Transcript Eric Prebys Fermi National Accelerator Laboratory NIU Phys 790 Guest Lecture E. Prebys, NIU Phy 790 Guest Lecture March 18 & 20, 2014

Eric Prebys
Fermi National Accelerator Laboratory
NIU Phys 790 Guest Lecture
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
1

Some Handy Relationships
Basic Relativity
pc
b=
E
E
g =
mc 2
pc
bg =
mc 2
v

c

1
1  2
moment ump  m v
t ot alenergy E  m c2
kinet icenergy K  E  m c2
E 
2

m c    pc
2 2
2
These units make
these relationships
really easy to
calculate
Units
 For the most part, we will use SI units, except
-19 J]
 Energy: eV (keV, MeV, etc) [1 eV = 1.6x10
2
 Mass: eV/c
[proton = 1.67x10-27 kg = 938 MeV/c2]
 Momentum: eV/c
[proton @ β=.9 = 1.94 GeV/c]
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
2
e
The simplest accelerators
accelerate charged particles
through a static electric field.
Example: vacuum tubes (or
CRT TV’s)
e
e
Cathode

V
Anode
K  eEd  eV
Limited by magnitude of static field:
- TV Picture tube ~keV
- X-ray tube ~10’s of keV
- Van de Graaf ~MeV’s
Solutions:
FNAL Cockroft-Walton
- Alternate fields to keep particles in
= 750 kV
accelerating fields -> RF acceleration
- Bend particles so they see the same accelerating field
over and over -> cyclotrons, synchrotrons
3

side view
A charged particle (v<<c) in a
uniform magnetic field will
follow a circular path of radius


f


s

B

B
mv
qB
v
2
qB
(constant!!)
2m
f
qB

2
m
top view
“Cyclotron Frequency”
For a proton:
fC  15.2  B[T ] MHz
Accelerating “DEES”
March 18 & 20, 2014
4
E. Prebys, NIU Phy 790 Guest Lecture

~1930 (Berkeley)
Lawrence and
Livingston
 K=80KeV

 1935 - 60” Cyclotron
 Lawrence, et al. (LBL)
 ~19 MeV (D )
2
 Prototype for many
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
5

The relativistic form of Newton’s Laws for a particle in an
electromagnetic field is:
dp
F=
= q E + v ´ B ; p = g mv
dt
(

A particle of unit charge in a uniform
magnetic field will move in a circle
of radius
T-m2/s=V
p
r=
eB
p
( Br ) =
e
pc
( Br ) c =
e
Beam “rigidity” =
constant at a given
momentum (even
when B=0!)
March 18 & 20, 2014
6
)
side view
top view
B

B
constant for
fixed energy
units of eV in our usual convention
( Br )[T-m] =
p[eV/c] p[MeV/c]
»
c[m/s]
300
E. Prebys, NIU Phy 790 Guest Lecture
Remember
forever!

If the path length through a
p
transverse magnetic field is short
compared to the bend radius
of the particle, then we can think of
the particle receiving a transverse “kick”

B
l
p  qvBt  qvB(l / v)  qBl
and it will be bent through small angle

In this “thin lens approximation”, a
dipole is the equivalent of a prism in
classical optics.
p
Bl
 

p ( B )

E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
7

Cyclotrons have a fixed magnetic field


As the energy grows, so does the radius
Since the minimum radius is limited by the strength of the
magnetic fields we can make, cyclotron get impractically large
pretty quickly.

On the other hand, we just showed the trajectory of a
particle will always depend on
B(x, y, z) B(x, y, z)
µ
p
( Br )

If, as the particles accelerate, we scale all magnetic
fields with the particle momentum, then the
trajectories will be independent of energy!
”Synchrotron”*
*Edward McMillan, 1945
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
8




Cyclotrons relied on the fact that
magnetic fields between two pole
faces are never perfectly uniform.
This prevents the particles from
spiraling out of the pole gap.
In early synchrotrons, radial field
profiles were optimized to take advantage of this effect, but in
any weak focused beams, the beam size grows with energy.
The highest energy weak
focusing accelerator was the
Berkeley Bevatron, which had
a kinetic energy of 6.2 GeV

High enough to make antiprotons
(and win a Nobel Prize)

It had an aperture 12”x48”!
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
9

Strong focusing utilizes alternating magnetic gradients to precisely
control the focusing of a beam of particles
The principle was first developed in 1949 by Nicholas Christofilos, a
Greek-American engineer, who was working for an elevator company in
Athens at the time.
 Rather than publish the idea, he applied for a patent, and it went
largely ignored.
 The idea was independently invented in 1952 by Courant, Livingston and
Snyder, who later acknowledged the priority of Christophilos’ work.


Although the technique was originally formulated in terms of
magnetic gradients, it’s much easier to understand in terms of the
separate functions of dipole and quadrupole magnets.
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
10
Strong focusing was originally implemented by building magnets with non parallel
A Main
Unitfaces
today to introduce a linear magnetic gradient
pole
(PFW and fig. 8 coils removed)
By (x) = B0 +
¶By
¶x
x
=
+
dipole
quadrupole
CERN PS (1959, 29 GeV)
Later synchrotrons were built with physically separate dipole and quadrupole magnets. The
first “separated function” synchrotron was the Fermilab Main Ring (1972, 400 GeV)
11
=
+
dipole
quadrupole
Fermilab
Strong focusing is also much easier to teach using separated functions, so we will…
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
11
By =
¶By
¶x
Bx =
x
y
x
Note: Ñ ´ B = 0 ®

¶Bx
y
¶y
¶By ¶Bx
=
º B¢
¶x
¶y
A positive particle coming out of the page off center in the
horizontal plane will experience a restoring kick
f
Bx ( x)l
Blx
  

( B )
( B )
focal length
*or quadrupole term in a gradient magnet
E. Prebys, NIU Phy 790 Guest Lecture
x (Br )
f=
=
Dq
B'l
March 18 & 20, 2014
12
Bx =
¶Bx
y
¶y
y
( B )
f 
B' l
Defocusing!
Luckily, if we place equal and opposite pairs of lenses, there
will be a net focusing regardless of the order.
pairs give net focusing in both planes -> “FODO cell”
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
13

We will work in a right-handed coordinate system with x horizontal,
y vertical, and s along the nominal trajectory.
xˆ
yˆ
sˆ
For now, assume
bends are always
in the x-s plane
Particle trajectory defined at any point in
s by location in x,x’ or y,y’ “phase space”
dx
º x¢ » q
ds
x
y¢
x¢
s
y
x
unique initial phase space point 
E. Prebys, NIU Phy 790 Guest Lecture
unique trajectory
March 18 & 20, 2014
14

The simplest magnetic “lattice” consists of quadrupoles and the spaces
in between them (drifts). We can express each of these as a linear
operation in phase space.
Quadrupole:
x
Dq = Dx¢ = f
x  x(0)
 x   11
    
1
x'  x' (0)  x(0)
x'   f


f
Drift:
x
s

x(s)  x(0)  sx' (0)
x' (s)  x' (0)
0  x(0) 

1  x' (0) 


 x(s)   1 s  x(0) 
  


 
 x' (s)   0 1  x' (0) 
By combining these elements, we can represent an arbitrarily complex
ring or line as the product of matrices.
M  M N ...M2M1
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
15

At the heart of every beam line or ring is the basic “FODO” cell,
consisting of a focusing and a defocusing element, separated by drifts:
f
-f
L
 1 L  11
 
 M  
 0 1  f
E. Prebys, NIU Phy 790 Guest Lecture
L
0  1 L  1

 1

1  0 1  
 f

Remember: motion is
usually drawn from left
to right, but matrices
act from right to left!
 L  L 2
0  1    

f f
1 
L
 
 2
f

L2 
2L 
f 
L 

1
f 
March 18 & 20, 2014
16

It might seem like we would start by looking at beam lines and them move
on to rings, but it turns out that there is no unique treatment of a
standalone beam line

Depends implicitly in input beam parameters

Therefore, we will initially solve for stable motion in a ring.

Rings are generally periodic, made up of more or less identical cells


In addition to simplifying the design, we’ll see that periodicity is important to
stability
The simplest rings are made of dipoles and FODO cells

Or “combined function magnets” which couple the two
 Our goal is to de-couple the problem into
Periodic
two parts
“cell”
N
Mring  McellMcell Mcell  Mcell
E. Prebys, NIU Phy 790 Guest Lecture

The “lattice”: a mathematical
description of the machine itself,
based only on the magnetic fields,
which is identical for each identical
cell

A mathematical description for the
ensemble of particles circulating in
the machine (“emittance”);
March 18 & 20, 2014
17

In the absence of degeneracy, an nxn matrix will have n
“eigenvectors”, defined by:
æ M 11
ç
ç
çè M n1

Eigenvectors form an orthogonal basis


æ V1 ö
M 1n ö æ V1 ö
֍
÷
ç
÷
=
l
iç
֍
÷
÷
çè Vn ÷ø
M nn ÷ø çè Vn ÷ø
i
i
That is, any vector can be represented as a unique sum of eigenvectors
In general, there exists a unitary transformation, such that
æ l1
ç
M¢ = UMU -1 = ç
çè 0

æ 0 ö
ç
÷
0 ö
ç
÷
÷
®
V
=
UV
=
¢
1
i
i
ç
÷
÷
ç
÷
ln ÷ø
ç 0 ÷
è
øi
Because both the trace and the determinant of a matrix are
invariant under a unitary transformation:
Tr ( M ) = M 11 + M 22 +
Det(M) = l1 ´ l2 ´
E. Prebys, NIU Phy 790 Guest Lecture
+ M nn = l1 + l2 +
+ ln
´ ln
March 18 & 20, 2014
18

We can represent an arbitrarily complex ring as a combination of
individual matrices
Mring  Mn ...M3M2M1

We can express an arbitrary initial state as the sum of the
eigenvectors of this matrix
 x
 x
   AV1  BV2  M   A1V1  B2 V2
 x 
 x 

After n turns, we have
 x
M    A1n V1  Bn2 V2
 x 
n

Because the individual matrices have unit determinants, the
product must as well, so
Det ( M) = l1l2 = 1® l2 = 1/ l1
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
19

We can therefore express the eigenvalues as
1  ea ; 2  ea ; wherea is in generalcomplex

However, if a has any real component, one of the
solutions will grow exponentially, so the only stable
values are
1  ei ; 2  ei ; where is real

Examining the (invariant) trace of the matrix
TrM  ei  ei  2 cos

So the general stability criterion is simply
absT rM  2
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
20

Recall our FODO cell
L
L
f

 L  L 2
1    
f

f
 
M 
L

 2
f

L2 
2L 
f 
L 

1
f 
-f
Our stability requirement becomes
  L 2 
abs 2      2  L  2 f
 f 


E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
21

We can express the transfer matrix for one period as the sum of an identity matrix
and a traceless matrix
 (s)  “Twiss Parameters”
1 0
  ( s)
  B

M(s  C, s)  A
0 1
   (s)   (s)  not Lorentz parameters!!

The requirement that Det(M)=1 implies
A2  B2 ( (s)2   (s) (s))  1
We can already identify A=Tr(M)/2=cosμ. Setting the determinant of the second
matrix to 1 yields the constraint
Normalization relationship
  (s)2   (s) (s)  1
 only two independent
We can identify B=sinμ and write

 ( s) 
1 0
  ( s)
  sin  
  I cos   J sin 
M (s  C, s)  cos  
0 1
  ( s)   ( s) 


Note that

 ( s )   ( s )
 ( s )    2 ( s )   ( s ) ( s )

0
  (s)

  I

  
J  
2
0
 ( s )   ( s ) ( s ) 
   ( s )   ( s )    ( s )   ( s )  
2

So we can identify it with i=sqrt(-1) and write
M(s  C, s)  eJ ( s )
Remember this! We’ll see it again in a few pages
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
22

If we know the transfer matrix or one period, we can explicitly
calculate the lattice functions at the ends
æ cos m + a sin m
ö æ 1 0 ö
æ a
b sin m
M=ç
÷ =ç
÷ø cos m + ç -g
g
sin
m
cos
m
a
sin
m
0
1
è
è
ø
è

b ö
÷ sin m = Icos m + Jsin m
-a ø
If we then know the transfer matrix from point a to point b, Mba,
we can evolve the lattice functions from a to b
Jb
sb
sa
E. Prebys, NIU Phy 790 Guest Lecture
æ m11 m12
Mba = ç
çè m21 m22
Ja
ö
÷
÷ø
-1
Jb = Mba Ja Mba
March 18 & 20, 2014
23
 m11
M ( sb , sa )  
 m21
m12 
 m
  M 1 ( sb , sa ) 22
m22 
  m21

Using

We can now evolve the J matrix at any point as
 ( sb )   m11
  ( sb )

  
J ( sb )  
   ( sb )   ( sb )   m21

 m12 

m11 
 ( sa )  m22
m12   ( sa )


m22    ( sa )   ( sa )   m21
 m12 

m11 
Multiplying this mess out and gathering terms, we get
  ( sb )   m11m22  m12 m21 

 
  ( sb )     2m11m12 
  ( s )    2m m 
b 
21 22


E. Prebys, NIU Phy 790 Guest Lecture
 m11m21   m12 m22   ( sa ) 
m 
m 
2
11
2
21
m 
m 
2
12
2
22


  ( sa ) 
  ( s ) 
a 

March 18 & 20, 2014
24

Drift of length L:
æ a (s) ö æ 1 0 -s ö æ a (0)
æ 1 s ö ç
÷ ç
2 ֍
M=ç
Þ
b
(s)
=
-2s
1
s
b (0)
÷
ç
÷
ç
ç
÷
0
1
è
ø
çè g (s) ÷ø è 0 0 1 ø çè g (0)

ö
a (s) = a 0 - g 0 s
÷
2
Þ
b
(s)
=
b
2
a
s
+
g
s
0
0
0
÷
÷ø
g (s) = g 0
Thin focusing (defocusing) lens:
 1
M   1

 f

 1




0   
    0
1   
       2
 f

E. Prebys, NIU Phy 790 Guest Lecture
1
f
1
1
f2

1

0       0   0
f
 0 
0   0       0

2
1
1   0 
    0   0  2 0
f
f

March 18 & 20, 2014
25

For the moment, we will consider curvature in the horizontal (x)
plane, with a reference trajectory established by the dipole fields.
yˆ
xˆ
sˆ
s is motion along
nominal trajectory

rx
Particle trajectory

Reference trajectory

General equation of motion (considering only transverse fields!)



  dp d

Transverse acceleration = γ doesn’t change!
F  ev  B 
 mR  mR
dt dt
xˆ
yˆ sˆ


e
e
 ev  B
 vs By xˆ  vs Bx yˆ  vx By  v y Bx sˆ
R

v x v y vs 
m
m
m
Bx B y 0

We must solve this in the curving coordinate system

Messy but straightforward
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
26
vs t  r
s
 


r
x
Transform independent
variable from t to s
s is measured along nominal
trajectory, vs measured along
actual trajectory
dx dx dt 1 dx
=
=
ds dt ds s dt
1 dy
y¢ =
s dt
x¢ º
æ
ds r
r
xö
s = = vs =
v » 1v
dt r
r + x s çè r ÷ø s
We will keep only the first order terms in the magnetic field
Bx ( x, y, s )  Bx 0,0, s  
Bx ( s )
B ( s )
B ( s )
x x
y no
coupling
 x
y
x
y
y
no x dipole
B y ( x, y, s )  B y 0,0, s  
B y ( s )
x
x
B y ( s )
y
y no
coupling
 B0 
B y ( s )
x

B  B y ( s)
x

x

x
Expanding in the rotating coordinate system and keeping first order terms…
“centripetal”
term
E. Prebys, NIU Phy 790 Guest Lecture
é1
1 ¶By (s) ù
x¢¢ + ê 2 +
úx = 0
ë r ( B r ) ¶x û
1 ¶Bx (s)
y¢¢ y=0
( Br ) ¶y
Looks “kinda like” a
harmonic oscillator
March 18 & 20, 2014
27

We have our equations of motion in the form of two “Hill’s Equations”
x¢¢ + K x (s)x = 0
K > 0 Þ "focusing"
y¢¢ + K y (s)y = 0
K < 0 Þ "defocusing"
K(s) periodic!

This is the most general form for a conservative, periodic, system in which deviations
from equilibrium small enough that the resulting forces are approximately linear

In addition to the curvature term, this can only include the linear terms in the magnetic
field (ie, the “quadrupole” term)
By =
¶By
¶x
x
¶B
Bx = x y
¶y
By
Bx
y
x
Note: Ñ ´ B = 0 ®
¶By ¶Bx
=
¶x
¶y

The dipole term is implicitly accounted for in the definition of the reference trajectory
(local curvature ρ).

Any higher order (nonlinear) terms are dealt with as perturbations.

Rotated quadruple (“skew”) terms lead to coupling, which we won’t consider here.
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
28

These are second order homogeneous differential equations, so the explicit
equations of motion will be linearly related to the initial conditions by
æ x(s) ö æ m11 (s) m12 (s) ö æ x0 ö
֍
÷
ç
÷ = çç
÷
ç
è x¢(s) ø è m21 (s) m22 (s) ø è x0¢ ÷ø

Exactly as we would expect from our initial naïve treatment of the beam
line elements.
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
29
x  K (s) x  0

Again, these equations are in the form

For K constant, these equations are quite simple. For K>0
(focusing), it’s just a harmonic oscillator and we write
x( s )  A cos
x' ( s )


Ks 


 
 

 a cos K s  b sin K s
  K a sin K s  K b cos
In terms if initial conditions, we identify a  x0 ; b 
and write



 x( s)   cos K s

  
 x( s)    K sin K s

E. Prebys, NIU Phy 790 Guest Lecture



x0
K
1
sin K s
K
cos K s



Ks

 x 
0
 x 
 0 

March 18 & 20, 2014
30

For K<0 (defocusing), the solution becomes
æ
æ x(s) ö ç
ç
÷ =ç
è x¢(s) ø ç
çè

cosh
(
K sinh
Ks
(
)
Ks
ö
sinh K s ÷ æ
x0 ö
K
֍
÷
ç
÷ è x0¢ ÷ø
cosh K s
÷ø
(
1
)
(
)
)
For K=0 (a “drift”), the solution is simply
x( s )  x0  x0 s
 x( s )   1 s  x0 
  
 
 
 x( s )   0 1  x0 

We can now express the transfer matrix of an arbitrarily complex beam line
with
M  M1M2M3...Mn

But there’s a limit to what we can do with this
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
31

Looking at our Hill’s equation
x  K (s) x  0; K (s  C )  K (s)

x( s ) 



x(s)  A cos K s   so try a solution of the form
assume w( s  C )  w( s), BUT
Aw(s) cos (s)   
 ( s  C )   ( s)
If K is a constant >0, then
If we plug this into the equations of motion (and do a lot of math), we find
that in terms of our Twiss parameterization
   (C )
Phase advance
over one period
w2
 ( s) 
k
ww
1 d  w2 
1 d ( s)
   
 ( s)  

k
2 ds  k 
2 ds
1   2 ( s)
 (s) 
 ( s)
k
1

  2
  (C )   
w
 (s)
E. Prebys, NIU Phy 790 Guest Lecture
s0  C

s0
1
ds
 (s)
Super important!
Remember forever!
March 18 & 20, 2014
32

Generally, we find that we can describe particle motion in terms of
initial conditions and a “beta function” β(s), which is only a function
of location in the nominal path.
b (s) µ w(s)2
x
Lateral deviation
in one plane
x(s) = A b (s) cos (y (s) + d )
s
Phase
advance
ds
 (s)  
 (s)
0
s
The “betatron function” βs is
effectively the local wavenumber
and also defines the beam
envelope.
Closely spaced strong quads -> small β -> small aperture, lots of wiggles
Sparsely spaced weak quads -> large β -> large aperture, few wiggles
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
33

The general expressions for motion are
x = A b cosf;f = y (s) + d
x¢ = -

We form the combination
A
b
(a cosf + sinf )
g x 2 + 2a xx¢ + b x¢ 2
= A 2 (g b cos 2 f - 2a 2 cos 2 f - 2a sinf cosf + a 2 cos 2 f + sin 2 f + 2a sin f cosf )
(
= A 2 (g b - a 2 ) cos 2 f + sin 2 f
)
= A 2 = constant

This is the equation of an ellipse.
x'
A 
A 
x
E. Prebys, NIU Phy 790 Guest Lecture
Area = πA2
Particle will return to a different
point on the same ellipse each time
around the ring.
March 18 & 20, 2014
34

If we evaluate the cell at the center of the focusing quad, it looks like
L
L
2f
2f
-f
Leading to the transfer Matrix
æ 1
1
M =ç
ç çè 2 f
æ 1
0 ö
÷æ 1 L öç 1
1 ÷ çè 0 1 ÷ø ç
÷ø
çè f
æ
L2
1- 2
ç
2f
=ç
ç
L
L2
ç - 2+ 3
çè 2 f
4f
E. Prebys, NIU Phy 790 Guest Lecture
æ 1
0 ö
÷æ 1 L öç
1
1 ÷ çè 0 1 ÷ø ç ÷ø
çè 2 f
æ
Lö ö
2L ç 1+
÷
è 2 f ÷ø ÷
÷
L2
÷
1- 2
÷ø
2f
0 ö
÷
1 ÷
÷ø
Note: some textbooks
have L=total length
March 18 & 20, 2014
35
We know from our Twiss Parameterization that this can be written as
æ
L2
1- 2
ç
2f
ç
ç
L
L2
ç - 2+ 3
çè 2 f
4f
æ
Lö ö
2L ç 1+
÷
ö
b 0 sin m
è 2 f ÷ø ÷ æ cos m + a 0 sin m
=ç
÷
2
÷
g
sin
m
cos
m
a
sin
m
L
0
0
ø
÷ è
1- 2
÷ø
2f
From which we see that the Twiss functions at the middle of the magnets are
a0 = 0
recall
L2
m
cos m = 1- 2 = 1- 2 sin 2
2f
2
m L
® sin =
2 2f
æ
è
b 0 sin m = 2L ç 1+
1 db
a =2 ds
E. Prebys, NIU Phy 790 Guest Lecture
1
b0
Flip sign of f to
get other plane
L ö
2 f ÷ø
mö
æ
çè 1+ sin ÷ø
2
® b 0 = 2L
= b max
sin m
gb - a 2 = 1® g 0 =
b min
mö
æ
1sin
çè
÷
2ø
= 2L
sin m
March 18 & 20, 2014
36

As particles go through the lattice, the Twiss parameters will vary
periodically:

x
β = decreasing
α >0
focusing
β = min
α=0
minimum
s
x
x
x
x
β = max
α=0
maximum
x
x
x
x
β = increasing
α<0
defocusing
x
Motion at each
point bounded by
x( s )  A  ( s )
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
37

It’s important to remember that the betatron function represents a
bounding envelope to the beam motion, not the beam motion itself
Trajectories over multiple turns
Normalized particle trajectory
x( s )  A ( s ) sin  ( s )   
1/ 2
s
ds
 (s)  
 (s)
0
βs is also effectively the local
wave number which determines
the rate of phase advance
Closely spaced strong quads  small β  small aperture, lots of wiggles
Sparsely spaced weak quads  large β  large aperture, few wiggles
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
38
Particle trajectory

Ideal
orbit
As particles go around a ring, they
will undergo a number of
betatrons oscillations ν
(sometimes Q) given by
1

2


ds
  (s)
This is referred to as the “tune”
We can generally think of the tune in two parts:
Integer :
magnet/aperture
optimization
E. Prebys, NIU Phy 790 Guest Lecture
6.7
Fraction:
Beam Stability
March 18 & 20, 2014
39
If the tune is an integer, or low order rational number, then the effect of any
imperfection or perturbation will tend be reinforced on subsequent orbits.

When we add the effects of coupling between the planes, we find this is also
true for combinations of the tunes from both planes, so in general, we want
to avoid
kx x  k y y  integer (resonantinstability)
“small” integers
Avoid lines in
the “tune plane”
fract. part of Y tune

fract. part of X tune

Many instabilities occur when something perturbs the tune of the beam, or
part of the beam, until it falls onto a resonance, thus you will often hear
effects characterized by the “tune shift” they produce.
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
40
x'




If each particle is described by an ellipse with a
particular amplitude, then an ensemble of particles will
always remain within a bounding ellipse of a particular
area:
x
Area = ε
Either leave the π out, or include it explicitly as a “unit”. Thus
• microns (CERN) and
• π-mm-mr (FNAL)
These are really the same
Are actually the same units (just remember you’ll never have to explicity
use π in the calculation)
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
41


Because distributions normally have long tails, we have
to adopt a convention for defining the emittance. The
two most common are

Gaussian (electron machines, CERN):

95% Emittance (FNAL):
In general, emittance can be different in the two
planes, but we won’t worry about that.
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
42

As we go through a lattice, the bounding emittance remains constant

x
x
x
x
large spatial distribution
small angular distribution
E. Prebys, NIU Phy 790 Guest Lecture
x
s
x
x
x
x
x
small spatial distribution
large angular distribution
March 18 & 20, 2014
43

In our discussions up to now, we assume that all fields scale with
momentum, so our lattice remains the same, but what happens to the
ensemble of particles? Consider what happens to the slope of a particle as
the forward momentum incrementally increases.
px
p0

x 
px
p0
x  x 
px
p0  p
 p 
px

 x 1  
p0  p  p0 
 x   x
p
p0
If we evaluate the emittance at a point where  =0, we have
“Normalized emittance”
=constant!
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
44

As a beam is accelerated, the normalized emittance
remains constant

Actual emittance goes down down

Which means the actual beam size goes down as well
betatron function
RMS emittance
95% emittance
v/c


The angular distribution at an extremum (α=0) is
We almost always use normalized emittance
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
45

First “separated function” lattice

1 km in radius

First accelerated protons from 8 to 400 GeV in 1972
1968
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
46

The Main Ring accelerated protons from kinetic energy
of 8 to 400 GeV*
Parameter
Symbol
proton mass
m [GeV/c2]
kinetic energy
K [GeV]
total energy
E [GeV]
momentum
p [GeV/c]
rel. beta
β
rel. gamma
γ
beta-gamma
βγ
rigidity
(Bρ) [T-m]
Equation
Injection
Extraction
0.93827
K + mc2
E 2 - ( mc2 )
( pc ) / E
2
E / (mc )
( pc ) / (mc2 )
p[GeV]/(.2997)
2
8
400
8.93827
400.93827
8.88888
400.93717
0.994475
0.999997
9.5263
427.3156
9.4736
427.3144
29.65
1337.39
*remember this for problem set
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
47



L=29.74 m

Phase advance μ=71°

Quad Length lquad=2.13 m
Beta functions (slide 36)
L
L
2f
From design report
-f
b max,min
2f
mö
æ
çè 1 ± sin ÷ø
2
= 2L
sin m
æ
71° ö
çè 1 ± sin 2 ÷ø
= 2(29.74)
sin 71°
= 99.4 m, 26.4 m

sin
m
2

f=
E. Prebys, NIU Phy 790 Guest Lecture
( Br )
lquad B¢
Magnet focal length
=
L
29.74
®f=
= 25.61 m
°
2f
2sin(71 / 2)
Quad gradient (slide 12)
® B¢ =
( Br ) = (1337.39 ) = 24.5 T/m
lquad f ( 2.13) ( 25.6 )
March 18 & 20, 2014
48

We could calculate α(s),β(s), and γ(s) by hand (slide 25) , but…

There have been and continue to be countless accelerator modeling programs;
however MAD (“Methodical Accelerator Design”), started in 1990, continues to be
the “Lingua Franca”
98.4m (exact) vs.
99.4m (thin lens)
main_ring.madx
!
! One FODO cell from the FNAL Main Ring (NAL Design Report, 1968)
!
beam, particle=proton,energy=400.938272,npart=1.0E9;
half quad
K1=1/(2f)
LQ:=1.067;
LD:=29.74-2*LQ;
qf: QUADRUPOLE, L=LQ, K1=.0195;
d: DRIFT, L=LD;
qd: QUADRUPOLE, L=LQ, K1=-.0195;
build FODO cell
fodo: line = (qf,d,qd,qd,d,qf);
use, period=fodo;
match,sequence=FODO;
24.7m
vs. 26.4m
force periodicity
SELECT,FLAG=SECTORMAP,clear;
SELECT,FLAG=TWISS,column=name,s,betx,alfx,bety,alfy,mux,muy;
TWISS,SAVE;
calculate Twiss parameters
PLOT,interpolate=true,,colour=100,HAXIS=S, VAXIS1=BETX,BETY;
PLOT,interpolate=true,,colour=100,HAXIS=S, VAXIS1=ALFX,ALFY;
stop;
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
49

We normally use 95% emittance at Fermilab, and 95% normalized emittance of the
beam going into the Main Ring was about 12 π-mm-mr, so the normalized RMS
emittance would be
We have divided out the “π”

We combine this with the equations (slide 45), beam parameters (slide 47) and
lattice functions (slide 48) to calculate the beam sizes at injection and extraction.
Parameter
Symbol
kinetic energy
beta-gamma
normalized emittance
Equation
Injection
Extraction
K [GeV]
8
400
βγ
9.4736
427.3144
2x10-6
beta at QD
bmax [m]
bmin [m]
x size at QF
s x [mm]
4.58
.68
y size at QF
s y [mm]
2.36
.35
x ang. spread at QF
s x¢
46.1x10-6
6.9x10-6
y ang. spread at QF
s y¢
89.5x10-6
13.3x10-6
beta at QF
E. Prebys, NIU Phy 790 Guest Lecture
99.4
26.4
March 18 & 20, 2014
50

In our definition and derivation of the lattice function, a closed path
through a periodic system. This definition doesn’t exist for a beam line,
but once we know the lattice functions at one point, we know how to
propagate the lattice function down the beam line.
Mout, in
  out 


  out 
 
 out 
  in 
 
  in 
 
 in 
  out   m11m22  m12 m21 

 
  out     2m11m12 
     2m m 
21 22
 out  
E. Prebys, NIU Phy 790 Guest Lecture
 m11m21   m12 m22  in 
m 
m 
2
11
2
21
m 
m 
2
12
2
22
 
  in 
  
 in 
March 18 & 20, 2014
51

When extracting beam from a ring, the initial optics of the beam line are
set by the optics at the point of extraction.
  in 
 
  in 
 
 in 

For particles from a source, the initial lattice functions can be defined by
the distribution of the particles out of the source
  in 
 
  in 
 
 in 
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
52

In spite of the name, g4beamline is not really a beam
line tool.




Does not automatically handle recirculating or periodic systems
Does not automatically determine reference trajectory
Does not match or directly calculate Twiss parameters
 Fits particle distributions to determine Twiss parameters and
statistics.
Nevertheless, it’s so easy to use, that we can work
around these shortcomings


Create a series of FODO cells
Carefully match our initial particle distributions to the
calculations we just made.
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
53

We’ve calculated everything we need to easily create a
string of FODO cells based on the Main Ring



Won’t worry about bends
Phase advance per cell = 71°, so need at least at least ~5 cells to
see one betatron period. Let’s do 8.
First, create a quadrupole
# Main Ring FODO cell
param L=29740.
param QL=2133.3
param aperture=50.
# Not really important
param -unset gradient=24.479
kill=1 saves time if you make a mistake!
param -unset nCell=8
genericquad MRQuad fieldLength=$QL ironLength=$QL apertureRadius=$aperture ironRadius=5*$aperture
kill=1
Doesn’t really look like a
real quadrupole, but the
field is right and that’s all
that matters.
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
54

Create 8 cells by putting 16 of
these, spaced 29740 mm apart,
with alternating gradients (good
practice for doing loops in g4bl).


If we put the first quad at z=0., the beam
will start the middle of it. Is this what we
want? Why or why not?
We create a Gaussian beam based on the
parameters we calculated on Slide 50
beam gaussian sigmaX=.682 sigmaXp=.00000686 sigmaY=.351 sigmaYp=.00001332 \
meanMomentum=$P nEvents=$nEvents particle=proton

We want to track the first hundred particles individually, so we add
trace nTrace=100 oneNTuple=1 primaryOnly=1

We want to fit the distributions at regular intervals to calculate the beam widths
and Twiss parameters. so we add the lines
param totlen=2*$nCell*$L
profile zloop=0:$totlen:100 particle=proton file=main_ring_profile.txt

Now run 1000 events

Need enough for robust fits
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
55

The individual track information is written to the standard root output file
in an Ntuple called “AllTracks”
TFile ft("g4beamline.root");
TNtuple *t = (TNtuple *) ft.FindObjectAny("AllTracks");

The profile information is written to a text file. This can be read directly
into histoRoot. I’ve provided a class (G4BLProfile) to load it into root*
G4BLProfile fp(filename);
TNtuple *p = fp.getNtuple();

We want to create a plotting space on which we can overlay several plots.
The easiest way I know do to this is an empty 2D histogram.
TH2F plot("plot","Track Trajectories",2,0,sMax,2,-xMax,xMax);
plot.SetStats(kFALSE); //turn off annoying stats box
plot.Draw();

Overlay a 3 sigma “envelope”, based on the fitted profiles
p->Draw("3*sigmaX:Z","","same");
p->Draw("-3*sigmaX:Z","","Same");
“same” option draws over existing
plot
*http://home.fnal.gov/~prebys/misc/NIU_Phys_790/
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
56
t->SetMarkerColor(kRed); // These are points, not lines
t->Draw("x:z","EventID==1","same");
...
~ 360° = betatron period
71°
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
57

If we overlay all 100 tracks (remove “EventID” cut), we see that although
each track has a periodicity of ~5 cells, the envelope has a periodicity of
one cell..
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
58

We can plot the fitted lattice functions and compare them to our calculations.
TH2F beta("beta","Horizontal (black) and Vertical Beta Functions",2,0.,sMax,2,0.,120000.);
beta.SetStats(kFALSE);
beta.Draw();
beta functions in mm!
p->Draw("betaX:Z","","same");
p->SetMarkerColor(kRed);
p->Draw("betaY:Z","","same");
slight mismatch because of thin lens approximation
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
59

In our previous discussion, we implicitly assumed that the distribution of
particles in phase space followed the ellipse defined by the lattice function
x'


x'


x
…but there’s no guarantee
What happens if this it’s not?
Area = 

Once injected, these particles will
follow the path defined by the lattice
ellipse, effectively increasing the
emittance
x
Injected
particle
distribution
x'
x
E. Prebys, NIU Phy 790 Guest Lecture
Lattice
ellipse
Effective
(increased)
emittance
March 18 & 20, 2014
60

In our example, we carefully matched our initial distributions to the
calculated lattice parameters at the center of the magnet. If we start
these same distributions just ~1m upstream, at the entrance to the
magnet, things aren’t so nice.
Individual track trajectories look
similar (of course)
E. Prebys, NIU Phy 790 Guest Lecture
Envelope looks totally crazy
March 18 & 20, 2014
61

If we fit the beta functions of these distributions, we see that the original
periodicity is completely lost now.

Therefore, lattice matching is very important when injecting, extracting,
or transitioning between different regions of a beam line!
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
62

Built at CERN, straddling the French/Swiss border

27 km in circumference

Started in 2008, broke, started again in 2010

Has collided two proton beams at 4000 GeV each

In 2015, will reach design energy of 7000 GeV/beam (~7 x Fermilab Tevatron)
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
63
Design:


7 TeV+7 TeV proton beams

Can’t make enough antiprotons for the
LHC

Magnets have two beam pipes, one
going in each direction.
Stored beam energy 150 times
more than Tevatron



27 km in circumference

2 major collision regions: CMS and ATLAS

2 “smaller” regions: ALICE and LHCb
E. Prebys, NIU Phy 790 Guest Lecture
Each beam has only 5x10-10 grams
of protons, but has the energy of a
train going 100 mph!!
These beams are focused to a size
smaller than a human hair to collide
with each other!
March 18 & 20, 2014
64

e+e- or proton-antiproton (opposite charge) colliders had particles going in
opposite directions in the same beam pipe

Because the LHC collides protons (same charge), the magnets have two
apertures with opposite fields
quadrupoles
dipoles (Bmax = 8.3 T)
Problem set is just our in class example with different numbers.
E. Prebys, NIU Phy 790 Guest Lecture
March 18 & 20, 2014
65