generalized three-dimensional equations for the emittance and field

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Real beams have to be considered in periodic quadrupole channels and thus .... For a uniformly charged ellipsoid, we calculate I from the potential and readily.
Particle Accelerators, 1987, Vol. 21, pp. 69-98 Photocopying permitted by license only © 1987 Gordon and Breach Science Publishers, Inc. Printed in the United States of America

GENERALIZED THREE-DIMENSIONAL EQUATIONS FOR THE EMITTANCE AND FIELD ENERGY OF HIGH-CURRENT BEAMS IN PERIODIC FOCUSING STRUCTURES INGO HOFMANN and JURGEN STRUCKMEIER GSI Darmstadt, P.O.B. 110541, 6100 Darmstadt, West Germany (Received December 2, 1985; in final form September 4, 1986) For high-current beams in three dimensions, a set of ordinary differential equations relating the rms emittances to the nonlinear field energy is derived. The equations are valid under constant or periodic focusing. They can be specialized to one and two dimensions as well and thus provide a general framework to study emittance growth. They allow one to estimate the expected total emittance growth if the change of nonlinear field energy can be predicted from general principles, like the homogenization of charge density in space-charge-dominated beams. The equations also lead to estimates for the maximum longitudinal-transverse emittance transfer in situations where equipartitioning occurs. A comparison with results from computer simulations of 3-D bunched beams yields excellent agreement with theory and confirms the emittance growth formulas derived from this theory.

1. INTRODUCTION Recently an equation was presented by Wangler et al. l relating the rms emittance growth of a beam to its electric field energy in the case of an azimuthally symmetric charge distribution and a continuous focusing force (a similar equation had been derived previously by Lapostolle2). This equation has been found useful for interpreting the change of rms emittance in numerical simulations under extreme space-charge conditions. In this limiting case, it has been possible to predict final emittance growth from the field energy stored in an initially nonuniform charge distribution. l Wangler et ale also confirm the approximate formula for emittance growth, which was previously derived heuristically by Struckmeier, Klabunde, and Reiser,3 assuming a balance between field energy and (transverse) kinetic energy. Real beams have to be considered in periodic quadrupole channels and thus deviate. from azimuthal symmetry. In linear accelerators, they are bunched longitudinally and allow for transfer of emittance between the transverse and longitudinal directions, as was shown previously by Jameson 4 and by Hofmann and Bozsik5 using computer simulations. The question has been raised whether it is possible to generalize Wangler's equation to a realistic multidimensional case with periodic focusing, and to obtain equations for emittance change that can be used for practical design, as well as a means for enhancing our general understanding of emittance growth. Here we show that it is indeed possible to derive such equations by forming moments of Vlasov's equation (Section 2). In 69

70

I. HOFMANN AND J. STRUCKMEIER

Section 3, we specialize the general equation to beams in one, two, or three dimensions; in Section 4, we present a minimum energy principle; in Section 5, we discuss Debye shielding; and in Section 6, we present applications of the general theory and a comparison with results from computer simulations of 3-D bunched beams.

2. DERIVATION OF THE BASIC EQUATION We assume equations of motion in x, y, z with linear, time-dependent external focusing forces and arbitrary space-charge forces, where time is replaced by the distance s ( = v · t) and q is the charge of the particles:

x" + kx(s)x -~Ex(x, y, z, s) = 0, my v

y" + ky(s)y - ~ Ey(x, y, z, s) = 0, my v

(1)

E follows from Poisson's equation: q

(2)

V·E=-n(x,y,z,s), £0

where n is the density, which is determined by projecting a distribution in 6-D phase space into real space:

n = f f ff(x, y, z, x', y', z', s) dx' dy' dz', with f satisfying Vlasov's equation (v == dx/ds) (3) We define second-order moments (with N the total number of particles)

x =N- j- -fx 2

xx'

=N-1j-

1

°

°

2f

ofXX'fdx

dx o ° odz', °

0

°

dz',

etc.,

EMITIANCE OF HIGH-CURRENT BEAMS

71

and obtain from Eq. (3), by computing the respective moments,

d- -x 2 -2xx' =0 ds '

d- q-xx' -X,2 + k xx 2 - - - x Ex = 0, ds my3v2

(4)

d2q-x'2+2kxXx' ---x'Ex=O, ds my3v2 and similarly in y, z. We then define the rms envelope,

i == (?)1/2, and rms emittance, (5) 6

We next follow the procedure by Sacherer and obtain the rms envelope equations _ E;(S) d2 _ q xEx (6) 2x+kx(s)X-16-3 ---3-2-_-=0 dS ~ my v x (and likewise in y, z) by eliminating x ,2 and xx' and ignoring the third part of Eq. (4) by virtue of Eq. (5). We observe that neither Eq. (4) nor Eq. (6) is a closed set of equations, since in general higher-order moments of Vlasov's equation are contained in xEx, x' Ex, and we obtain an infinite set of coupled moment equations. Hence, Eq. (6) contains E;(S) as an unknown function, whose derivative can be readily derived from Eq. (5) (likewise in y, z):

d Ex2_ (2-'- - , - ) - - 32q - 3-2 X X Ex - xx xEx , my v

ds

d

ds

32q (2-'-,-) Y Y E y -yy yEy , my v

2_ Ey - - - 3-2

(7)

32q (2-'- -'-E) z - - - 3-2 Z z E z - zz z z· -d E2_ my v ds The difficulty now lies with the unknown moments involving E. In the following, we attempt to replace these by the electric field energy as a quantity that promises more physical insight. To this end we require that

!-

XlEx=N- 1 where nvx == f

0

-JX'Exf dxo odz' =N-1f f fExnvxdxdYdZ,

f f x'f dx 'dy' dz'

0

(8)

is the local averaged velocity of beam particles

72

I. HOFMANN AND J. STRUCKMEIER

(in the moving frame). With the local current given by j =qnvv,

(9)

we obtain (10)

and similarly for y, z. By integrating Eq. (3), we derive the continuity equation

an as

.

- + (qv) -1 V • J = 0

(11)

and write (E = -V ~ En da) = O. Eo

(16)

s

For constant focusing, we can write this as a total energy conservation law, T+

1 yN

+-2- W

Vex

= const.,

(17)

+ y,2 + Z'2)

(18)

with 2

T

== myv

(X,2

2

the kinetic energy in the beam frame; and myv 2

-

~ex == -2- (kXV"y2

-

-

+ kY y2 + k z Z2)

(19)

the potential energy due to external focusing forces. Here we have neglected the boundary integral, which is justified if the boundary is far away and thus dEn I ds ~ o.

3. GENERALIZED EMITTANCE EQUATIONS IN DIFFERENT DIMENSIONS 3.1. 3-D Equation In order to relate the third term on the right-hand side of Eq. (15) to the field energy of a uniformly filled ellipsoid, we must calculate its potential and field energy. In Appendix A, we derive the potential of a uniform rotationally symmetric ellipsoid with semiaxes a (in x, y) and c (in z) and show that the field energy inside a large sphere of radius R is given by

N2q

[6(1-/+-/ --5] , a R

2 Wu = - 40nEo c

2 C ) 2

(20)

where /(cla) is a geometry factor (= 1/3 for a spherical bunch). By allowing a and c to vary with s, we can calculate the time derivative of Wu and find (Appendix A) dWu = ds

_ 3N q 2 (deL + da 1 - f\ . 2

20nEo ds a 2

(21)

ds ac - )

In order to show the relationship of dWulds to the remaining term in Eq. (15), we

74

I. HOFMANN AND J. STRUCKMEIER

define

1 ( 1 dx 2 _ 1 dy 2 _ 1 dz 2 - ) 1="2 x2--;j;xEx+ y2d;yEy + z2--;j;zEz

.

(22)

For a uniformly charged ellipsoid, we calculate I from the potential and readily find

1 = _~dWu u Nq ds ·

(23)

For a mo~eneral ellipsoid, it can be shown6 that the averaged quantities xEx , yEy , and zEz depend only weakly on the actual charge distribution, if bunches with identical rms dimensions ("rms equivalent bunches") and ellipsoidal symmetry are compared. The latter requires a particle density of the kind 2 y2 Z2 ) X (24) n(x, y, z, s) = n ( a 2 + b 2 + c 2 ' S , for which one finds 6 -

xEx = yEy

3Nq

= 20VS JrEO(Z2)1/2 3Nq

zEz

1-[ 2

)..3'

Z2

(25) (26)

= 20VS JrEo(Z 2) 112 X 2f)..3'

with the value of A3 depending on the profile in Eq. (24): 1

(uniform) ,

~ 1.006

(parab 0 1·IC) ,

5V5 = 1.051 6Vii:

(Gaussian),

25 8V3 Jr = 1.018

[hollow (r

25VS 210

Note that our A3 corresponds to Thus, we can generally write

2



Gaussian)].

5V5 A3 in Ref. 6. (27)

and obtain from Eq. (15) the generalized 3-D emittance equation (here actually proven under the constraint of Ex = Ey and x 2 = y2):

.!! E

+ 12 .!! E2 + 12 .!! E 2 = _

(dW -)..3 dWu ) ds ds·

(28)

The integration in W is performed over a large enough volume that hence we may neglect the surface integrals.

dEn/ds~O,

1 x 2 ds

2 x

y ds

y

z ds

z

32 my3v2N

EMITTANCE OF HIGH-CURRENT BEAMS

75

Keeping in mind that the right-hand side vanishes for a uniformly charged bunch, we have thus shown that the change of emittance is directly related to the change of the nonlinear field energy. We observe that we have on the left-hand side the weighted sum of de 2 /ds for x, y, and z, as a result of the coupling introduced by the space-charge force. This equation therefore promises an estimate of the change of the weighted sum of emittances if the change in nonlinear field energy can be estimated from general principles. It then also allows an estimate of the maximum emittance transfer if "equipartitioning" should occur as a result of a coherent instability5 or single-particle resonance. The actual dynamics of coherent instabilities requires solving equations for higherthan-second-order moments, which is beyond the framework of our derivation. Equation (28) also indicates the possibility of a slowly growing emittance due to "structure resonances" in a periodic focusing system, if W oscillates with a period close to the focusing period. This can happen for certain values of the phase advance ao and a (see Ref. 7 for the 2-D analogue). We thus suggest that Eq. (28) gives a rather general framework for describing emittance growth. Its practical value still depends on the possibility of estimating changes of the nonlinear field energy, without actually calculating its time dependence, which is, in fact, possible in many situations. Equation (28) is supplemented by the rms envelope equations derived by Sacherer. 6 With Eqs. (6), (25), and (26), we obtain for the rotationally symmetric ellipsoid: (29)

(30)

with i == (X 2 )1/2 (= a/V5 for uniform ellipsoid), i == (Z2)1I2 (= c/V5 for uniform ellipsoid), and (31) (32)

Note that for a spherical bunch we have f = 1/3, hence hx = h z = A3 nearly spherical bunch (0.8:$ i/i:$ 5), we have approximately, with f (see Appendix A), h

x

i)

=1+-21 ( I - -i

hz =l.

=1. For a =1/(3i/i) (33) (34)

76

I. HOFMANN AND J. STRUCKMEIER

3.2. 2-D Equation In Appendix B, we show that, for a continuous beam with uniform elliptic cross section, the field energy calculated within a large circle of radius R is given by Wu

2 N q2

= 16nEo

(

2R ) 1 + 4ln a + b '

(35)

where a and b are the semiaxes in x and y. Thus, we find

d dW ds(a+b) u - d =- 4wo s a+ b

(36)

where we have introduced N 2q 2 Wo=-161l'Eo

(37)

as the field energy within the actual beam volume. The 2-D equivalent of I in Eq. (22) is readily seen to obey again

Iu =

-

:q

d;u.

(38)

The main difference with the 3-D case is that here xEx and yEy are independent of the density profile, as long as elliptical symmetry is satisfied: 6 X2

y2

n(x, y, s) = n ( a2 + b 2'

) S



(39)

We then have 1= Iu and obtain (again neglecting a surface integral, if R is sufficiently large) (40)

It is appropriate to introduce the in-beam field energy Wo as the normalization constant and rewrite the 2-D generalized emittance equation as

1 d 2 1 d 2 dW-Wu . 2ds Ex + 2ds Ey =-4Kds % y

x

(41)

Note that dlds Wu has to be replaced by A2 dlds Wu , with A2 a correction factor close to unity, if Eq. (39) is not satisfied. 8 Here we have introduced the generalized perveance given by (following the notation of Ref. 1) N q2 K=----

(42)

21l'Eomy3v2 ·

For a round beam (x 2 = y2 and

ex

!!:- Ex -

2 _

ds

= Ey ), -2K Z

we readily obtain

!!:- W -

x d

s

Wo

Wu

'

(43)

EMITTANCE OF HIGH-CURRENT BEAMS

77

which agrees with the equation derived in Ref. 1 for continuous focusing (note that x 2 = X 2 /4, with X the radius of an equivalent uniform beam as used in Ref. 1).

Equation (41) is again a promising tool to describe changing emittances, if the change of the nonlinear field energy is known, or if it can be estimated from general principles. Note that (W - Wu)/wo depends only on the type of nonuniform density, not on the actual size of the beam. For a parabolic density profile, we show in Appendix D that it has the value 0.0224, whether of spherical or elliptic cross section. A Gaussian profile yields 0.154 for a round beam 1 (numerical results indicate the same value for an elliptic cross section). For completeness, we give the rms envelope equations following from Eq. (6): d2 £2 K 1 -i+k (s)i--3 - - - = 0 2 ds x 16x 2 i +Y , 2 d £2 K 1 -y+k (s)y-_Y - - - = 0 . ds 2 y 16y3 2 i + Y X

(44) (45)

3.3. I-D Equation

For a uniformly charged sheet (Izl ~ c and infinitely extended in x, y), the field energy within Izi :5 L is found in Appendix C (N particles per unit area in x, y) to be

w: = N

2

q2

£0

u

(~ _ ~) 4

(46)

6 '

and dWu N 2 q 2 dc - = ---=-Nql ds

6£0

ds



(47)

zEz has been evaluated in Ref. 6 to be slightly dependent on the density profile, hence (with? = c 2 /3 for a uniform sheet)

-E z

z

where A1 is given by (differing by

Nq

(2)1/2'1

= 20 £0 . Z

0

A1'

(48)

from Ref. 6)

1

(uniform),

0.996

(parabolic),

0.977

(Gaussian),

0.987

[hollow (Z2 · Gaussian)].

We thus have (49)

78

I. HOFMANN AND J. STRUCKMEIER

and the 1-D generalized emittance equation 1

!!:.- e 2 =

Z2 ds

_

z

32 my3v2N

(dW _ Al dWu ) ds ds·

The corresponding envelope equation is [z =

(50)

(Z2)1I2]

d2 _ e;(s) 1t _ ds 2 Z + kz(s)z - 16i3 - K V3 Al = O.

(51)

4. MINIMUM FIELD ENERGY FOR A UNIFORM-DENSITY BEAM Practical evaluation of the generalized emittance equations requires estimates of the possible change of nonlinear field energy. Numerical calculations for a continuous, round beam have indicated that a uniform density profile has lower field energy than a variety of peaked or hollow profiles with the same rms radius. 1 ,3 Here we show that this is generally true in any dimension. Let us thus consider a 3-D bunched beam with the field energy W given by Eq. (14). We require that the variation of S == W

be zero, with Hence, tJS

a'i

+ a'1X2 + a'2Y 2 + a'3Z2

(52)

Lagrange multipliers to keep the rms dimensions constant.

= III [lOoE bE + N- I(lYIX2 + lY2y2 + lY3Z2) tJn] dx dy dz = O.

(53)

v

By partial integration, we can write this as tJS =

III[q> + N- I(lYIX 2 + lY2y2 + lY3Z2)] tJn dx dy dz + 10 IIq> tJE 0

v

n

da = O.

(54)

s

The boundary integral can be neglected for a large enough integration volume, since the total charge is kept constant. The variation of density, 6n(x, y, z), cannot be arbitrary, since N must be constant and n + 6n ~ 0 everywhere. We thus define it by an arbitrary displacement {6x(x, y, z), 6y(x, y, z), 6z(x, y, z)} of the position vector: 6n(x, y, z)

= n(x + 6x, y + 6y, z + 6z) -

n(x, y, z)

= Vn · 6x,

(55)

and obtain tJs =

III[q> + N- I(lYIX 2 + lY2y2 + lY3z2)](Vn • tJx) dx dy dz = 0,

(56)

which is satisfied by either l/J = -N-1(a'1X2 +a'2y 2 + a'3Z2) (interior of beam) or n = constant = 0 (outside), corresponding to a uniformly charged ellipsoid. The same proof holds for 2-D and 1-D beams in full space, or with a distant boundary. The important conclusion from this is that relaxation of a distribution with nonuniform density towards one of uniform density is accompanied by a growth of emittance to compensate for the reduced nonlinear field energy.

EMITIANCE OF HIGH-CURRENT BEAMS

79

5. SHIELDING NEAR THE SPACE-CHARGE LIMIT For a practical evaluation of the generalized emittance equations, we will find it useful to study analytically the nonlinear field energy of stationary distributions in high-current beams. This will enable us to relate the nonlinear field energy of a matched beam to its tune depression vIvo (or alao with periodic focusing), which we will use as a dimensionless parameter describing intensity. For small v I vo, we expect the well-known "plasma shielding" effect, where the external potential is shielded from the beam interior by the space-charge potential. This collective behavior of an intense beam leads to the development of a practically uniform density for v I vo~ 0, regardless of the shape of the distribution function, provided that the external focusing force is linear. In computer simulations, the general observation has been that an intense beam relaxes to such a more uniform self-consistent density profile if injected with a nonmatched profile. The relaxation is accompanied by a change of emittance, which we intend to calculate from our theory. For the sake of simplicity in the following, we assume round (spherical or cylindrical) beams with time-independent (continuous) focusing forces. 5.1. 3-D Case We assume first a Gaussian distribution function of the single-particle Hamiltonian (thus automatically a stationary distribution): n(O)

[

f = (2JlIl )312 exp where

ep follows from Poisson's equation, V2 ep

q)/Il,J

(X,2 + y,2 + Z,2 k 2 2 + 2" r + my3v2 ep

(57)

obtained by calculating the density

= _!L n(O) exp [- (~r2 + - q - ep)/ItJ ' Eo 2 my3v2

(58)

with n(O) the density on axis and It =X,2 the average of X,2 as a measure for the beam temperature. Equation (58) can be solved explicitly in the low-current limit, where ep in the density expression is negligible. Here we are interested in the high-current limit and assume that the nonlinear dependence on ep can be expanded as a power series: V2 ep

= _!L n(0)[1 - (~r2 + - q - ep)/It + · · Eo 2 my3v2

.J '

(59)

where we retain only the first-order term in the total potential. We make the substitution (60)

with (61)

I. HOFMANN AND J. STRUCKMEIER

80

and obtain the familiar equation V2~ = C~,

(62)

which is solved by the modified spherical Bessel function

~~/ZIn+ln for n

= 0 and assuming z == VC r: . h -VfiCT/ 2/ Z /1/2 = Z -1 sIn Z.

(63)

Using Eq. (60), the density follows from Poisson's equation as n = n(O)[

aIA aIA erIADr7;.:-rIAD _e :7;.:-aIAD]/[ 2 _ e :7;.:-aIAD] ,

(64)

where we have introduced the Debye length according to

A1>= C- 1

(65)

and defined a as the beam edge. Introducing the plasma frequency (on axis), with time replaced by distance, as

w2 = q2n (O) p

(66)

Eomy3v2'

we can write (67)

in agreement with the usual definition of the Debye length. Equation (64) reveals the shielding behavior, which can be seen more easily by rewriting the expression, using AD «a (avoiding r = 0), as e(r-a)/AD]

n = n(O) [ 1-

/

(68)

.

r a

The density is thus uniform except for a sheath of thickness AD, where it drops to zero. Comparison of Eq. (68) [more accurately Eq. (64)] with Eq. (59) shows that the solution found is consistent with the series truncation as long as \r - at> AD and AD« a. Inside the boundary sheath the truncation is invalidated when approaching, = a, where the full solution cannot have a sharp edge (see Fig. 1). For a waterbag (i.e., step function) distribution defined as

f

n(O)

=

1Jr (3", )3n ()

(X,2 + y,2 + Z,2

2

k 2

q

)

+ 2r + my3v2