscillation with the frequency
*=u
Superimposed is a small longitudinal oscillation which will be
ignored here, it leads to higher harmonics in the radiation.
In the moving system the electron emits dipole radiation with the
frequency *=u and the wavelength *
u = u/
K=1 K>>1
Lab frame
z pz
Interference
For interference to occur between wavefronts emitted by the
same electron the electron must slip behind the first wavefront by a whole number of wavelengths over one period.
The time for the electron to travel one full period is λu/cβ and
during this time the first wavefront (moving at velocity c of
Some steps
The separation,
d
, between the two wavefronts will be This separation must be a whole number of
Bazinga!!
A magnet with a period of a few 10s of mm produces light with a
wavelength on the order of nm because of the huge 2 term
At the maximum magnetic field value, K is also maximum and so
the output wavelength is longer than when K is small.
In other words, the output wavelength of an undulator gets longer
as the magnetic field increases. This is different to the synchrotron
radiation emitted by a dipole where we saw that higher magnetic
fields are used, especially in wavelength shifters, to produce shorter
wavelength radiation
The wavelength observed changes significantly with observation
angle but since there is a θ2 term the wavelength always lengthens
and K dependence
The variation of undulator output first harmonic wavelength with the deflection
parameter and the observation angle for a 3 GeV electron passing through a 50 mm period undulator
Number of periods
An electron passing through an undulator with
N
uperiods produces a wave train with
N
uoscillations
Spectrum
Due to its finite length, this wave train is not
monochromatic but contains a frequency spectrum
which is obtained by Fourier transformation
Higher harmonic wavelength
Energy
Example of a computed photon energy spectrum of undulator radiation
for an undulator with 10 periods.
The spectral energy of the mth harmonic that is emitted into the solid
angle ΔΩm. = 1000, the undulator has the period λu = 25 mm and the parameter K = 1.5. Note that the energy ratios Um/U1 depend only on the harmonic index m and the undulator parameter K, but not on nor on λu
Bibliography
G. Margaritondo, Y. Hwu and G. Tromba
“Synchrotron light: From basics to coherence and
coherence-related applications”
Again on Brilliance
A source can be made brighter by increasing the flux, by decreasing the
size or by enhancing the angular collimation
The average flux emitted by each single circulating electron is fixed by
the electron motion parameters.
One could, however, increase the number of circulating electrons, i.e., the
stored current in the ring. Unfortunately, the improvements in that sense practically saturated at 1 ampere in the 1980's
Coherence
A point-like single-wavelength (monochromatic) source is a
coherent source.
But what happens if the source is no longer monochromatic, or
no longer point-like, or both?
The fringes will be blurred and, beyond a certain point, no longer
visible. This point marks the difference between coherent and non-coherent sources
Two point source
Note that (d/D)z, where z is the illumination angle of each of the two
slits.
Thus, the condition for spatial coherence can be written: Sz z<
This equation implies that while reducing the source size Sz we improve
not only the source brightness but also the spatial coherence.
The theory of diffraction tells us that
the angular distance between two adjacent fringes is about /d radians.
If this value is substantially larger than
Sz/D, then the superposition of the two
patterns gives a somewhat blurred but still clearly visible set of fringes.
The condition for source coherence,
known as “lateral coherence" or
“spatial coherence“, is Sz/D < /d, or Sz
Angular divergence
Suppose that each one of the two point source
has an angular divergence
z.
Only a portion of this angular emission can be
used to produce a detectable fringe pattern.
This portion is /S
z. This implies that of the entire
emission over the angular range
zonly a fraction
(/S
z)/
z= /(S
z
z) can be used to produce
coherence-requiring phenomena
By increasing the source collimation, i.e., by
Coherent power
The coherent power of the source is the fraction of
the emitted power that can be used to produce
coherence-requiring phenomena
When the brightness is increased, then the
coherent power is also enhanced.
Also note that the coherent power increases with
the square of the wavelength.
Reaching high spatial coherence is thus more
Diffraction limit
How much spatial coherence can be obtained?
If S
z
z(or S
y
y) equals the wavelength, the
source is fully coherent
This is the so-called diffraction limit.
Sources of the class of ELETTRA, BESSY-II are fully
coherent down to wavelengths of the order of 100
nm. The Swiss Light Source arrives down to 10 nm.
Longitudinal or temporal coherence
The first-order fringe for occurs at the angle /d radians,
and that for + at (+ )/d radians.
These fringes are difficult to observe in the superposition
pattern if they are too much shifted from each other. On the contrary, if /d (+)/d then they are blurred but visible.
This implies << 1
This is the coherence condition related to the
Coherence length
In some applications of longitudinal coherence, what
matters is the so called coherence length
This notion can be understood by noting that two waves of
wavelengths and + , which happen to be in phase at a certain point in space, will become out of phase beyond this point.
Specifically, they will be totally out of phase (i.e., the
maximum on one wave coincides with the minimum of the other) after a distance Lc such that
Lc/ - Lc/(+ ) = 1/2,
which for a small gives Lc /2 = 1/2 , or
Usually, depending also from applications, the
coherence length of synchrotron radiation is not enough and a monochromator is needed
Model for a chaotic source (no coherence!)
Consider a particular excited atom radiating light of
frequency
0 We can consider a wave train of electromagnetic
radiation steadily emanating from the atom until it suffers a collision.
During a collision, the energy levels of the radiating
atom are shifted by the forces of interaction between the two colliding atoms. Thus the radiated wave train is interrupted for the duration of the collision.
When the wave of frequency
0 is resumed after thecollision, its characteristics are identical to those that it had prior to the collision, except that the phase of the wave is unrelated to the phase before the collision
Phase
the phase (t) remains constant during periods of
free flight but changes abruptly each time a
collision occurs
Relevance of coherence for
crystallography
Introduction to SPARC_LAB 117
A. Cianchi
Application of coherent radiation
Application of coherent radiation
Coherent emission due to bunching!
Synchrotron radiation is emitted into a broad spectrum
with the lowest frequency equal to the revolution
frequency and the highest frequency not far above the critical photon energy.
At low photon frequencies we may observe an
enhancement of the synchrotron radiation beyond intensities predicted by the theory of synchrotron radiation as discussed so far.
For photon wavelengths equal and longer than the
bunch length, we expect therefore all particles within a bunch to radiate coherently and the intensity to be
proportional to the square of the number
N
e ofparticles rather than linearly proportional as is the case for high frequencies. This quadratic effect can greatly enhance the radiation since the bunch population can be from 108–1011 electrons.
Coherent emission II
Generally such radiation is not emitted from a storage ring beam because
radiation with wavelengths longer than the vacuum chamber dimensions are greatly damped and will not propagate along a metallic beam pipe.
Much shorter electron bunches of the order of 1-2 mm and the
associated coherent radiation can be produced in linear accelerators where a significant fraction of synchrotron radiation is emitted
spontaneously as coherent radiation.
