&AC-PUB-342
Au&St 1967
ASYMPTOTIC THEORY OF BEAM BREAK-UP
IN LINEAR ACCELERATORS*
W. K. H. Panofsky and M. Bander**
Stanford Linear Accelerator Center
Stanford University, Stanford, California
(To be submitted to Review of Scientific Instruments)
*
Work supported by U. S. Atomic Energy Commission
**
Now at University of California, Irvine, California
I
I. GENERAL DESCRIPTION OF OBSERVED PHENOMENA
The observed beam current of the SLAC SGO-section linac appears to obey the
results of independent particle dynamics at low intensities. However, as was first
observed on April 24, 1966, the pulse length of the transmitted beam appears to
shorten provided the beam current exceeds a threshold value at a given distance
along the accelerator; the g-rester the distance,
the lower the threshold. This general
behavior is illustrated in Fig. 1. Further tests clearly indicated that the phenom-
enon responsible is the sudden onset of a radial progressive instability conven-
tionally called beam break-up (BBU).
Observation of radial instability in high current linear accelerators is not
new,lW6
and the phenomenon has been conclusively associated with the excitati.on
of transverse deflecting modes.
However, one should clearly recognize that we
are dealing with two quite distinct mechanisms by which such modes can lead to an
amplifying action. The first mechanism discussed in the above references results
from the negative group velocity of the TEMll
mode of the conventional disk-loaded
structure.
This negative
goup
velocity will feed transverse energy from the end of
a given acceleratin,
m section t.o the front, thus leading to the regenerative action
involved in the “backward-wave oscillator. ”
This phenomenon of regeneration
within a given section characteristically occurs at currents of several hundred
milliamperes at pulse lengths of several microseconds. The second mechanism
which is dominant in a multiscction relatively low current accelerator (such as
SLAC or the liharkov 2-GeV accelcrator)7involvcs amplification from section to
section, coupled only by the electron beam without backward propagation of elcctro-
magnetic energy.
- 1 -
In this paper
we will give the theory of the second mechanism only, which is
the dominant cause of the BBU phenomena occurring at SLAC. As will be seen, this
mechanism is very gcncral, being quite independent of the d&ailed. structure of the
accelerating sections.
II. THE MULTICAVITY MODEL
A. The Model
We will represent each section of the accelerator by a single cavity; the cavity
geometry constitutes a free parameter which can be choskn to fit the experimental
behavior.
We will assume:
(a) Only one resonant mode at a frequency w0
and loss factor Q is of
significance.
(b) The cavity has axial symmetry and the axial electric field vanishes
along the axis of symmetry.
(c) The rate of build-up of oscill.ation giving rise to the radial modulation
of the beam is small compared to w
0’
th
Consider a particle of charge e to cross at a t.ime t the n of N cavities
at a distance x from the z-axis, taken to be an axis of symmetry. Let L be the
distance between cavities, and let the particle velocity be v M c = 1 (see Fig. 2).
B.
Equation of Motion
Let the electric field F in the n
th
cavity be derivable from a vector potential
A, and let each cavity be excited near a single resonant frequency uo. We obtain
from the deflection theorem* for the change in transverse momentum px in the
nth
cavity :
Apx =
e
s
aAZ
- dz
3X
(1)
-2 -
This transverse momentum Ap,
results in a difference in displa6ement of
(Apx/mor) L bet7veen the (n f l)th and nth
cavity where moY
is the particle
energy. We can thus write a difference equation which can be approslmated as a
transverse differcnt,ial equation of motion as follows :
c. Exe
itation of Cavities
Equation (2) gives the radial equation of motion as governed by the transverse
gradient of the longitudinal component of the vector potential and hence the electric
field. No special assumptions as to mode structure are assumed. If the particle
passes at a distance x (assumed constant in each cavity) from the symmetry axis,
then $11 general, work is done against the longitudinal field.
Each cavity excited at a frequency w near w.
loses energy U to the current
j at the rate j
J
j? + d; and loses energy to wall losses at the rate
w U/Q. The
rate of build-up is therefore given by (averaged over many cycles as designated by the
symbol)
a=-j
~s&-!$.
at
s
(3)
If the field z = 7 on the axis and varies linearly from. the axis,
J-
z l
dp can
be approximated by:
-
J
J
aEz
$-.d;-x _
dz
ax
(4)
where x is the transverse coordinate at which the beam carrying the current j
passes from the axis.
The cavity excitation is thus proportional to x; on the other
hand the cavity, once excited, will affect the motion in x according to Eq. (2).
-3-
As a result the beam will receive a transverse structure so that x will be modu-
lated at a frequency w near wO.
Note that the field integrals appearing in Eq. (2) and in Eq. (4) are simply
related if we can assume that oscillations take place near w = wO and that the
rates of build-up or damping
that all field amplitudes vary
placement to vary as e
+iwt
amplitudes carrying both the
Using this convention we
are slow relative to w
0’
We adopt the convention
+iwt
as e
and that we consider the transverse dis-
also. The quantities x,
E, A thus become complex
phase and the (slowly varying) amplitude information.
can write the field integral in (4)) using Ez = - aAz/at :
s
aE
-? dz $5 - iw
s
aAZ
I(t) =
- dz
(5)
3X
3X
giving the equations of motion and the energy build-up equations
a
ax
( 1
ieL
dn
Ydn =-0 I
m W
au
wU=-J$j
at'
Q
.
(6)
In general U
and I are related quadratically through a (generally complex)
impedance. In general we can write
(7)
and
(8)
(9)
where * denotes the complex conjugate and Re the real part. Hence (7) becomes,
*
* aI
noting that I g + I at =
21*
aI
at :
31
~+$p4~x.
-4 -
(10)