INSTABILITIES IN LINEAR ACCELERATORS
A. Pisent
INFN Laboratori Nazionali di Legnaro, Padova, Italy
Abstract
Coherent instabilities are one of the effects that limit the performances of
modern linear accelerators. Beam break up in electron linacs and, in more
details, envelope instabilities in proton linacs are discussed in this lecture.
1.
INTRODUCTION
Linear accelerators are used in many different fields, both as stand alone machine and as injectors of
other machines (mainly synchrotrons). The current (or bunch population) in modern linacs is generally
high, so to increase the performances in terms of luminosity, or efficiency, beam quality....As a
consequence the collective behavior of the beam can become an issue, and collective instabilities can
be observed, or determine some design constraints.
In this paper we treat two examples of transverse instabilities, one for electron linacs and one for
proton linacs. The transverse motion can give losses in a more intuitive way, and can be analyzed
separately from longitudinal motion in both cases. For electron linacs (after few MeV) the
longitudinal motion is decoupled since all particles have the same velocity. For proton linacs we
considered the stability of a continuous monoenergetic beam; this is generally a good approximation
of a linac beam, even if sometimes longitudinal and transverse beam frequencies are similar and the
three dimensional problem has to be solved.
1.1 Definition of instability
Particles in the linac are driven by magnetic lenses and RF cavities; the corresponding forces are
generally periodic, sometimes with adiabatic changing parameters. In this way the accelerator
designer determines a beam with the same periodicity, in order to have a simple transport all over the
structure.
In other words it is required as a nominal (or unperturbed) situation:
 
 
f 0 ( x, p, t  T )  f 0 ( x, p, t )
(1)

 
with f ( x , p, t )d 3 xd 3 p number of particles in the elementary volume of phase space, since x are the

three spatial coordinated and p are the associated moments; t is time and T is the period.
This is the result achieved for single particle dynamics in the first lessons, where the periodic beta
function defines the envelope equal to itself period after period. But if the beam intensity increases the
electromagnetic forces determined by the beam cannot be neglected, the condition (1) has to be
preserved in presence of external plus internal beam forces. In the next section we shall say something
more about internal forces.
But even when a periodic distribution f0 , which satisfies condition (1), exists and is known, in real life
the beam can deviate from the periodic distribution by a small amount f1, and evolve, in linear
approximation, according to:
 
 
f ( x , p, t )  f 0 ( x , p, t )  f1e it
(2)
with f1 perturbation amplitude. If  is real, it represents the frequency of the beam mode; if it is
complex, the beam is unstable and   1 Im   0 is the instability rise time.
Moreover we define the threshold of the instability as the minimum beam current that gives 
complex. In the following of the lecture we will show in some detail how this scheme works in
practice, since for the envelope instability we shall determine
 the periodic solution,
 the beam modes due to a perturbation, in section 4.1.1 and 4.3
 the possible instability of such modes in section 4.1.2 and 4.4
 the threshold at which the instability occurs in section 4.5.
1.2
Electromagnetic fields generated by the beam
Each particle in the beam feels the effect of the electromagnetic field generated by the other particles
in the field boundaries due to the accelerator elements. The mathematical problem corresponds to the
solution of the Maxwell equations in the presence of the beam charges in motion, with accelerator
boundary conditions. In practice the problem can often be simplified, in consideration of beam energy
and acceleration frequency.
1.2.1 Space charge
We already saw in the previous lecture that the space charge, i.e. the direct interaction between
particles, scales dramatically with energy. In the case of a continuous and homogeneous beam, with
round cross section of radius a, the space charge contribution to the equation of motion inside the
beam can be calculated using Gauss and Ampere laws:
e( E x  B y c) x
Fx
d 2x


 2
2
2
2
ds
mc  
mc 2  2
2a
(3)
with s longitudinal space coordinate, and

I
I C  3 3
(4)
space charge parameter, Ic=0mc3/e characteristic current (7.8 MA for protons, 4.2 kA for electrons).
This contribution is of course repulsive, but at high energy decreases due to the compensation of the
magnetic field (1/2 contribution), the increased beam rigidity (1/) and the lower longitudinal
charge density (1/). Direct space charge is generally interesting for low energy proton linacs. The
corrections due to the external boundaries, introduced in the space charge lecture, are generally small
since the RF frequency is low and the structures rather big (the vacuum pipe is large).
1.2.2 Wake fields
For beam instabilities in linacs we are also interested to the opposite case, where high energy electrons
travels in very high frequency structures. In this case direct space charge can be neglected, but the
electromagnetic waves caused by the charges in the structure can heavily interact with the particles
that follows.
This is a complex topic that is extensively discussed in other CAS lectures [1][2][3]. We just give here
a pictorial view and introduce the minimum formalism necessary to give an example of instability.
The presence of beam charges perturbs the field and generates waves inside the accelerating structure;
this is like when we turn on the engine of a boat in the middle of a calm lake and more or less circular
waves propagate around us. The (phase) velocity of the waves is independent from the perturbation,
and is (about1) c for the electromagnetic waves, and much lower for the waves in the lake. When the
boat starts to move and accelerates, at a certain moment its velocity overcomes the wave velocity and
the wake forms behind it. Before entering this new regime there is a critical moment, when the boat
has the same velocity as the wave, and it has to overcome all its waves at the same time (sonic bang
for supersonic airplanes).
Electrons traveling in a high energy structure are very close to this situation, since both their velocity
and wave velocity are close to c. The fields left behind the particle are called wake fields.
For example a (source) charge Q1 traveling with a (small) offset x1 respect to the center of the RF
structure perturbs the accelerating field configuration and leaves a wake field behind (Fig. 1). A
following (test) particle will experience a transverse field proportional to the displacement and to the
charge of the source particle:
Fx  e
w
Q1 x1
L
(5)
The proportionality factor w, mainly function of the delay  between the two particles, is called
transverse wake function per period (of length L), and expresses the electromagnetic aspects of the
problem (see appendix 1). It turns out for example that it has a strong dependence on RF frequency.
The contribution of this force to the motion of the test particle is of dipole kind (independent from test
particle coordinate x) and reads:
x "
eQ1w
x.
mc 2 L 1
(6)
As it happens for the space charge also this contribution decreases with energy, but much slowly.
Knowing w() the effect of a more general particle distribution applying the superimposition principle
can be calculated; in mathematical terms w is the solution expressed as a Green function, and in the
general case one has to integrate over the source particle distribution.
w
Fx  e Q1 x1
L
x
1
Q1
s
Fig. 1 Definition of the transverse wake function w: wake fields generated by the source charge Q1, and force
acting on the following test particle.
The wake function can be calculated and measured with various methods. It depends on the details of
3
the resonator geometry, but has always a strong dependence on RF frequency ( w   ).
1
The phase velocity is c for a wave in free space, but in a wave guide can be different (see the lecture on
accelerating structures).
We can try a very schematic statement: beam generated forces, relevant for high current beams, are
mainly due to direct space charge for low energy low frequency hadron linacs, and mainly due to
wake fields in high frequency high energy electron linacs. In the following of this paper we will
outline how beam instabilities can arise in both regimes.
This statement is as schematic as needed for a lecture; a counter example can be given immediately
and is represented by the RF electron guns; they are RF structures where the electrons are generated
(for laser driven photoemission) in a high electric field region and rapidly accelerated. The RF
frequency can be around 3 GHz, and the current in the pulse is very high. As a result the direct space
charge (dominating just in the first millimeters after the cathode) and the effect of the structure cannot
be decoupled so that the complete electromagnetic problem has to be solved.
2.
ELECTRON LINACS
2.1
Beam breakup
A very well known instability affecting beam quality in electron linacs is the beam breakup (BBU).
When the bunch current is high it can happen that the transverse oscillation of the head of the bunch
causes a resonant growth of the transverse oscillation of the tail, with consequent deformation of the
bunch, emittance growth and possibly beam losses. The emittance growth is particularly detrimental
in the case of a linear collider where the luminosity is affected.
We sketch here the basic mechanism by considering two particles, representing the head and
the tail of the bunch. The head particle (with transverse coordinate x1) undergoes transverse
oscillations due to magnetic quadrupoles, organized in some regular pattern (for example a
FODO). Let’s approximate its motion with a pendulum equation:
x1"Kx1  0
(7)
where K is constant. This approximation, called “smooth approximation” is valid if the phase advance
per period is small (see Appendix 3).
The tail particle, with transverse coordinate x, will experience the same external focusing force, plus
the force due to the wake field generated by the head particle. Namely, making use of equation (6):
x " Kx 
eQ1w
eQ1w
x1 
A cos K s
2
mc L
mc 2 L
(8)
with A amplitude of the head particle oscillation. The motion of the tail particle, forced at its own
natural frequency, is resonant:
x 
eQ1 wA
z sin K s  A cos K s
2mc 2 L K
(9)
and will experience a linear growth (fig.2). The unstable motion of the tail gives rise to a “banana”
shape of the bunch, that increases beam cross section at the experimental point.
Problems similar to head tail effect can occur between one bunch and the following ones, so that if the
range of wake fields is sufficiently long the motion of the bunch that follows can grow; this
phenomenon is called multibunch beam breakup, and can be very dangerous for the next generation of
linear colliders.
R
F
s
tr
u
c
t
u
r
e
4
2
x1(z)
x2(z)
R
F
s
tr
u
c
t
u
r
e
0
2
4
0
5
10
z
15
20
Fig. 2 Evolution of bunch head and tail along the linac; as a result the beam shape is deformed, as sketched in
the upper part of the figure.
2.2 Limitations for new linear colliders
The new linear collider will be a very large, complicated and costly machine. The main options in this
moment are three, one based on a superconducting 1.3 GHz (TESLA), one on a normal conducting 11
GHz linac (NLC) and one on a 30 GHz normal conducting linac (CLIC). There are many substantial
differences in the three approaches, different time schedule and physics goal [4][5][6].
In general the higher operating frequency allows gaining in compactness, but wake fields worsen with
the third power of frequency, and can compromise the beam quality and luminosity. For this reason in
the last decade a big effort has been put in the study of beam manipulation approaches able resist
BBU, and in the development of new structures with lower wake fields.
The approaches to limit the wake fields are mainly addressed to dump and detune the dipole modes. In
the first case particular structures has been studied where the unwanted modes are coupled to lossy
materials. For the second approach the geometry is changed in such way that the frequency of dipole
modes vary along the linac giving an attenuation of the wake fields due to decoherence of the various
components. The two methods are indeed complementary, since the decoherence gives a rapid
decrease, but on the long range the various components can compose again. In fig. 3 we show the
dumped and detuned structure developed at CERN.
Some silicone carbide is located in a position reached by the dipole mode field, but not by the
accelerating mode field (TM01).
Fig. 3 Detuned and dumped structure developed at CERN (Curtesy of I. Jensene).
3.
PROTON LINACS
Modern high intensity proton linacs, like the one in construction for SNS in the United States, are
based on superconducting cavities for the main part of the linac. The operating frequency is much
lower then for linear colliders (800 MHz for SNS, but 352 MHz are proposed for CERN SPL) and
large beam holes, so to avoid that halo particles hit and activate the structure. As a consequence wake
fields are very weak and only direct space charge is important ( is less than 2).
Direct space charge can drive an instability, called envelope instability, that causes exponential
growth of some of the beam modes. The necessity to avoid this phenomena enters in the design
constrains.
3.1
The basic mechanism
3.1.1 Beam modes
Beam modes can be classified according to their pattern on the transverse x y plane. In figure 4
dipole, odd and even mode are shown.
Metallic boundary
y
x
dipole mode
even mode
odd mode
Fig. 4 Transverse pattern for three beam modes; in green is the initial configuration, while dotted is the
configuration after half a period.
In dipole oscillations the center of mass of beam distribution varies. In odd and even oscillations the
center of mass of the distribution remains on axis and the beam shape changes, keeping the x and y
decoupling of motion.
In the case of beam break up (previous chapter) the dipole oscillations of the tail particle where made
unstable by wake fields. In the case of proton linacs, where only direct space charge is important, it is
very unlikely that dipole oscillations become unstable.
Indeed space charge forces are internal forces, and as such they cannot influence the motion of the
center of mass (due to the third principle of Newtonian dynamics). So space charge cannot make
unstable dipole oscillations. We shall therefore look deeper just on odd and even modes.
3.1.2 Envelope oscillations
If we look to the pattern of the odd mode it can look somehow familiar, since it is the deformation that
the beam shape undergoes during a FODO period; in fig 5 we show the beam envelopes in the two
planes, the focusing function K(s) and the phase space representation.
In the first lectures we learned (in absence of space charge) how to find the periodic envelopes a and b
(defined using betatron function as (xx)1/2). As a result the beam distribution in phase space f0
between the beginning and the end of the period is conserved, but each particle advances along its
invariant ellipsis of an angle  (phase advance, half of the trace of the transport matrix); the subscript
0 means absence of space charge.
2
2
0  3  
180

 220.04084
xmax
F
0.002
b
a
x
ax, ay K(s) arbitra ry sc ale
y
Zi
D
F
a
1
 Zi
3
kkk
i
kkk
0
K
0

s
axT
2
b
0.002
Real space
 xmax
0
0.1
0.2
0.3
0.4
0.5
0
Zi
0.6
0.7
0.8
0.9
P
0
T
length s/L
x’
x’
a
f0(s)
a
f0(s+L)
x
0
a
x
Phase space
Fig. 5 Periodic channel of FODO kind (Focusing zero Defocusing zero). In the top of the figure the beam
configuration in real space and beam envelopes a(s) ,b(s) and focusing function K(s) are plotted. In the bottom
part the transformation in phase space is indicated.
When current cannot be neglected, one still transports the beam following a nominal solution f0,
corresponding to periodic envelopes, and phase advance per period . Following the scheme of
chapter 2, we want to check the stability of f0.
The modes that enter into play are those determined by envelope oscillations: in real life the beam will
follow the periodic solution with a certain error. This circumstance (called mismatch) will determine
some envelope oscillations that for our example are the modes f1 we want to analyze.
What happens with no current can be understood from phase space picture (Fig. 6). Since the ellipsis
that represents our beam is not invariant, it will rotate period after period; in particular since every
point advances of an angle 0 after a period on the invariant ellipsis, the thick point in Fig. 6 (left) will
come back to its initial phase after 20 periods. Indeed due to the symmetry of the ellipsis, the
distribution is identical when the thick point has done half a turn.
0
x’
180

 29.85764

180


 29.85764
 180  59.71528
2
2
2 0   

2
2
0  3  
180

 59.71528
f(s+L)
ax, ay K(s) arbitrary scale
0.002
a
invariant curves
f(s)
x
0
Zi 1
 Zi 3
0
kkki axT

kkk0 2
0.002
Envelope wavelength=L*2envelope
0
2
4
6
8
10
12
14
Zi 0
T
length s/L
Fig. 6 Envelope oscillations in the zero current case; the motion of the red point determines the rotation of the
mismatched beam ellipsis in phase space.
We can therefore conclude that in absence of space charge the envelope phase advance is 20, since
after half turn of the red point the particle distribution is identical. Vertical and horizontal motion are
decoupled so that odd and even mode of the envelope oscillation will have the same frequency.
In presence of space charge the situation is a little modified: the single particle phase advance per
period  is depressed respect to 0, due to the space charge defocusing. When mismatched the beam
envelope will oscillate as described above for zero current case, but now the space charge
discriminates the odd and even mode, that have different frequency. Namely (in smooth
approximation, see appendix 3):
 odd  3 2   02
(10)
 even  2( 2   02 )
(11)
that in the limit case of small current (=0) reduces to the known result 20.
0
180

 29.85764

180


 14.72113
 180  47.07845
2
2
2 0   

2
2
0  3  
180

 39.26339
ax, ay K(s) arbitrary scale
0.002
Zi 1
 Zi 3
0
kkki axT

kkk0 2
0.002
2
2
0  3  
0
2
4
6
8
10
12
180

 39.26339
14
Zi 0
xmax
T
length s/L
ax, ay K(s) arbitrary scale
0.002
Zi 1
 Zi 3
0
kkki axT

kkk0 2
0.002
Envelope wavelength=L*2envelope
 xmax
0
2
4
6
0
8
10
2
12
2
0  3  
Zi 0
T
length s/L
180

14
 39.26339
P
xmax
ax, ay K(s) arbitrary scale
0.002
Zi 1
 Zi 3
0
kkki axT

kkk0 2
0.002
 xmax
0
0
2
4
6
8
Zi 0
10
12
14
P
T
length s/L
Fig. 7 Envelope oscillations in presence of space charge (0=300 and =150); the matched envelopes, the odd
and even modes are plotted.
3.1.3 Envelope instability
In Fig. 8 we plot the phase advance of the odd and even mode for a given external focusing (0=1040)
decreasing the depressed phase advance . This corresponds for example to an increase of beam
current. At a certain current one of the modes arrives to 180 deg phase advance, i.e. to a half integer
resonance (called also parametric resonance). This resonance gives a fast exponential growth of beam
envelopes.
In fig 8, lower part, we show how in resonance conditions the beam envelope explodes due to the
numerical errors of the computer.
even  104  0  208
208
220
odd kk 0
0=104 deg
0=208 deg
200
180
even kk 0 160
180
140
120
Increasing beam current
104 100
120
100
80
0
 180

60
40
20
0
kk
105
 104.07954

 180

0
Depressed tune 
 69.62981
2  0
2

2
 180

 177.09241
0
2
2 180
3  
 159.30311

ax, ay K(s) arbitrary scale
0.004
0.002
Z
i 1
Z
i 3
0
kkk
i  axT
kkk 2
0
0.002
0.004
0
5
10
15
Z
20
25
30
i 0
T
length s/L
Fig. 8 Odd and even mode phase advance for increasing beam intensity and unstable envelope pattern in
correspondence of a semi integer resonance (0=1040 and =700).
In the next chapter we will explain in more formal terms this phenomenon, starting from the
development of the envelope equations with space charge. The tools we shall introduce have much
wider applications than envelope instability analysis.
3.2
Envelope equations with space charge
Linear accelerators, can operate in much heavier space charge condition respect to rings. Even if both
kinds of machine use similar focusing scheme (for example FODO), a linac can operate with a
stronger tune depression due to space charge. Indeed in a linac is possible to have 0=0.5; in a ring
instead the tune shift Q-Q0 is limited to about half unit, so that Q/Q0 is smaller (Q0 can be large, and is
always more than 1).
The fact that a linac is more robust is mainly related to the fact that the beam passes only once in the
focusing structure. To calculate correctly the beam envelopes in linacs (and transfer lines) one has to
add the space charge term, while for most applications in rings it is enough to consider unperturbed
envelopes and to calculate the tune shifts2.
For a homogeneous charge distribution the forces are linear:
Fx
mc  
2
2

x
a ( a  b)
.
(12)
This is the extension of eq. (3) from round to elliptical beam. Here and in the following we shall omit
to write the equation for the plane y when the extension is clear.
y
Fx
mc  
2
b
2

x
a ( a  b)
a x
Fig. 9 Transverse force (electric and magnetic field effect) seen by a particle inside a homogeneous charge
distribution with elliptical cross section.
The single particle equations of motions are:

 
x"  K ( s) 
x0
a(a  b) 


 
y"  K ( s) 
y0
b(a  b) 

(13)
that are linear, but with a term that depends upon beam dimensions. The solution can be written in
terms of Floquet functions, like for the zero current case:
2
The bare and space charge depressed tunes are Q0   0 n 2 and Q   n 2 where n number of FODO
periods in the ring can be large. The space charge effects in the ring depends on Q  Q0  Q while the space
charge effect on envelope depends on /0.
x  a( s) exp( i x ( s)),
(14)
y  b( s) exp( i y ( s))
which satisfy the phase and the envelope equation:
 x'
a" Ka 
x
x
a2
2
a3

.

ab
(15)
0
These two last equations correspond to the real and imaginary part when the solutions (14) are
substituted in the equations (13).
~ ( s  L)  a~( s) of the envelope equation (15), the beam is matched,
Choosing the periodic solution a
and the phase advance per period is:
   ( s  L)   ( s )  
sL
s

ds
a~ 2
(16)
Indeed the approach we have sketched gives a consistent solution of the beam evolution only if it
exists a particle distribution that projected in the x x’ and y y’ planes is an invariant ellipsis, while
corresponding to an homogeneous charge distribution in x y plane. Such distribution exists (is called
Kapchinsky Vladimirsky) and is discussed in Appendix 2.
3.3 Envelope modes
The real beams will be matched with a tolerance [7][8]:
a ( s )  a~ ( s )   x ( s )
~
b( s )  b ( s )   y ( s )
(17)
and the evolution of s can be calculated with linearized envelope equations:
2

3 x
 

 x "  K  ~ 4 
 
2 x
~
~ 2 y  0
a
a~  b 
a~  b





2

3 y
 

 y "  K  ~ 4 
 
2 y
~
~ 2 x  0
b
a~  b 
a~  b




(18)

In particular in smooth approximation (see Appendix 3):
 
K  0
 L
L2
 2   02   2
2a
~
a~  b  const  a

the system (17) becomes:
L
a2
2
(19)


x   y "
2
 even
  y "
2
 odd
x
L2


L2
x
x
y  0
(20)
 y  0
with
 odd  3 2   02 and  even  2( 2   02 ) ,
that is the result used in the initial outline. It is interesting to observe that the mode configuration is
such because the external focusing strength is the same in x and y, or in other words because F and D
quadrupoles have the same strength.
This is most of the time correct in linacs, while in storage rings, due to curvature, the horizontal
focusing is generally stronger. This procedure can be extended to the general case [9], and if the two
zero current tunes are very different the envelope oscillations in x and y are decoupled.
3.4 Parametric resonance and envelope instability
The linear (and canonical) system of the perturbed envelopes:
2

3
 

 x "  K  ~ x4 
 
2 x
~
~ 2 y  0
a

a~  b 
a~  b



2

3 y

 y "  K  ~ 4 
~
b

a~  b




 
2 y
~ 2 x  0

a~  b



is not formally different respect to the linear equations of motion for a single particle with non
negligible coupling between the two degrees of freedom. The solution, i.e. the evolution toward one
period, is a linear transformation:
 x 
 x 
 '
 
 x   M  x '
 y 
 y 
 
 
 y ' s  L
 y ' s
(21)
where M is a 4*4 matrix real and symplectic. This last word expresses the internal relation between
matrix coefficients due to the canonical characteristic of the problem; namely:
MJM T  J
with
0
 1
J 
0

0
0
0
1

0  1 0
1
0
0
0
0
0
For one degree of freedom this condition corresponds simply to detM=1 (Liouville theorem).
The evolution of beam envelopes for N periods is calculated according to:
(21)


 s  NL  M N  s
with
 x 
  x '
  
 y 
 
 y '

In particular for an eigen vector  of M, defined by:


M  
with  complex number called eigenvalue, the time evolution is determined by the powers of :


M N   N 
If   1 the envelope is unstable (exponential growth period after period), if   1 the solution is
oscillatory and stable. In this sense the eigenvector is the mode configuration and the eigen-value 
give the mode frequency.
 i
 i
For example in smooth approximation   e even and   e od d . This means that the envelopes are
always stable.
In the most general case the four eigen values of the matrix M will lay on the complex plane following
the conditions:
1.  eigenvalue implies * eigenvalue, since M is real
2.  eigenvalue implies 1/ eigenvalue since M is symplectic3.
The cases of fig. 10 can therefore occur. In the left part of the figure, the two couple of eigenvalues
separately satisfy the two condition above, either laying on the unit circle (stable case) or being on the
real axis (unstable case). The circle (stable) and the coordinate axes (unstable) do cross for =1800,
that is indeed the phase angle where the parametric resonance and the instability occurs.
In the right part of fig 10 is also shown that an other unstable condition can occur (involving all the
four eigenvalues) as analyzed in ref [7].
3
Poof: The spectrum of M (i.e. the set of eigenvalues) is the set of solutions of the characteristic equation (in ):
det( M  I )  0
with I identity matrix. By complex conjugating this relation we prove that  eigenvalue implies * eigenvalue,
since M is real. Moreover by transposing the argument of the determinant we prove that M T has the same
spectrum of M, and by inverting the argument of the determinant we prove that 1/ is eigenvalue of M-1. Finally
by using condition (21):
det( M T  I )  det( J 1M 1 J  I )  det( J 1 ) det( M 1  I ) det( J )  0
proves that  eigenvalue implies eigenvalue.
Im 
unstable solution
1
Im 
=eienv
1
Re 
Re 
Stable solution
confluence:
2 deg of freedom
Parametric resonance:
1 deg of freedom near 180 deg
Fig. 10 Eigenvalues of the envelope transport matrix.
3.5
Threshold of the instability
The threshold of the instability can be explicitly evaluated in smooth approximation. Indeed from the
two relations:

L
   
2
a2
2
0
L2
2a 2
(22)
one can get explicitly the depressed phase advance:
    2  4 02
 ( , 0 ) 
2
(23)
with

L I
L

2 I C 2 N  2 2
(24)
that depends on beam energy, current and normalized emittance N. But in smooth approximation we
know explicitly the beam frequency, and we can determine the threshold as the value of  that
corresponds to the parametric resonance (see fig. 11 ). Therefore:
   Threshold ( 0 )
(25)
determines the stability. For example in fig. 11 is shown that the threshold for 0=1200 is Taking
as an example L=1 m,  (protons at 5 MeV) and normalized emittance 10-6 m, the instability
threshold is 320 mA. Note that this is valid for a continuous beam, to be compared with the peak beam
current in a linac (about 200 times the average). The threshold is therefore 16 mA.
If should also be noticed that if 0 is smaller than 90 deg the beam is always stable. This is indeed
used as a design criterion for high current machines. A low phase advance per period, and
consequently a smooth focusing, has some costs in terms of beam dimensions (see appendix 3), but
guarantees the stability of envelope oscillations.
It should be added that even when 0 is smaller than 90 deg the envelope oscillations have an impact
on the beam dynamics since they can drive resonances on single particle dynamics, as mentioned in
the next chapter.
2 40
2 40
2 20
2 00
 
 
o dd    0 

  1 80  0


1 80
1 80
1 60
  1 80
 

ev en    0 
 0

1 80 
 

)
1 40
1 20
 

   0 

1 80

1 80
1 00

80
1 80
60
40
20
2 .51 21 7
0
0 .1
0 .1
2 40
Threashold
1
10

2 40

1 00
1 00
2 20
2 00
 
 
o dd    0 
  1 80
 1 80
 0

1 80 
 1 60
ev en    0 
 
   0 

  1 80
 0 1 40

1 80 

1 20
  1 80

1 80 
)
1 00
80
1 80
60
40
20
1 .41 33 7
0
0 .1
0 .1
1
10


1 00
1 00
Fig. 11 Space charge depressed phase advance and envelope mode phase advance as a function of the parameter
, proportional to beam current.
4.
BEAM HALO
New high intensity proton linacs require an extremely good control of beam losses. Typically the
required 1 W/m of beam losses means 10-7/m for 1 GeV 10 mA. The reasons one can think to
determine such small losses are various, and indeed their complete control requires probably a
knowledge of beam dynamics deeper than presently available. Nevertheless some interesting
phenomena are observed in computer simulations even in the simplified hypothesis of a continuous
beam. It can generally be observed the formation of a cloud of few particles around the beam core
(beam halo) particularly important when the beam is mismatched.
This suggests a physical explanation that involves the envelope oscillations we have studied in the
previous sections.
It should also be noted that the development of superconducting structures with large bore apertures
allows a good margin between beam dimensions and structure bore hole, reducing the risk of beam
losses and structure activation.
Matched beam
40% Mismatched beam
Fig. 12 Beam halo for a mismatched beam
4.1
Single particle effects due to space charge
Some essential points can be understood if analyzing the beam halo formation as a single particle
effect in the space charge field.
The idea is that the number of particles that evaporate from beam core is anyway so small that they do
not contribute to the electromagnetic field in which the particles evolve, field due to external forces
and space charge forces generated by beam core. The halo particles are driven far from the core by
non linear resonances (driven mainly by space charge), with some analogy with what happens in
hadron colliders (the number of iterations is smaller, the non linearity needs to be larger).
A very good approximation of the problem is to consider core of the beam following the known
dynamics, and then analyze the dynamics of the few halo particles.
For two degrees of freedom the analysis of core motion is the one we have developed in the previous
chapter, so that here we can give an explanation of how beam mismatch enters into single particle
dynamics.
The equations of motion are determined by space charge and external forces, and have the form:

 
x" F (a , x , s )  0
(26)
where the coordinates a corresponds to envelope.
If the beam is matched the envelopes (and consequently the force) is periodic:
 
 
F (a , x , s  L)  F (a , x , s )
(27)
so that the single particle dynamics can be analyzed with the iteration and stability analysis of the one
turn map (Poincarè map) similarly to what is done for external forces in rings.
If the beam is mismatched instead the force will contain not only the single particle frequencies, but
all the envelope frequencies so that there will be resonances when:
h1 * x  h 2 * odd  h 3 * even  2 * h 4
h1 , h 2 , h 3 , h 4 integer
(28)
In fig 13 we plot the transverse frequency (phase advance per period in 2 units) for a particle with
different initial conditions. The frequency is calculated with a Fast Fourier Transform of the simulated
particle trajectory.
In all cases we took x’=0, while the initial coordinate moved from inside to outside the beam. Inside
the beam the frequency is given by the space charge depressed phase advance. Increasing the
transverse coordinate the frequency decreases, reaching asymptotically the zero current phase advance
for trajectory so external to make space charge unimportant.
The curve between these extreme points is not smooth, but have a structure that gives a
characterization of the non linear behavior of the various trajectories. This method is called Frequency
Map analysis, and permits to associate smooth behaviour to regular trajectories, the plateaus to non
linear resonances, the fuzzy behavior to chaotic trajectories [9][10]. In particular the plot in fig. 13,
that correspond to the unmatched beam of Fig. 12, is rather reach; it shows the resonance 1/5
(explicitly 5 x  2 ), and two strong resonances with beam frequency, namely 2 odd  2 and
2 even  2 .
The presence of these resonances and their possible overlap, make a mismatched beam less stable than
a matched beam, and allow beam halo formation. The practical consequence of this is that in a high
power linac one has a maximum mismatch allowed, that translates in prescriptions in the way beam
transitions have to be done, and in mechanical tolerances in the alignment of lenses.
Single particle frequency
+/4

0/2
-/4
/2
Beam core
Initial particle coordinate x/a
Fig. 13 Frequency map analysis: the Fast Fourier of the motion of particle for different initial coordinate x and
initial momentum x’=0 is calculated.
5.
CONCLUSIONS
Linacs are single pass machines where the attainment of the maximum intensity is generally (i.e.
always) required. This limitation is often given by instabilities.
In electron linacs wake fields drive beam breakup and multi-bunch instabilities.
In proton linacs direct space charge determines envelope modes, that can either become unstable
(envelope instability) or drive beam halo formation.
ACKNOWLEDGEMENTS
...............................
REFERENCES
[1] A. Mosnier “Instabilities in linacs” CERN 95-06 p.437.
[2] L. Palumbo, V.G. Vaccaro, M. Zobov “Wake fields and impedances” CERN 95-06 p.307.
[3] A.W.Chao “Physics of Collective beam instabilities in High energy Accelerators”. WileyInterscience Publication (1993)
[4] M. Dohlus “Accelerating structures for Multibunches” Conference Linac ’96, CERN96-07 p.
565.
[5] R.H. Miller “Accelerator structures for Linear Colliders” Linac 2000 Conference, SLAC-R561 p. 366
[6] G. Carrot et al Linac 2000 Conference, SLAC-R-561 p. 416
[7] J. Struckmeier, M. Reser “theoretical studies of Envelope Oscillations and Instabilities of
Mismatched Intense Charge Particle beams in periodic focusing channels” Particle
Accelerators Vol. 14 pp. 227-260 (1984)
[8] Qian Qian, C. Davidson “ Non linear dynamics of intense ion beam envelopes” Phys. Rev. E,
Vol 53,Num. 5 (1996).
[9] A. Bazzani, M. Comunian, A. Pisent “Frequency Map Analysis of an Intense Mismatched
beam in a FODO channel.” Particle Accelerators Vol. 63 pp 79-103 (1999)
[10] J. Laskar Physica D Vol 67 p 257 (1993)
APPENDIX 1 – DEFINITION OF WAKE FIELDS
Following reference [2] the transverse wake function due to the passage through a structure is defined,
for a dipole mode, according to:
1
w (r , r1 ; ) 
eQ1r

F

(r , z , r1 , z1 , t )dz [V/Cm]
(1)

where F is the transverse force on the test particle, with the subscript 1 are denoted the coordinates
of the source particle that generates the field,  denotes the delay between source and test particle.
As an effect of the integration (1) in a periodic structure only the field components synchronous to the
particle give a significant contribution. In other words, in perfect analogy to what happens for the
accelerating field, due to the periodic loading of the wave guide the dipole modes will form some pass
bands; as a consequence some field component have phase velocity equal to particle velocity and
contribute to the integral (1). For a single mode

w ( )  2ke
 r
2Q
sin  r [V/Cm]
(2)
with k kick factor of the mode, r frequency and Q quality factor of the mode. The factor k scales as
the third power of the operating frequency.
In the more general case the wake field is given by the superposition of more modes. The detrimental
effect of transverse mode con be diminished by decreasing Q, or changing the dipole mode frequency
cell by cell (approximately 10% along the structure) so that the superposition of more modes can give
a faster decay of w.
APPENDIX 2 – KV (KAPCHINSKY-VLADIMIRSKY) DISTRIBUTION
In chapter 4 we calculated a periodic solution for a periodic focusing channel in presence of space
charge. The periodic distribution must:
1. follow the invariant ellipsis of linear motion, uncoupled in x x’ and y y’ phase planes.
2. correspond to an homogeneous charge density in xy plane.
The first characteristic is necessary for the periodicity of the solution, the second for the linearity of
the forces. If both the conditions are satisfied the solution is self consistent.
The solution with the characteristic required is the KV distribution:


f 0 ( X , Y , X ' , Y ' )   X 2  Y 2  X '2 Y '2 1
X
x
a
Y
y
b
X '
ax'
Y '

(1)
ax'


with  Dirac function (  ( z )  0 if z  0 and  ( z )  1 for any domain including 0). The
distribution has been written for a position where a’=0 and b’=0 (typically the axis of symmetry of a
quadrupole in a FODO).
To prove the first propriety we have to project the distribution on the plane Y and Y’:

2

0
0


f 0 ( X , Y , X ' , Y ' )dYdY '    d  RdR  X 2  X '2  R 2  1 




(2)

   dR  X  X '  R  1  H X  X ' 1
2
2
2
2
2
2
0
with R 2  Y 2  Y '2 and H(z) step function ( H ( z )  0 if z  0 and H ( z )  1 if z  0 ). Indeed the
value of the integral is unity if and only if the argument of  can be 0 for any R positive. The resulting
distribution on horizontal phase space is therefore an homogeneous ellipsis with semi axis a and /a.
The second propriety can now be proved by projecting in X’ and Y’.
f
0
( X , Y , X ' , Y ' )dX ' dY ' 





   dR 2 X 2  Y 2  R 2  1  H X 2  Y 2  1
(3)
0
with R  X ' Y ' . The charge density is therefore a homogeneous ellipsis with semi axis a and b.
2
2
2
q.e.d.
The existence of such a distribution is not trivial. For example is easy to verify that it is not possible to
find the analogous distribution for one or three degrees of freedom.
On the other hand the existence of such a solution allows an approximate solution for transport of
beam envelopes with strong space charge. Indeed for a real distribution one can calculates the r.m.s
(root mean square) envelope and emittance, defined as
2
a  2 x 2 and   4 x 2 * x'2  xx' .
These beam characteristics evolve in good approximation as for a KV beam with same r.m.s.
parameters. This allows for example to calculate the periodic solution, for a given lattice, solving the
envelope equations without a multiparticle (or Montecarlo) simulation of the beam particles.
APPENDIX 3 – SMOOTH APPROXIMATION
A3.1 NEGLIGIBLE SPACE CHARGE
The general idea is that for a Hill equation:
x" K ( s ) x  0
(1)
we want an approximate solution of pendulum kind:
z

x  a0 sin   0 
 L
(2)
by applying some averaging. The problem is not completely trivial since the average of the force is
null:
L
1
K ( z )dz  0
L 0
(3)
We know that the exact solution can be written in terms of Floquet functions:
x  a( s) sin(  ( s))
L
L
 0   ' dz  
0
0

a
2
(4)
dz
so that in the limit in which the envelope is almost constant (smooth limit) the (4) gives:
0 
L
(5)
a2
or, using the betatron function ( a 2   ( z ) ):
0 
L
(6)

This approximation is valid when 0 is small. The prove can be found in [] for a sinusoidal variation of
K (Mathieu equation) and in [] for the general case.
We verify here the impact of smooth approximation for a FODO with thin lenses:
K ( z) 
1
( ( z )   ( z  L / 2))
f
(7)

with f focal length and  Dirac function (  ( z )  0 if z  0 and  ( z )  1 for any domain including
0). In this case the exact solution is known:
sin
0
2

L
4f
and
 
1  sin  0 
 2 
  L
sin  0
(8)
where   correspond to the maximum and minimum envelope along the period. The average envelope
dimensions and the deviation from a constant envelope are therefore related to
 
L
sin  0
  
 
2 
 sin  0 

  
 2 

.
(9)
The first of these relation guaranties that for small 0 the approximation (6) is valid, the second shows
that the envelope variation vanishes for small 0:
a 1   0


a 2 
4
(10)
In Fig. 1   ,   ,  are plotted as a function of the phase advance, together with the smooth
approximation (6). For a small phase advance the envelopes are smooth, but large (and this can be an
important increase of the accelerator cost). The smooth approximation is valid up to about 600.
10



max
 ( )
 ( )
min
 ( )  min( )
max
L 2
10
8
6

4
1 180

 

2

Smooth approximation
.01
0
20
40
60
80
100
120

0 [deg]
0
140
160
180
180
Fig. 1 Betatron function and its smooth approximation in a thin lens FODO
A3.2 SPACE CHARGE
In the case with linear space charge (KV distribution, discussed in Appendix 2) the single particle Hill
equation is:

 
x"  K ( s) 
x0
a(a  b) 

(11)
In chapter X we showed that the solution can be expressed in terms of Floquet function, and so that
for smooth envelopes (a and b almost constant and equal):
L
 
0

a
2
dz 
L
a2
(12)
The external focusing strength can be approximated from the dynamics without space charge:
 
K  0
 L 
2
(13)
so that from equation (11) we have
 2   02 
L2
(14)
2a 2
From the system (12) plus (14) one can determine the depressed phase advance  and beam
dimensions given the external focusing, the beam emittance and current. This system is also useful
when other combination of parameters are known, like depressed tunes, emittance and current.
In Fig. 2 we show both the exact envelope and the smooth approximation for a long FODO channel
(50 period). The smooth approximation ignores the cell by cell oscillations but describes correctly the
long range oscillations due to the initial mismatch.
Indeed smooth approximation is a rather powerful tool to get fast prediction, and we used it whenever
possible, like to calculate envelope frequency. What smooth approximation cannot predict is the
envelope growth for envelope instability.
Indeed we showed that the envelope explodes if the phase advance per period is 1800: this means that
the resonance is between cell frequency and envelope frequency, and a perfectly smooth envelope
(ignoring cell periodicity) the instability cannot develop. That is the reason why in chapter 4.3 we did
not use the smooth approximation.
0
180


180

 14.48187

2
 180  46.54566
2
2 0   

2
2
0  3  
180

 38.76466
0.002
Zi 1
ax, ay K(s) arbitrary scale
 29.55545
 Zi 3
kkki axT

kkk0 2
 Zi 0 

 T 
0
asmooth
 Zi 0 

 T 
 bsmooth
0.002
0
5
10
15
20
25
Zi 0
T
length s/L
Fig. 2 Envelope oscillations and their smooth approximation.
30
35
40
45
50