Naval Research LaboratoryWashington, DC 20375-5000

NRL Memorandum Report 6378

Beam Breakup Instabilities in Linear Accelerators:Transition, Phase Mixing and Nonlinear Focusing

D. G. COLOMBANT AND Y. Y. LAU

Plasma Theory BranchPlasma Physics Division

(N

1 December 13, 1988

DTIC

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Beam Breakup Instabilities in Linear Accelerators: Transition, Phase Mixing and

Nonlinear Focusing12 PERSONAL AUTHOR(S)

Colombant, D.G. and Lau, Y.Y.13a TYPE OF REPORT 13b TIME COVERED 14 DATE OF REPORT (Year Month Day) '5 pa :

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FIELD GROUP SUB-GROUP - Beam break/-up, Linear accelerators,Electron beam instability Beam-circuit interaction, ,'

/ I #

19 ABSTRACT (Continue on reverse if necessary and identify by block number)

The temporal and spatial evolution of the cumulative beam breakup instability (BBU) is analyzed numer-ically using a continuum model. It is found that neither phase mixing nor linear transverse focusing is suffi-cient to render BBU stable in a long pulse machine, when external damping is absent. A sufficiently strong

nonlinear octupole focusing field may limit the BBU growth, however. , - Y r

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BEAM BREAKUP INSTABILITIES IN LINEAR ACCELERATORS:TRANSITION, PHASE MIXING AND NONLINEAR FOCUSING

1

The control of beam breakup instabilities remains a major concern for

accelerator builders. Various remedies have been examined2: lowering the

quality factor Q of the deflecting modes, increasing the focal strength,

phase mixing, nonlinear focusing, and detuning of the cavities, etc. All1-7

of these may readily be incorporated in the continuum models, from which

asymptotic formulas have been obtained for certain cases. These formulas

have frequently been used to compare with experimental observations and

with other more elaborate theoretical works. 6,8 At the moment, they are

the only ones available to describe the transition of BBU from the

cumulative type to the regenerative type.7'9

In spite of the versatility in the continuum model, a direct numerical

integration of the governing partial differential equations does not seem

to have been published, although the evolution of BBU has been calculated

in great detail by other means, in some limiting cases.1,3,10 Here, we

analyze the evolution of the cumulative BBU by a direct numerical

integration, intending to look into the effects of phase mixing and

nonlinear focusing which defies analytic treatment. This work was also

motivated by the recent observation 7 that the asymptotic growth of the

cumulative BBU is independent of the linear transverse focal strength. It

is then of interest to determine in what way nonlinear focusing affects the

temporal and spatial evolution of BBU.

To isolate the mechanism of phase mixing and of nonlinear focusing, we

ignore the damping associated with finite Q. Setting Q equal to infinity

should not be construed as this damping effect being unimportant. It

serves the additional purpose of providing a more stringent test for the

numerical scheme. For without a finite Q, the soldtion grows much more

Manuscript approved October 17. 1988.

1

rapidly, making the accuracy in the numerical integration more demanding.

In any case, the effect of finite Q enters only through the well-known

decay factor exp (-w0t/2Q) in the amplitude of the solutions, where W0 is

the breakup mode frequency.

Our numerical results reveal the following. 'hen ) is infinite, the

cumulative BBU evolves asymptotically (at a fixed distance from the

injector) at a rate which is independent of the linear transverse focal7

strength. The excellent agreement with the analytic formula adds to our

confidence in our numerical calculation. [In a separate test, the

numerical solution agrees with the analytical results to within 20% over

nine orders of magnitude variations in the beam's transverse displacement.]

When the analysis is extended to include a finite spread in the betatron

frequency, we find that this finite spread has virtually no effect on the

above-mentioned asymptotic behavior. While neither a linear focusing force

nor a finite spread in the betatron frequency, by themselves, can provide

long-term stabilization of BBU, a sufficiently strong nonlinear focusing

force may limit the amplitude of the beam's transverse displacement, even

in the limit Q 4 -.

For a continuous coasting beam, the evolution of the cumulative BBU is1,3

governed by the following normalized equations

(a/3T + a/Z) 2X + 2X = A (1)

a2 A/ T 2 + A 2 2eX. (2)

Here T = wot(>O), Z = 0z/c(>O), Q = k ac/w where k is the betatron

wavenumber, s is the dimensionless coupling constant7 which is proportional

to the beam current and shunt impedance but is independent of the focal

2

strength, Q(Z,T) is a measure of the strength of the deflecting mode which

drives the BBU and X(Z,T) is the normalized transverse displacement of the

beam. In all numerical calculations, we assume initial rest condition and

homogeneous boundary conditions; only one non-trivial boundary condition is

imposed at Z = 0: X(0,T) = 1 for 0 < T < 0.2 and is zero otherwise. Thus,

X is normalized to the initial displacement, imparted upon the beam at

Z = 0 for a short time (similar to a delta function excitation. 1,3-8)

Shown in Fig.1 is the spatial distribution of X(Z,T) at four different

times, obtained from a fourth order numerical scheme in space and time. We

set £ = 0.000413 and Q = 0.14. The dotted curves in this figure are.7

obtained from the asymptotic solution :

IX(Z,T)I - B Z exp(3(T_Z)3/4 1

where x = c(T-Z)/(wZ) > 0, B = C/[Max(1,x/2.81)J1 112 ,

1.1573 1 - 0-21Ti for x > 5p(T) = I 16 'T2/ (4a, b)

11 - 0.0556x + 0.004 2 for -c < 5

and C is a constant to be determined by a "one-point fit" with the

numerical solution. Equation (3) describes the transition from the strong/

8~focusing solution8 [p(r) 4 1, Eq. (4b)] to the weak focusing solution1 [Eq. Oz

(4a)] as x increases.' This transition occurs when r = 0.5. The excellent

agreement, over a wide range of x, between the numerical solution and the

analytic theory shown in Fig. 1 may be regarded as a validation for both or

the numerical scheme and the analytic result that X evolves at a later time7 0]

as if the linear focusing force were absent. The numerical data shows

By--

D i sitributiofl/_I Availability Codes

A- il and/or

Dist special

that the spatial variation of X is biased toward the "slow beam-cyclotron

mode".7

The effect of phase mixing due to a spread in the betatron frequency

may be examined by pretending that the electron beam consists of N groups

of particles, the ith group bearing its individual betatron frequency 9i"

Thus, we replace Eq. (1) with N equations

(3/3T + a/Z)2 Xi + 9i2Xi . A, i 1 1, ... N (5)

and the quantity X in the right-hand side of Eq. (2) is replaced by <X> =

GIX 1 + G2X 2 + ... + GNXN where Gi represents the fraction occupied by the

ith group (G1 + G2 + ... + GN = 1). In the "spread mass" model,11 4 Q1 i =

Q(i/N)1 /2 and G. = 6i(N - i)/N(N - 1). The solution <X> is shown in Fig.

2 for c - 0.000413, 2 = 0.14, N = 8. A virtually identical figure was

obtained when we increased N to 16. From this figure, we see that a finite

spread in the betatron frequency has no effect on the long-time growth of

the cumulative BBU. This is due to the fact that the cumulative BBU grows

asymptotically at a rate independent of the linear focusing force, and a

finite spread in the betatron wavelength cannot alter this long-time

behavior.

A nonlinear octupole focusing force may be handled also by solving

Eqs. (1) and (2), except that the 92X term in Eq. (1) is now replaced by

AX3 in a simplified model. The evolution of X is shown in Fig. 3 for the

case A = 0.0392. Here, we see that the amplitude of X is limited at large

T, in marked contrast to the cases of linear focusing and phase mixing. It

is possible that the change in the phase relationship between A and X due

to nonlinearity might have contributed to the stabilization shown in Fig.

3, but this issue remains to be examined in the future.

4

In summary, an accurate numerical scheme waz devised to analyze the

classic model of cumulative BBU. It is suggested that this line of attack

could be fruitfully applied to a great variety of problems in a rather

straightforward manner. Interesting areas such as phase mixing and

nonlinear focusing vere analyzed in this initial study, and some unexpecced

features reported.

Helpful discussions with S. Zalesak and D. Chernin are gratefully

acknowledged. We also thank M. Friedman, C. W. Roberson and P. Sprangle

for their support. This work was supported by the Department of Energy,

Contract No. DE-AI05-86-ER13585 and by the Office of Naval Research.

5

References

1. W. K. H. Panofsky and M. Bander, Rev. Sci. Inst. 39, 206 (1968).

2. R. Helm and G. Loew, Linear accelerators, Eds. P. M. Lapostolle and .

L. Septier, (North Holland, Amsterdam, 1970), Ch. B.l.4, p. 173.

3. V. K. Neil and R. K. Cooper, Part. Accel. 1, 111 (1970).

4. R. J. Briggs, University of California Report. No. UCID-18633 (1980),

unpublished.

5. G. J. Caporaso, Proc. of the LINAC '86, Stanford Linear Accelerator

Center p. 17 (1986).

6. R. L. Gluckstern, R. K. Cooper and P. J. Channell, Part. Accel. 16,

125 (1985); R. L. Gluckstern and F. Neri, Proc. 1987 Particle

Accelerator Conference, p. 1017.

7. Y. Y. Lau, Naval Research Laboratory Memo Report No. 6237 (1988).

8. V. K. Neil, L. S. Hall and R. K. Cooper, Part. Accel. 9, 213 (1979).

9. R. L. Gluckstern, Bull. Am. Phys. Soc. 33, 918 (1988).

10. D. Chernin and A. Mondelli, to be published in Particle Accelerators.

11. E. P. Lee, Phys. Fluids 21, 1327 (1978).

6

.04 eg 2

.10 I-

.02 C

0C0

-Jo- T= 200-. 02 T=1OO

-. 04' L .2

0 20 40 0 20 40

.4 - .8 -0x x.2 04-

00

Tz 300 T=4001

Z -. 2- Z-. 6 1 -- -

0 20 40 0 20 40

Fig. 1 Evolution of X(Z,T) and comparison with the analytic theory

(dotted curves).

7

4

Z40

2 ( X>

0

-2

T-41

0 100 200 300 400Fig. 2 Temporal evolution of <X> at Z =40, including the effet ut phase

mixing.

z:x 2-XA0

0

T= 200 T 0

-. 843

0 20 40 0 20 40

3- 3-

2- x 2 xI I

0 0

'T= 800 ' z T 1000-2- -2-

-3Z -3 -ZfIIrI I I 1

020 40 0 20 40

Fig. 3 Evolution and saturation of X(Z,T) in the presence of nonlinear

octupole focusing.

9

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