The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

NUMERICAL DESIGN OF A BEAM-ENERGY-SPREAD MONITOR
Tsuyoshi SUWADA
KEK, High Energy Accelerator Research Organization
1-1 Oho, Tsukuba, Ibaraki, 305 Japan

Abstract
The KEKB injector linac is now being upgraded for
the KEKB project. A beam-energy-spread monitor is underdevelopment in order to easily handle the beam orbits ofprimary high-current electron beams for producing asufficient number of positrons. Paticularly, it is very usefulto nondistructively control the energy spread at the entranceof the arc line in the linac. The author has proposed amulti-stripline-type energy-spread monitor in order toreinforce this purpose. This report describes the basic
design of the monitor using a numerical analysis while
taking account of the capacitive couplings between thebeams and electrodes. The analysis result shows that theresolution(LiaJE) of the energy spread is expected to be
3xl0-4 for the nominal operation conditions.

0",=7mm and O"v=2mm on the nominal design[3]. The
longitudinal length and width of the electrodes need to be
decided from the viewpoint of the signal-to-noise ratio.
The principle of the monitor is simple. Let us assume
two types of beams: one is a line beam and the other is ahorizontally spread beam. If they pass through the chamber
center, assuming that the total charge for each beam is thesame, the electric potential on the central electrode inducedby the line beam is larger than that induced by the spreadbeam, because the charge density of the line beam is larger
than that of the beam spread around the central region. Thismeans that if the width of the potential distribution inducedon the multiple electrodes is measured at a bending section,
the energy spread can be known because the horizontal
beam width changes according to the horizontal dispersion.
Pick-up

1. Introduction........

The injector linac is required to stably accelerate
primary high-current electron beams (6xl0 10e1bunch) inorder to produce a sufficient number of positrons for theKEKB rings[ I]. The beam-position monitor is useful inorder to easily handle the orbits of such high-current beams
so as to suppress any beam breakup caused by largetransverse wake fields. On the other hand, the beamenergy-spread monitor (BESM) is also important tosuppress longitudinal wake fields and to stably transport thebeams, especially at the arc line of the linac. In the linac
design[2] for the KEKB, the new beam line is extended byabout 225m in order to increase the beam energy by joining
the old positron production line with an arc line. The
primary high-current electron beams are accelerated to abeam energy of 1.5 GeV at the arc. The nominal value ofthe energy spread (cr E/E) is designed to be 1. I% at theentrance of the arc. It is important to control this energy
spread to be as small as possible in real-time operation, andto control the energy spread within a resolution on the orderof I o- 3 at least, for operators to easily handle the beamorbits.

2. Basic design and principle of the BESM
A multi-stripline-type BESM was proposed in order toeasily monitor the energy spread at the arc and to realize acompact and simple monitor. The basic design of themonitor is shown in Fig.!. The monitor comprises a
rectangular chamber with a horizontal length longer thanthe vertical one. Multiple electrodes are stretched on theinner surface of the chamber, where each electrodecomprises a parallel stripline (50Q) with the beam axis.
The transverse sizes of the chamber have been decided to
fully include transverse beam sizes. In the arc design, thechamber sizes are required to be 70x30mm 2 , taking into
account beam sizes of 5 a, where the maximum sizes areChamber

Striplinc....

·.: ·.. ·BeamFig. I. Cross-sectional schematic view of the beam-energyspread monitor. The beam-induced electric-field lines arcalso shown schematically.

3. Brief overview of the numerical method
A numerical analysis was carried out on the basis of acharge-simulation method[4,5]. Here, only a brief overview
of this method is given in order to prepare for a discussionof the BESM analysis in the next section.
The electromagnetic fields generated by relativistic
charged beams inside the vacuum chamber are almost
boosted in the transverse direction to the beam axis due toLorentz contraction. This phenomenon shows that theelectromagnetic coupling of the electrodes to the beams canbe well treated as a two-dimentional electrostatic potentialproblem. Thus, the BESM analysis is simply attributed tothe electrostatic potential problem on the transverse plane.
The charge-simulation method is based on the boundary
element method for analyzing electrostatic potentialproblems. In this method, some boundary elements andimaginary charges are introduced in order to analyze anelectrostatic-field system. All of the conductor surfaces inthe system arc divided into many domains, which are called''boundary elements"; imaginary charges are also arrangednear to the boundary elements in a one-to-one manner. The-264-

The II th Symposium on Accelerator Science and Technology, Harima Science Garden City, I997

electrostatic potential of each conductor can be calculated
so as to satisfy the boundary conditions of the system, thatis, so as that the calculated equipotential surfaces
correspond to the conductor surfaces by using the linear
superposition of the electrostatic field contributed by all of
the imaginary charges.

4. Application to the BESM
All of the surfaces of the chamber and electrodes areThe boundary elements
segmented as shown in Fig.2.
on the chamber are arranged by using nxn elements withequal intervals in each vertical and horizontal direction,respectively. Each electrode surface is segmented by mxl
elements (for the sake of simplicity, and the verticalelement is fixed to be 1=2). The segmentation numbers (nand m) are chosen based on the convergence of theelectrostatic-field analysis. This method is guaranteed by
the uniqueness of the solution on the electrostatic theorem.

Fig.2). For the electrodes, the parameters we(~w), he(~h)
and mx/( ~nxn) are used. The parameter 8 is given byo=ex f,(4- 5)

where e is the interval between the adjoining boundary
elements by a factor of 2 and f is a free parameter whichmust to be tuned (to be generally chosen within 0.2-1.5).
The electrostatic potential (vk;) of the i-th boundary elementin the k-th electrode (k=O is assigned to the chamber) can becalculated by linearly superposing the potentials (11.;;)
generated by all of the imaginary charges(q;) and beam
charge(q"), as follows:(4- 6)/ I = ""'£..J V··U'j=

I ~jqi +I ~iqhi·.i.it::(4 -7)

II
P-=--ln-+C'u 2-"oli;'""'(4- 8)(4- 9)

hweFig. 2. Segmentation of the BESM chamber and electrodes.
The black and open circles show the boundary elements andimaginary charges, respectively.
The rectangular coordinates (x;. y;) of the boundary
elements on the chamber are given as:

where Pii and r;; are the potential coefficient approximatedfor a line charge with infinite length, and the length
between the i-th boundary element and the j-th imaginary(or beam) charge, respectively; e0 is the dielectric constantin the vacuum, and C the integration constant. Theboundary conditions in this analysis are given as follows:(a) The chamber potential (V0 ) is equal to zero (groundpotential),(4- 10)V0 = 0.
(b) The charge summation on the k-th electrode is equal tozero,(4- II)(c) The summation on the chamber charge (q/ 1) and thebeam charge is conserved to be zero by the charge
conservation,(4- 1)(4-12)
j(4- 2)(d) The electrostatic potentials of the boundary elements inthe same electrode are equal to the electrode potential(Vk),

The coordinates (Xi, Y) of the imaginary charges are givenas:

Xj=w(~-;j).

(4-3)

Y; = h(_!_±~-_!_j). (I~ j~ n).

(4-4)

2 h n
·Here, w and h indicate the width and height for the innersurface of the chamber, respectively, and 8 is the interval
between the boundary element and the paired imaginary
charge; the sign (±) is selected so that the imaginary
charges are arranged inside the boundary elements(seek_vk- V;.

II(4-13)

Based on the above discussion, the electrode potentials arecalculated by solving the above simultaneous linear
equations (4.7) obeying boundary conditions((4.10)-(4.13)).

5. Check of the numerical analysis
The goodness of the numerical analysis wasinvestigated by analyzing the convergency of the-265-

The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

electrostatic potentials calculated for each electrode invarious segmentation .numbers (n and m ), where a Iinccharge with infinite length was assumed to be at thechamber center. - The parameter f was tuned so as toproduce good symmetrical and constant electrostatic
potentials on the electrode surfaces. The segmentation
numbers of n=40 and m=5 were obtained by a convergency
calculation, the accuracy of which was deduced to be better
than I%. The parameters chosen based on the calculation
are summarized in the following table.
Table 1. Several parameters used in the BESM analysis.
Chamber width w (mm)Chamber height h (mm)Chamber segmentation nxn
Electrode width w,. (mm)Electrode thickness h,. (mm)Electrode segmentation mxl
Number of electrodes
Gap length /~ (mm)Chamber potential V0 (Volt)
Beam charge (nC/mm)
Parameter f703040x40

Figure 3 shows the variation of the sensitivity and theresolution (Aa,jE) depending upon the charge width. Theerrors were derived by assuming errors of I% in theelectrode-potential measurement.
For example, an 8-bit digital oscilloscope is available toperform it. The resolution of the energy spread was derived
from the horizontal dispersion of 1],=9mm/%. It is
understood that the resolution strongly depends upon thehorizontal beam size.
Thus,the energy-spreadmeasurement is advantageous at a large dispersion section.
The horizontal beam size at a dispersion-maximum location
around the center of the arc line is about crx-7mm; for thisbeam size the resolution is expected to be 3x I o- 4•

7. System design
A data-acquisition system (DAQ) is briefly described
here. Figure 4 shows a schematic drawing of the DAQsystem. The pick-up signal from each electrode is sent to asignal combiner after delaying a suitable time using cabledelay lines. Then, the combined signal is fed to a digital
oscilloscope, which measures the peak voltage for thepulse-train signal.

3I5x2102010.6

Digital Oscilloscope

6. Results of a numerical analysis
For the sake of simplicity, a numerical analysis wasperformed based on the variation in the width(] cr) of thehorizontal spread charge at the chamber center. The spreadcharge was replaced by ten line charges arranged at regularintervals on the y-axis. The charge distribution wasassumed to be a Gaussian function. It is useful to introduce
the sensitivity (S) as follows:(6 -14)
where a.,. is the width of the electrode potential distribution
using the least-squared fitting of a Gaussian function, V;
and V peok are the i-th electrode potential and the peakpotential fitted.
620.16600.14-...

580.12560.1..="'

540.08"' 520.06

500.04;..,...:=:."'

2345Charge Width(mm)Pick-up Signals
Fig. 4. Block diagram of the data-acqusition system of theBESM.
8. Conclusion
A new beam-energy-spread monitor was proposed by
the author. The basic design of the monitor was formulatedusing numerical anaysis. The resolution of the energyspread measurement is expected to be 3xHl'4• The
resolution is sufficiently small to stably transport highcurrent beams and for operators to easily control them.

References[I] S. Kurokawa, et al., KEK Report 90-24 ( 1991 ).
[2] I. Sato, et al., KEK Report 95-18 (1996).
[3] T. Kamitani, private communication .
[4] T. Kohno and T. Takuma, Numerical Calculation
Method of Electric Field (CORONA, Tokyo, 1980).
[5] H. Singer, et al., IEEE Trans. Power Apparatus Syst.
PAS-93, (1974) 1660.

=....
::;·

=~480

CableDelays~~'-'6Fig. 3. Variations in the sensitivity and resolution of theenergy spread depending upon the horizontal width of thecharge.

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