CERN Proton Linac

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16 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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1

CERN Proton Linac

2

Multi
-
gaps Accelerating Structures:

B
-

High Energy Electron Linac

-

When particles gets ultra
-
relativistic (v~c) the drift tubes become
very long unless the operating frequency is increased. Late 40’s the
development of radar led to high power transmitters (klystrons) at very
high frequencies (3 GHz).

-

Next came the idea of suppressing the drift tubes using traveling
waves. However to get a continuous acceleration the phase velocity of
the wave needs to be adjusted to the particle velocity.

solution
: slow wave guide with irises
iris loaded structure


3

Iris Loaded Structure for Electron Linac

4.5 m long copper structure, equipped
with matched input and output couplers.
Cells are low temperature brazed and a
stainless steel envelope ensures proper
vacuum.

Photo
of a

CGR
-
MeV

structure

4

Other types of S.W. Multi
-
cells Cavities

side coupled

nose cone

5

Energy
-
phase Equations

-

Rate of energy gain for the synchronous particle:

s
s
s
eE
dt
dp
dz
dE

sin
0


-

Rate of energy gain for a non
-
synchronous particle, expressed in
reduced variables, and :

s
s
E
E
W
W
w




s

















small
eE
eE
dz
dw
s
s
s
.
cos
sin
sin
0
0




-

Rate of change of the phase with respect to the synchronous one:



s
s
RF
s
RF
s
RF
v
v
v
v
v
dz
dt
dz
dt
dz
d

























2
1
1




Since:





3
0
2
2
2
s
s
s
s
s
s
v
m
w
c
c
v
v












6

Energy
-
phase Oscillations

one gets:

w
v
m
dz
d
s
s
RF
3
3
0





Combining the two first order equations into a second order one:

0
2
2
2





s
dz
d
3
3
0
0
2
cos
s
s
s
RF
s
v
m
eE





with

Stable harmonic oscillations imply:

real
and
s
0
2


hence:

0
cos

s

And since acceleration also means:

0
sin

s

One finally gets the results:

2
0




s
7

The Capture Problem

-

Previous results show that at ultra
-
relativistic energies (


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瑲癥汩湧睡癥v獴牵捴畲敳u捡扥慤攠楤敮a楣慬i⡰桡獥⁶敬潣楴礽挩

-

䡥湣攠瑨攠煵敳瑩潮楳㨠捡i睥⁣慰畲攠汯眠歩k整楣e敬e捴牯猠敮敲e楥猠(


㰠1⤬
as they come out from a gun, using an iris loaded structure matched to c ?

v

=
c

e
-


0
< 1

gun

structure



t
E
E
z

sin
0

The electron entering the structure, with velocity v < c, is not synchronous
with the wave. The path difference, after a time dt, between the wave and
the particle is:



dt
v
c
dz


Since:

c
v
k
factor
n
propagatio
with
kz
t
RF
RF
RF









one gets:






d
d
c
dz
g
RF
2










1
2
c
dt
d
g
and

8

The Capture Problem (2)

2
1
0
0
2
0
0
1
1













g
e
c
m
E
2
1
1
2
cos
cos
2
1
0
0
0
2
0
0

















E
e
c
m
g







d
eE
c
m
d
g
2
0
2
0
sin
cos
1
2
sin



dt
d
d
d
dt
d








2
0
0
sin
sin
c
m
eE
dt
d












sin
1
0
2
1
2
0
0
eE
dt
d
c
m
dt
d
c
m
mv
dt
d














From Newton
-
Lorentz:

Introducing a suitable variable:



cos

the equation becomes:

Using

Integrating from t
0

to t


(from

=

0

to

㴱=

Capture condition

9

Improved Capture With Pre
-
buncher

A long bunch coming
from the gun enters an
RF cavity; the
reference particle is
the one which has no
velocity change. The
others get accelerated
or decelerated. After
a distance L bunch
gets shorter while
energies are spread:

bunching effect
.
This
short bunch can now
be captured more
efficiently by a TW
structure (v

=挩

10

Improved Capture With Pre
-
buncher (2)

The bunching effect is a space modulation that results from a velocity
modulation and is similar to the phase stability phenomenon. Let’s look at
particles in the vicinity of the reference one and use a classical approach.

Energy gain as a function of cavity crossing time:



0
0
0
0
2
0
sin
2
1
eV
eV
v
v
m
v
m
W













0
0
0
v
m
eV
v



Perfect linear bunching will occur after a time delay

Ⱐ捯牲敳潮摩朠瑯
a distance L, when the path difference is compensated between a
particle and the reference one:

RF
v
t
v
z
v



0
0
.





(assuming the reference particle
enters the cavity at time t=0)

Since
L = v


潮攠整猺

RF
eV
W
v
L

0
0
2

11

The Synchrotron
(Mac Millan, Veksler, 1945)

The synchrotron is a synchronous accelerator since there is a synchronous RF
phase for which the energy gain fits the increase of the magnetic field at each
turn. That implies the following operating conditions:

B
e
P
B
cte
R
cte
h
cte
V
e
r
RF
s














sin
^
Energy gain per turn




S
ynchron
ous particle



RF s
ynchronism




Constant orbit



Variable magnetic field

If

v
=

c
,



r

hence


RF

remain constant (ultra
-
relativistic e
-

)

12

Energy ramping is simply obtained by varying the B field:

v
B
R
e
r
T
B
e
turn
p
B
e
dt
dp
eB
p















2
)
(
Since:



p
v
E
c
p
E
E






2
2
2
0
2

The number of stable synchronous particles is equal to the harmonic
number h. They are equally spaced along the circumference.


Each synchronous particle satifies the relation p=eB

⸠.桥h⁨慶攠瑨攠
nominal energy and follow the nominal trajectory.


The Synchrotron (2)








s
s
turn
V
e
RB
e
W
E
sin
ˆ
'
2





13

The Synchrotron (3)

During the energy ramping, the RF frequency
increases to follow the increase of the
revolution frequency :

hence :


)
,
(
s
RF
r
R
B
h





)
(
)
(
2
2
1
)
(
)
(
2
1
2
)
(
)
(
t
B
s
R
r
t
s
E
ec
h
t
RF
f
t
B
m
e
s
R
t
v
h
t
RF
f









Since , the RF frequency must follow the variation of the




B

field with the law : which asymptotically tends


towards when
B
becomes large compare to (m
0
c
2

/ 2

r) which corresponds to


v c (pc >> m
0
c
2

). In practice the
B
field can follow the law:

2
2
2
0
2
c
p
c
m
E


2
1
2
)
(
2
)
/
2
0
(
2
)
(
2
)
(








t
B
ecr
c
m
t
B
s
R
c
h
t
RF
f

R
c
f
r

2

t
B
t
B
t
B
2
2
sin
)
cos
1
(
2
)
(





14

Zero gradient synchrotron

retour

15

Alternating gradient synchrotron

Non parallel pole faces shows a
field gradient. If focusing in one
plane, defocusing in the other
one. Alternating focusing and
defocusing magnets can lead to
global focusing.

16

Separated functions synchrotron

retour

Quadrupole

+

Dipole

=

FODO system

Super Protons Synchrotron at CERN

17

Single Gap Types Cavities

Pill
-
box variants

Coaxial cavity


noses disks

Type

/4


18

Ferrite Loaded Cavities

-

Ferrite toroids are placed around
the beam tube which allow to reach
lower frequencies at reasonable size.

-

Polarizing the ferrites will change
the resonant frequency, hence
satisfying energy ramping in protons
and ions synchrotrons.

19

High Q cavities for e
-

Synchrotrons

20

LEP 2: 2x100 GeV with SC cavities

21

Dispersion Effects in a Synchrotron

E+

E

E

If a particle is slightly shifted in
momentum it will have a different
orbit:

dp
dR
R
p


This is the “
momentum compaction

generated by the bending field.

If the particle is shifted in momentum it will
have also a different velocity. As a result of
both effects the revolution frequency changes:

dp
df
f
p
r
r


p=particle momentum

R=synchrotron physical radius

f
r
=revolution frequency

cavity

Circumference


2

R

22

Dispersion Effects in a Synchrotron (2)

dp
dR
R
p



x

0
s
s
p
dp
p

d


x







d
x
ds
d
ds



0
The elementary path difference
from the two orbits is:


x
ds
dl
ds
ds
ds



0
0
0
leading to the total change in the circumference:

m
m
x
dR
xds
ds
x
dR
dl








0
0
1
2



Since:

p
dp
D
x
x

we get:

R
D
m
x


< >
m

means that
the average is
considered over
the bending
magnet only

23

Dispersion Effects in a Synchrotron (3)

dp
df
f
p
r
r


R
dR
d
f
df
R
c
f
r
r
r








2














d
d
d
p
dp
c
E
mv
p
1
2
2
1
2
2
1
2
0
1
1
1












p
dp
f
df
r
r










2
1





2
1

=0 at the transition energy



1

tr
24

Phase Stability in a Synchrotron

From the definition of


it is clear that below transition an increase in
energy is followed by a higher revolution frequency (increase in velocity
dominates) while the reverse occurs above transition (v


c and longer path)
where the momentum compaction (generally > 0) dominates.

Stable synchr. Particle
for

<0



> 0

25

Longitudinal Dynamics

It is also often called “ synchrotron motion”.

The RF acceleration process clearly emphasizes two coupled
variables, the energy gained by the particle and the RF
phase experienced by the same particle. Since there is a
well defined synchronous particle which has always the same
phase

s
,
and the nominal energy E
s
, it is sufficient to follow

other particles with respect to that particle. So let’s
introduce the following reduced variables:


revolution frequency :

f
r

= f
r



f
rs



particle RF phase :


=


-


s


particle momentum :

p = p
-

p
s


particle energy :

E = E


E
s


azimuth angle :


=


-


s


26

First Energy
-
Phase Equation



R









dt
with
h
hf
f
r
r
RF




For a given particle with respect to the reference one:





dt
d
h
dt
d
h
dt
d
r




1
1








Since:

s
r
rs
s
dp
d
p










one gets:









rs
s
s
rs
s
s
rs
h
R
p
dt
d
h
R
p
E






and

c
p
E
E
2
2
2
0
2


p
R
p
v
E
s
rs
s






27

Second Energy
-
Phase Equation

The rate of energy gained by a particle is:




2
sin
ˆ
r
V
e
dt
dE

The rate of relative energy gain with respect to the reference
particle is then:

)
sin
(sin
ˆ
2
s
r
V
e
E














leads to the second energy
-
phase equation:



s
rs
V
e
E
dt
d




sin
sin
ˆ
2













E
T
dt
d
E
T
T
E
E
T
T
E
T
E
rs
rs
r
rs
r
r
















Expanding the left hand side to first order
:

28


Equations of Longitudinal Motion



s
rs
V
e
E
dt
d




sin
sin
ˆ
2

















rs
s
s
rs
s
s
rs
h
R
p
dt
d
h
R
p
E






deriving and combining



0
sin
sin
2
ˆ









s
rs
s
s
V
e
dt
d
h
p
R
dt
d





This second order equation is non linear. Moreover the parameters
within the bracket are in general slowly varying with time…………………

29


Hamiltonian of Longitudinal Motion



s
V
e
dt
dW


sin
sin
ˆ


W
R
p
h
dt
d
s
s
rs



2
1


Introducing a new convenient variable, W, leads to the 1
th

order
equations:

p
R
E
W
s
rs













2
2
These equations of motion derive from a hamiltonian H(

,圬琩

W
H
dt
d









H
dt
dW






W
p
R
h
V
e
t
W
H
s
s
rs
s
s
s
2
4
1
sin
cos
cos
ˆ
,
,













30

Small Amplitude Oscillations



0
sin
sin
cos
2




s
s
s






(for small

)

0
2






s


s
s
s
rs
s
p
R
V
e
h



2
cos
ˆ
2







tr



㸠〠†††††〠0

s

<

⼲††††††獩s

s

> 0






tr



<0††††

⼲<

s

<

††††††
獩s

s

> 0

with

Let’s assume constant parameters R
s
,

p
s
,


s

and



















s
s
s
s
cos
sin
sin
sin
sin
Consider now small phase deviations from the reference particle:

and the corresponding linearized motion reduces to a harmonic oscillation:

stable for and

s

real

0
2


s