Dynamics of intense particle beam in axial-symmetric magnetic field

Nuclear Instruments and Methods in Physics Research A 772 (2015) 93–102

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Dynamics of intense particle beam in axial-symmetric magnetic field Yuri K. Batygin Los Alamos National Laboratory, Los Alamos, NM 87545, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 26 August 2014 Received in revised form 12 October 2014 Accepted 17 October 2014 Available online 6 November 2014

Axial-symmetric magnetic field is often used in focusing of particle beams. Most existing ion Low Energy Beam Transport lines are based on solenoid focusing. Modern accelerator projects utilize superconducting solenoids in combination with superconducting accelerating cavities for acceleration of highintensity particle beams. Present article discusses conditions for matched beam in axial-symmetric magnetic field. Analysis allows us to minimize power consumption of solenoids and beam emittance growth due to nonlinear space charge, lens aberrations, and maximize acceptance of the channel. Expressions for maximum beam current in focusing structure, beam emittance growth due to spherical aberrations and non-linear space charge forces are derived. & 2014 Elsevier B.V. All rights reserved.

Keywords: Beam Emittance Acceptance Solenoid Aberration Space charge

0 1. Lattice of periodic solenoid channel

¼@

Consider a focusing lattice consisting of a periodic sequence of focusing solenoids of length D, field Bo, distance between lenses l, and period L ¼ l þD (see Fig. 1). A matched beam reaches its maximum size in the center of the solenoids, and minimum size in the middle of drift space (see Fig. 2). The transformation matrix in a rotating frame through a period of the structure between centers of solenoids is given by [1] 0 1 0 1 D D cos θ2 cos θ2 sin θ2  1 l  sin θ2 θ θ @ A @ A 0 1  Dθ sin θ2 cos θ2  Dθ sin θ2 cos θ2 0

l cos θ  2D θ sin θ ¼@  θ 2 2 θ  D sin θ þ l D sin θ2

where

θ¼

D

θ sin

θ þl cos 2 θ2

l cos θ  2D θ sin θ

1 A;

ð1:1Þ

θ is the rotational angle of particle trajectory in a solenoid:

qBo D : 2mcβγ

1

 Dθ sin θ

2 l cos θ  l4Dθ sin θ þ D sinθ θ

l cos θ  2D θ sin θ

1 A

ð1:3Þ

From the matrices, Eqs. (1.1) and (1.3), the value of betatron tune shift per period, μo , is determined by cos μo ¼ cos θ  θ sin θ

ðL  DÞ : 2D

ð1:4Þ

Adopting the expansions cos ξ ¼ 1  ξ =2 þ ξ =24 and 3 sin ξ ¼ ξ  ξ =6, the value of betatron tune shift per period reads: ffi rffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    L θ2 1 D L þ : ð1:5Þ 1 μo ¼ θ 1 D 2 L D 6 2

4

Thus, the maximum and minimum values of the beta-function βmax=min ¼ m12 = sin μo in the channel are given by:

βmax ¼

h

i θ=2 L cos 2 θ2 1  DL 1  tan ðθ =2Þ sin μo

;

ð1:6Þ

ð1:2Þ

The matrix of transformation through the period of the structure between centers of drift space is: ! ! ! D cos θ 1 l=2 1 l=2 θ sin θ 0

l cos θ  2D θ sin θ

 Dθ sin θ

cos θ

0

E-mail address: [email protected] http://dx.doi.org/10.1016/j.nima.2014.10.034 0168-9002/& 2014 Elsevier B.V. All rights reserved.

1

βmin ¼

DÞ2 θ ðL  DÞ cos θ  ðL 4D sin θ þD sinθ θ : sin μo

ð1:7Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eqs. (1.6), (1.7) determine pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the maximum Rmax ¼ βmax 3 and minimum Rmin ¼ βmin 3 matched envelope of the beam with unnormalized emittance, э, and negligible beam current, I ¼0. Acceptance of the channel with aperture radius, a, is given by

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Y.K. Batygin / Nuclear Instruments and Methods in Physics Research A 772 (2015) 93–102

A ¼ a2 =β max :

2. Thin lens approximation

a2 sin μo

i : A¼ h θ=2 cos 2 θ2 L 1  DL 1  tan ðθ=2Þ

ð1:8Þ

The acceptance, Eq. (1.8), is maximized at a betatron tune shift within the range 0 o μo o 180o (see Fig. 3b, solid line).

If the thickness of the lens is significantly smaller than the period of the structure, D/L≪1, focusing properties of the solenoid can be represented by the thin lens matrix with focal length f: M¼

1

0

 1=f

1

! ;

f¼

D

θ2

:

ð2:1Þ

Consequently, the betatron tune shift per period of the structure is determined from Eq. (1.4) as: cos μo ¼ 1  θ

2

L L ¼ 1 ; 2D 2f

ð2:2Þ

from which the value of μo is

Fig. 1. Periodic structure of focusing solenoids.

rffiffiffiffi sffiffiffi L L : μo  θ ¼ D f

ð2:3Þ

From the condition jcos μo j r 1, the stability criteria for particle oscillations is expressed as 0 r L r4f [2]. Accordingly, the acceptance of the channel is simplified from Eq. (1.8) as A

a2 sin μo L

ð2:4Þ

and has a maximum at the value of μo  π =2. In this case, cos μo ¼ 1  L=2f ¼ 0, and f ¼ L=2 which expresses a condition of symmetry of the channel. The values of beta-function from Eqs. (1.6) and (1.7) are approximated as

Fig. 2. A matched beam in a periodic focusing structure.

βmin ¼

  L μ2 1 o ; sin μo 4

βmax ¼

L : sin μo

ð2:5Þ

Fig. 3. (a) Minimum and maximum beam sizes in a periodic solenoid structure with D/L ¼ 0.034: (solid line) solution from matrix analysis, Eqs. (1.6) and (1.7), (dotted line) smooth approximation to the beam envelope, Eq. (3.26), (b) acceptance of the channel: (solid line) determined by matrix method, Eq. (1.8), (dotted line) determined from envelope equation, Eq. (3.29).

Y.K. Batygin / Nuclear Instruments and Methods in Physics Research A 772 (2015) 93–102

From Eq. (2.2), the value of sin μo ¼ ressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   L L 1 : sin μo ¼ f 4f

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  cos 2 μo can be exp-

The dynamics of the slow variable is governed by the Hamiltonian H¼

ð2:6Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Correspondingly, p the values of maximum Rmax ¼ β max 3 and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi minimum Rmin ¼ β min 3 matched beam sizes are given by [2] sffiffiffiffiffiffiffiffiffiffiffiffiffi " #1=4 pffiffiffiffiffiffiffiffi 4ðf =LÞ2 L ð2:7Þ ; Rmin ¼ Rmax 1  : Rmax ¼ 3 L 4f 4ðf =LÞ  1

95

ðdRaver =dtÞ2 1 1 F 2n ðRaver Þ þ ∑ þ UðRaver Þ; 4 n ¼ 1 ω2n 2

ð3:8Þ

while the oscillatory correction is determined entirely by the properties of the rapidly varying part of the field: 1

ξðtÞ ¼  ∑

n¼1

F n ðRaver Þ

ω2n

cos ωn t:

ð3:9Þ

The equation of motion for the slow variable Raver ðtÞ is derived from the Hamiltonian, Eq. (3.8):

Under the maximum acceptance condition, L ¼ 2f , the matched beam sizes, Eq. (2.7), evaluate to rffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 3L Rmin ¼ ð2:8Þ ; Rmax ¼ 3 L: 2

Substitution of Eqs. (3.5), (3.6) in Eq. (3.10) gives

from which pthe ffiffiffi ratio of the beam sizes in a matched beam is Rmax =Rmin ¼ 2.

3 2 ðβcÞ2 2Ic2 2 þ ; R€ aver ¼ Raver Ωr þ 3 3 I R c aver βγ Raver

3. Dynamics of space-charge-dominated beam

where the transverse frequency of single particle oscillations is !2   2 1 sin 2 π nD ðθβcÞ2 1 θ Lβc 2 L þ Ωr ¼ : ð3:12Þ ∑ 2 π 2 D2 LD n4 n¼1

Analysis presented above gives us the ability to match a beam with negligible current. For analysis with non-zero current, I a 0, let us use the KV envelope equation for a round beam envelope RðzÞ in an axially symmetric channel [2]: d R 3 2 ðβcÞ2 2Ic2 þ ω2L R   ¼ 0; 2 3 3 I dt c Rβγ R

1 1 F n dF n ∂U  : R€ aver ðtÞ ¼  ∑ 2 2 n ¼ 1 ωn dRaver ∂Raver

ð3:1Þ

where ωL ðzÞ ¼ qBðzÞ=ð2mγ Þ is the Larmor oscillation frequency in magnetic field, and I c ¼ 4πεo mc3 =q ¼3:13  107 A=Z [Amp] is the characteristic beam current. The magnetic field along the structure can be expanded in Fourier series:      D 2 1 1 π nD 2π nz cos : ð3:2Þ B2 ðzÞ ¼ B2o þ ∑ sin L πn¼1n L L Substitution of expansion, Eq. (3.2), into envelope eq. (3.1) gives:       2 d R R qBo 2 1 1 π nD 2π nβ ct cos ¼ ∑ sin 2 2π mγ L L dt n¼1n   RD qBo 2 ð 3 βcÞ2 2Ic2  þ þ : ð3:3Þ 4L mγ I c Rβγ 3 R3

ð3:11Þ

The sum in Eq. (3.12) can be rewritten as sin ðnζ Þ 1 1 1 1 1 cos 2nζ ¼ ∑ 4 ∑ ; 2n¼1n 2n¼1 n4 n4 2

1

∑

n¼1

2

ð3:10Þ

ð3:13Þ

where ζ ¼ π D=L. Calculation of separate sums reads [4]: 1 π4 ¼ ; 4 90 n¼1n 1

∑

1

∑

n¼1

cos nξ π 4 π 2 ξ πξ ξ  þ  : ¼ 90 12 12 48 n4 2

3

4

ð3:14Þ

Therefore, Eq. (3.13) reads:     "   # 1 sin 2 π nD π4 D 2 1 D 1 D 2 L  þ ¼ : ∑ 2 L 2 L 3 L n4 n¼1

ð3:15Þ

Consequently, substitution of Eq. (3.15) into Eq. (3.12) yields the transverse oscillation frequency: #  2 " 4  4 4 βc L θ L 2 θ L θ þ Ω2r ¼ θ2 þ  : ð3:16Þ L D 12 D 6 D 12

Note that the envelope equation, Eq. (3.3), describes the oscillatory motion in a combination of two oscillating fields:

The expression in the square brackets in Eq. (3.16) is a square of the betatron tune shift, Eq. (1.5), which is related to the transverse oscillation frequency by

∂UðRÞ ; R€ ¼ ∑ F n ðRÞ cos ðωn tÞ  ∂R n¼1

μo ¼ Ωr

1

ð3:4Þ

where the first term describes oscillatory field due to variation of magnetic field B(z) along the channel.     R qBo 2 D 2π nβ c F n ðRÞ ¼  ; ð3:5Þ sin π n ; ωn ¼ 2 π n mγ L L and the second term determines oscillating function U(R), which depends on variable beam radius R:   ∂UðRÞ RD qBo 2 3 2 ðβcÞ2 2Ic2 ¼   : ð3:6Þ 3 3 ∂R 4L mγ I c Rβγ R However, for μo {2π variation of beam radius at the period of the structure is small, and function U(R) can be approximately assumed to be z - independent. Under these assumptions, according to the averaging method [3], envelope oscillations can be decomposed into a combination of slow variable Raver ðtÞ and a small-amplitude, rapidly oscillating component ξðtÞ: RðtÞ ¼ Raver ðtÞ þ ξðtÞ:

ð3:7Þ

L

βc

:

ð3:17Þ

Therefore, the averaging method gives the same value for the betatron tune shift as the matrix method. Upon changing the independent variable in Eq. (3.11) from t to z, the equation for the slow envelope variable becomes d Raver 3 2 μ2 P2  3 þ 2o Raver  ¼ 0; R dz2 aver Raver L 2

ð3:18Þ

where P2 is the generalized beam perveance: P2 ¼

2I Ic β γ 3 3

:

Retaining

ð3:19Þ only

the

leading-order

term

in

Eq.

(3.9),

ξðzÞ   F 1 ðRaver Þ cos ω1 t=ω21 , the rapidly oscillating component of the beam envelope is given by:     z

R ðzÞ L 2 D cos 2π : ξðzÞ ¼ aver3 θ sin π D L L 2π

ð3:20Þ

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Y.K. Batygin / Nuclear Instruments and Methods in Physics Research A 772 (2015) 93–102

Finally, the solution of the envelope equation can be expressed as z

ð3:21Þ RðzÞ ¼ Raver ðzÞ 1 þ υmax cos 2π ; L

decrease in the depressed betatron tune shift μ [5] qffiffiffiffiffiffiffiffiffiffiffiffiffi  2 μ ¼ μo 1 þ bo bo ;

where the amplitude of envelope oscillations around average value Raver ðzÞ is     θ2 L 2 D : ð3:22Þ υmax ¼ 3 sin π L 2π D

and the smooth approximation progressively approaches the exact solution of the envelope equation (see Fig. 4). Space-charge-induced beam emittance growth in a focusing channel is limited by the value (free energy effect [8]): ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u 3 ef f u 2 I Raver ΔW t ¼ 1þ ; ð3:32Þ Wo 3 3 ðβγ Þ3 I c

The solution (3.21) describes the characteristics of a general unmatched beam, with slow variation of average radius Raver ðzÞ superimposed on a fast oscillation with the period equal to the period of the structure. The relative amplitude of variation of beam envelope, υmax , Eq. (3.22), does not depend on either the beam current or the beam emittance [5]. Analysis of space-charge limited beam current in transport systems was performed in Refs. [5–7]. A matched beam corresponds to a constant value of the average beam envelope Raver ðzÞ ¼ Raver and can be determined from Eq. (3.18) by setting R″aver ðzÞ ¼ 0 : rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Raver ¼ Raver ð0Þ

bo þ

2

1 þ bo ;

ð3:23Þ

where Raver ð0Þ is the matched average beam size with negligible space charge, sffiffiffiffiffiffiffiffi 3L Raver ð0Þ ¼ ; ð3:24Þ

μo

and bo is the space charge parameter: !2 1 I Raver ð0Þ : bo ¼ 3 ðβγ Þ3 I c

ð3:25Þ

The minimum and maximum matched beam envelope sizes in presence of space charge forces are given by: Rmax = min ¼ Raver ð1 7 υmax Þ:

ð3:26Þ

Maximum beam current is achieved when maximum beam size is set equal to the aperture of the channel Rmax ¼ a , which is determined from Eqs. (3.23)–(3.26) as sffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi 3L 2 a¼ ð3:27Þ bo þ 1 þ bo ð1 þ υmax Þ:

μo

For negligible beam intensity, bo ¼ 0 , Eq. (3.27) describes the beam with maximum possible emittance (acceptance of the channel) approximated by envelope equation 3 ¼ Aenv : sffiffiffiffiffiffiffiffiffiffiffi Aenv L a¼ ð1 þ υmax Þ: ð3:28Þ

μo

Eq. (3.28) gives the following approximation to the acceptance of the channel (this equation should be compared to the exact expression for acceptance, Eq. (1.8)): Aenv ¼

a2 μo

Lð1 þ υmax Þ2

:

From Eqs. (3.27)–(3.28) the maximum beam current is: "   # I c μo 3 2 3 Aenv ðβγ Þ 1  I max ¼ : Aenv 2 L

ð3:29Þ

ð3:30Þ

Fig. 3 illustrates matched beam sizes, Rmax = min for a beam with negligible current, and acceptance of the channel, A, obtained from matrix analysis as well as that obtained utilizing the averaging approximation to the envelope equation,Aenv . It is clear that validity of averaging method (smooth approximation) is limited by values of μo r60o . Enhancement of the beam current leads to a

ð3:31Þ

where the free energy parameter ΔW=W o depends on the beam distribution (see Table 1). From Eq. (3.32) it is clear that minimization of the beam radius in the channel is the way to reduce space-charge induced beam emittance growth [8, p. 315].

4. Maximum space-charge limited beam current Eq. (3.30) gives an approximate value of the maximum beam current in a periodic structure. To determine a more exact value for space - charge limited beam current, consider the beam transport in drift space between lenses described by the envelope Eq. (3.1) without a focusing term: 2

d R 3 2 P2   ¼ 0: dz2 R3 R Eq. (4.1) can be integrated to yield [5]: !  2  2  2  2 dR dR 3 R2 R ¼ þ 1  o2 þ P 2 ln : dz dz o Ro Ro R

ð4:1Þ

ð4:2Þ

Now consider the space charge dominated regime, where the beam emittance can be neglected, 3  0. At the middle point between lenses, z¼ zo, the beam has a waist size Ro ¼ Rmin and zero divergence,R0o ¼ 0 . Thus, in this case, Eq. (4.2) can be rewritten as !2 pffiffiffi dR R 2Pz ¼ ln R; R ¼ ; Z¼ : ð4:3Þ dZ Rmin Rmin Consequently, expansion of the beam radius in drift space from R ¼ 1 to Rmax ¼ Rmax =Rmin is determined by the integral: Z Rmax pffiffiffi ðz  zo Þ 1 dR pffiffiffiffiffiffiffiffiffi ¼ 2P : ð4:4Þ Rmax Rmax 1 ln R The left hand side of Eq. (4.4) has a maximum value of 1.082 for Rmax ¼ 2.35 [9]. As already alluded to above, the maximum radius is achieved in the channel at the distance equal to half of the spacing between lenses, z zo ¼ L=2 , which in turn yields P L pffiffiffimax ¼ 1:082: 2Rmax

ð4:5Þ

From this expression, the maximum transported current in the channel is   Rmax 2 : ð4:6Þ I lim ¼ 1:17I c ðβγ Þ3 L The divergence of the beam at the lens can be estimated from Eq. (4.2): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   dRmax 4I lim Rmax Rmax ¼ : ð4:7Þ 2 ln dz Rmin L I c ðβγ Þ3 Total change in the slope of the beam envelope at the lens has to be equal to twice the value of dRmax =dz determined by Eq. (4.7). Therefore, the required focal length of the lenses is f  L=4, and the maximum space charge limited beam current is achieved in a

Y.K. Batygin / Nuclear Instruments and Methods in Physics Research A 772 (2015) 93–102

97

Fig. 4. Minimum and maximum beam sizes in a periodic solenoid structure with D/L ¼0.034 for a space-charge dominated proton beam with energy W ¼35 keV, beam current I¼ 3.5 mA and beam emittance 3 ¼ 9:262 π cm mrad: (solid line) exact solution of envelope equation, Eq. (3.1), (dotted line) smooth approximation to beam envelope, Eq. (3.26). Table 1 Characteristics of beam distributions. Distribution Definition ρðx; x'; y; y'Þ ¼ ρðIÞ I ¼ r 2x þ r 2y r 2x ¼ γ x x2 þ 2αx xx'þ βx x'2

Distribution in phase space ρðx; x0 Þ ¼ ρðr 2x Þ

Space charge field

C α R4 Eq. f 3 (8.4)

Coeff. K

0

0.0556 0

0

8 I z 9 Ic ðβγÞ3 3

0.114

0.094

0.01126

3 I z r2 3I r r2 r4 ð1  2 Þ2 ð1  2 þ Þ 3 4πεo βc R2 2πR2 βc 2R 2R 12R4 2 Ic ðβγÞ 3 I z 2I r2 I r2 4 ½1  expð  2 2 Þ expð  2 2 Þ 2πεo βcr I c ðβγÞ3 3 R R πR2 βc

0.164

0.187

0.02366

0.541

0.55

0.077

Space charge density

r 2y ¼ γ y y2 þ 2αy yy' þ βy y'2 KV Water Bag Parabolic Gaussian

1 δðI  F o Þ π2 F o 2 ; I r Fo π2 F o 2 6 I ð1 Þ Fo π2 F o 2 1 π2 F o 2

expð

I Þ Fo

1 π 3x 4 2 r 2x ð1 Þ 3π 3 x 3 3x 3 r2 ð1 x Þ2 2π 3 x 23x 2 r2 exp ð  2 x Þ π 3x 3x

I πR2 βc 4I 3πR2 βc 3I

ð1 

2 r2 Þ 3 R2

I r 2πεo R2 βc 2I r r2 ð1  2 Þ 3πεo βc R2 3R

Coeff. K

Coeff. ΔW [8] Wo

structure where μo  180o . Such transports are usually unstable, and can be used only with a limited number of focusing elements. From Eq. (3.32) one can estimate space-charge induced beam emittance growth in this case. Assuming initial beam emittance is negligible, 3  0, and taking into account that average radius of matched beam is   1 Rmax Rmax þ ¼ 0:712Rmax ; Raver ¼ ð4:8Þ 2 2:35 the space-charge induced beam emittance for beam transport with limited beam current, Eq. (4.6), is sffiffiffiffiffiffiffiffiffiffi R2max ΔW 3 ef f ¼ 1:09 : ð4:9Þ Wo L Ratio of space charge term to emittance term in the envelope equation, Eq. (4.1) gives an estimation of dominance of space charge forces over emittance in beam dynamics: !2 2 I lim Raver 1 : ð4:10Þ ¼ b¼ ðΔW=W o Þ ðβγ Þ3 I c 3 ef f Because the value of free energy parameter is within the range of

ΔW=W o ¼ 0::::8 U10  2 , the value of parameter b, Eq. (4.10), is larger than unit, bc 1, which indicates dominance of space charge in beam dynamics and allows us to use simplification of negligible emittance for the considered case.

5. Application to injector LEBT Most existing ion Low Energy Beam Transports (LEBT) utilize 2 or 3 solenoids with intermediate equipment (deflectors, bending magnets, Wien filters, emittance stations) to match the beam from the exit of ion source column to the subsequent RF structure.

Fig. 5. LEBT with two focusing solenoids.

Consider a LEBT comprised of 2 solenoids, separated by a distance L (see Fig. 5). The beam is characterized by a certain emittance э and effective current I ¼ I o ð1  ηÞ, where I o is the total beam current and η is the space-charge neutralization factor. Initial envelope parameters Rs , R0s are determined by extraction conditions from the ion source column. Final beam parameters Rf , R0f are determined by the matching conditions at the front end of the RF accelerator. The purpose of the analysis is then to find appropriate solenoid parameters, and distances d1 , d2 . Analysis of the previous sections allows us to select a beam envelope, corresponding to minimal values of the beam size at the center of solenoid, Rmax. In turn, minimization of the beam size Rmax allows us to minimize solenoid power consumption, effect of spherical aberrations, beam losses, and space-charge induced beam emittance growth. A two-solenoid system can be regarded as a part of a periodic focusing structure. Consider a beam with negligible current, but with a finite value of beam emittance (emittance-dominated beam). Evolution of the beam radius R along drift space z between solenoids as a function of initial radius Ro and slope of the envelope R0o is obtained by integration of

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Y.K. Batygin / Nuclear Instruments and Methods in Physics Research A 772 (2015) 93–102

Eq. (4.2) assuming I¼ 0 [5]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 ffi u 2 R u R ' 3 o ¼ t 1þ z þ z2 : Ro Ro R2o

ð5:1Þ

From the symmetry point of view it is clear, that a matched beam has a minimum size, or waist, Rmin ¼Ro in the mid-point of the drift space between lenses, and maximum size Rmax inside focusing elements. At the waist point, R0o ¼ 0 . Therefore from Eq. (5.1)   3L 2 : ð5:2Þ R2max ¼ R2min þ 2Rmin Equating ∂Rmax =∂Rmin ¼ 0 determines the minimal value of Rmax as a function of beam emittance and distance between lenses "   # ∂Rmax 1 1 3L 2 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0: ð5:3Þ

2ffi Rmin  R3 2 ∂Rmin min L R2min þ 2R3min The solution of Eq. (5.3) is rffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 3L Rmin ð0Þ ¼ ; Rmax ð0Þ ¼ 3 L; 2

ð5:4Þ

which coincides with Eqs. (2.8) for a periodic solution of a matched beam with zero current at a phase advance of μo  π =2. Eq. (5.4) determines the minimum value of Rmax at given value of beam emittance and given distance between solenoids L. Now consider the space-charge dominated regime, where beam emittance is negligible. Analysis presented in Section 4 determines the condition for transporting a beam with maximum current through drift space restricted by aperture Rmax and distance L. From Eq. (4.5) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L I Rmax : ð5:5Þ Rmax ¼ ; Rmin ¼ 1:082 I c ðβγ Þ3 2:35 Eq. (5.5) determine conditions for minimizing the value of Rmax of the beam with given current I in the structure limited by distance L. In a more general case, when both beam emittance and beam current are not negligible, the precise value of Rmax is determined by the variation of the value of Rmin at the mid -point between solenoids, z ¼zo, and a subsequent search for the smallest beam size at the center of the solenoids Rmax through an exact solution of the envelope equation in drift space. Dominance of emittance or space charge on beam dynamics can be estimated through space charge parameter bo, Eq. (3.25), where optimal value of average zero-current beam radius, Eq. (3.24), corresponds to μo  π =2 rffiffiffiffiffiffiffiffiffiffiffi 3L Raver ð0Þ ¼ 2 : ð5:6Þ

π

The value of bo  0 corresponds to the emittance-dominated regime, while values of bo 4 1 correspond to the space-charge dominated regime. After determination of the minimal value of Rmax, the distances d1, d2 are defined by integration of eq. (4.2) to establish points where the beam radius evolves from initial value of Ro to Rmax [5]: z¼

R2o 23

Z

ðRmax =Ro Þ2 1

ds ffi: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2  2 Ro R'o ½1 þ s þ PRo s ln s  1 3

ð5:7Þ

3

In Eq. (5.7), the values of Ro , Ro' correspond to either Rs ; R0s or Rf ; R0f . The slopes of beam envelopes at solenoids R01 , R02 can be found from Eq. (4.2): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u  2 #  2 "  2 u 0 3 Ro 2I R 0 2 t R ¼ ðRo Þ þ : ð5:8Þ 1 ln þ Ro Ro R I c ðβγ Þ3

The values of R01d , R02d are determined by Eq. (4.2) assuming Ro ¼ Rmin , R0o ¼ 0. Then, the focal lengths of the solenoids f1, f2, are determined by the total change in the slope of the beam at each solenoid: f1 ¼

Rmax ; jR1d' j þ jR1' j

f2 ¼

Rmax : jR2d' jþ jR2' j

ð5:9Þ

Subsequently, the magnetic field within each solenoid is determined from Eqs. (1.2), (2.1) as 2mcβγ Bo ¼ pffiffiffiffiffiffi : q fD

ð5:10Þ

The described procedure allows us to perform an optimization of Low Energy Beam Transport from point of view beam size minimization and the reduction of effects associated with beam emittance growth.

6. Spherical aberration of axial-symmetric magnetic lens Dynamics in an axial-symmetric field may be severely affected by spherical aberrations, which result in beam emittance growth. Spherical aberration coefficients are typically expressed through field integrals [10]. Analysis of the effect of spherical aberrations can be simplified by assuming particle radius does not change significantly inside the lens. Consider a single- particle equation of motion in a magnetic field   qBz 2 r€ þ r ¼ 0: ð6:1Þ 2mγ The magnetic field along the structure can be represented as Bz ðr; zÞ ¼ BðzÞ 

r2 B''ðzÞ; 4

ð6:2Þ

where BðzÞ is the longitudinal component of magnetic field at the axis. Usually it can be approximated as [11] BðzÞ ¼

Bo ; 1 þ ðz=dÞn

ð6:3Þ

where Bo is the maximum value of magnetic field, and d is the characteristic length. Parameter n ¼2 corresponds to well-known Glazer model [12], for short length/diameter lenses, while in many cases the field of a solenoid is better approximated by n¼ 4, which provides a more flat distribution. Let us integrate Eq. (6.1) assuming the particle radius remains constant during lens-crossing. The change in slope of particle trajectory follows from Eq. (6.1):  2 Z 1 q r 0 ¼ r 0o r B2z dz: ð6:4Þ 2mcβγ 1 In the near-axis approximation, the longitudinal field component is a weak function of radius, therefore one can assume: h  z n i r2 nBo zn  2 ð1  nÞ þ ð1 þ nÞ d 2 2 Bz  B ðzÞ  BðzÞB″ðzÞ; B″ðzÞ ¼ : h n  n i3 2 d 1þ z d

ð6:5Þ Under this approximation, Eq. (6.4) can be rewritten in more common way: r r 0 ¼ r 0o  ð1 þ C α r 2 Þ; f

ð6:6Þ

where the focal length f and spherical aberration coefficient C α are determined by the expressions R1  2 Z 1 1 qBo dξ 1  1 BðzÞB''ðzÞdz ¼d : ð6:7Þ ; C ¼  R1 2 α n 2 f 2 2mcβγ  1 ð1 þ ξ Þ  1 B ðzÞdz

Y.K. Batygin / Nuclear Instruments and Methods in Physics Research A 772 (2015) 93–102

More specifically, the focal length is determined by Eq. (2.1) with the effective length of the solenoid, D: (π Z n ¼ 2; 2 d  1:57 d; dξ ¼ 3pπffiffi ð6:8Þ D¼d n d  1:666 d; n ¼ 4: ð1 þ ξ Þ2 4 2

Calculation of the spherical aberration coefficient gives [11]: Cα ¼

Cα ¼

1 d

2

2 d

2

R1

ð1  3ξ Þ  1 ð1 þ ξ2 Þ4 d R1 dξ  1 ð1 þ ξ2 Þ2 2

R1

ξ

¼

ð3  5ξ Þ 2 d  1 ð1 þ ξ4 Þ4 R1 dξ  1 ð1 þ ξ4 Þ2 4

ξ ξ

1

n ¼ 2;

2

4d

¼

5 12d

2

n ¼ 4:

ð6:9Þ

99

plane ðr; r 0 Þ is given by: r2 R2

3 þ

 2 R2 0 r r3 r þ þ Cα ¼ 3: f 3 f

Let us introduce new variables (T, tion:

ð7:3Þ

ψ) arising from the transforma-

r pffiffiffi ¼ T cos ψ ; R

ð7:4Þ

  r R pffiffiffi ¼ T sin ψ : r0 þ f 3

ð7:5Þ

ð6:10Þ

The performed analysis corresponds to a weak focusing regime, when the radius of particle changes insignificantly while passing through the lens. In many applications, the spherical aberration coefficient can be expressed through solenoid sizes as [13] Cα ¼

5 ðS þ 2aÞ2

;

ð6:11Þ

where 2a is the pole piece diameter and S is the solenoid pole gap width.

Fig. 7. Action-angle dependence determined by Eq. (7.6).

7. Effect of spherical aberration on beam emittance growth Let us estimate the emittance growth of a beam during its passage through the lens. We assume that the position of the particle is not changed while crossing the lens, and only the slope of the particle trajectory is altered. The transformation of particle variables before ðr o ; r o' Þ to after ðr; r ' Þ lens-crossing is given by: r ¼ ro ;

r r 0 ¼ r 0o  ð1 þC α r 2 Þ: f

ð7:1Þ

Suppose, initial phase space volume is bounded by the ellipse r 2o 2

R

3 þ

r 02 o 2 R ¼ 3: 3

ð7:2Þ

To find the deformation of the boundary of the beam phase space after passing through the lens, let us substitute the inverse transformation r o ¼ r, r 0o ¼ r 0 þ ðr=f Þð1 þ C α r 2 Þ into the ellipse equation, Eq. (7.2). The boundary of the new phase space volume, occupied by the beam after passing through the lens at phase

Fig. 8. Effective beam emittance growth due to spherical aberration as a function of the parameter υ: (sold line) Eq. (7.9), (dotted line) approximated by Eq. (7.10).

Fig. 6. Distortion of beam emittance due to spherical aberrations, Eq. (7.6): (left) υ¼ 0, (right) υ¼ 1.6.

100

Y.K. Batygin / Nuclear Instruments and Methods in Physics Research A 772 (2015) 93–102

KV

Water Bag

Parabolic

Gaussian

Fig. 9. Effective emittance growth due to spherical aberrations for beams with different initial distributions in a lens with υ ¼1.6, Eq. (7.7).

Y.K. Batygin / Nuclear Instruments and Methods in Physics Research A 772 (2015) 93–102

KV

Water Bag

Parabolic

Gaussian

Fig. 10. Emittance growth of a 150 keV proton beam with current I¼ 1.11 mA and beam emittance 3 ¼ 0:039π cm mradin drift space of length z ¼100 cm.

101

102

Y.K. Batygin / Nuclear Instruments and Methods in Physics Research A 772 (2015) 93–102

In terms of new variables, the shape of the beam emittance after lens-crossing is T þ 2υT 2 sin ψ cos 3 ψ þ T 3 υ2 cos 6 ψ ¼ 1;

ð7:6Þ

where the following notation is used

υ¼

C α R4 : f 3

ð7:7Þ

without nonlinear perturbations, υ ¼ 0 and Eq. (7.6) describes a circle in phase space. Conversely, if υ a 0, Eq. (7.6) describes an S – shape distorted figure of beam emittance (see Fig. 6). Filamentation of beam emittance in phase space is a fundamental property of a beam affected by aberrations. Being symplectic in nature, the transformation, Eq. (7.1), conserves phase-space area. Conservation of phase space area enclosed by the deformed circle is also confirmed numerically: Z 3 2π Tðψ Þdψ ¼ π 3 : ð7:8Þ 2 0 While phase space area occupied by the beam before and after the lens are the same, the effective area, occupied by the beam, increases as a result of the encounter. Let us determine the increase in effective beam emittance as a square of the product of minimum and maximum values of T: 3 ef f pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ T min T max : ð7:9Þ 3 The values Tmax, Tmin are determined numerically from Eq. (7.6) (see Fig. 7). Dependence of the emittance growth on the parameter υ is shown in Fig. 8. Qualitatively, this relationship can be approximated by the function: 3 ef f pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ¼ 1þ Kυ ; ð7:10Þ 3 where the parameter K E0.4. Substitution of Eq. (7.7) into Eq. (7.10) gives the following expression for effective beam emittance growth: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u 3 ef f u C α R4 t ¼ 1þK : ð7:11Þ 3 f 3 Eq. (7.11), was tested numerically for a round beam with different particle distributions (see Fig. 9). Simulations were performed with macroparticle code BEAMPATH [14]. As a measure of effective beam emittance, the four-rms beam emittance was used and a two-rms beam size was used as a measure of the beam radius: ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi   ffi 3 ¼4 x2 x02  hxx0 i2 ; R ¼ 2 ð7:12Þ x2 : Simulations confirm that dependence, Eq. (7.11), is valid for four-rms beam emittance, although the coefficient K depends on the beam distribution (see Table 1). Generally, the value of K is smaller than that determined above, except for the case of a Gaussian distribution. 8. Space charge induced beam emittance growth in drift space Non-linear space charge forces inherent to a non-uniform beam act on the beam as a non-linear lens. Analysis developed in Section 7 can be applied to determine space-charge induced beam emittance growth. Consider the initial stage of beam drift in free space at a certain distance z, where radial particle positions are not changed significantly, but the momentum distribution is already affected by the space charge field of the beam, Eb(r). Change in the radial slope of the particle trajectory is given by r 0 ¼ r 0o þ

qzEb ðrÞ mc2 β γ 3 2

:

ð8:1Þ

Consider a Gaussian beam in drift space. The space charge field of the beam is approximated as:     I r2 I r r2 Eb ðrÞ ¼ 1  expð  2 2 Þ  1  2 þ :::: : ð8:2Þ 2 2πεo β cr πεo βc R R R Substitution of Eq. (8.2), into Eq. (8.1) results in a transformation, Eq. (6.6), where 1 I z ¼4 ; f I c ðβγ Þ3 R2

Cα ¼ 

1 R2

:

ð8:3Þ

Parameter υ, Eq. (7.7), which determines the effect of spherical aberration on the beam emittance is therefore C α R4 4 I z : ¼ 3 f 3 β γ 3 Ic 3

ð8:4Þ

Substitution of Eq. (8.4) into Eq. (7.11) results in the following expression for space charge induced beam emittance growth in free space: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u 3 ef f u I z t ¼ 1þK : ð8:5Þ 3 3 Ic β γ 3 3 The parameter K was determined numerically for different distributions (see Fig. 10). Results are summarized in Table 1. As follows from Eq. (8.5), initial emittance growth does not depend on initial beam radius. The same result was obtained in Ref. [15] for a waterbag distribution with э ¼ 0.

9. Summary In this work, we have determined matched beam transport conditions for a periodic structure of focusing solenoids in both, emittance-dominated and space-charge-dominated regimes. A closed-form expression for the maximal limited beam current and beam emittance growth in the considered structure were obtained. The developed analysis can be applied to beam matching in a typical solenoid focusing channels.

Acknowledgments Author is indebted to Konstantin Batygin for useful discussions and help in preparation of manuscript.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

K.L. Brown, F. Rothacker, D.C. Carey, Ch. Iselin, SLAC-91, Rev. 3 (1983). M. Resiser, Theory and Design of Charged Particle Beams, John Wiley, 1994. L.D. Landau, E.M. Lifshitz, Mechanics, third ed., Butterworth-Heinemann, 1976. I.S. Gradshtein, I.M. Ryzhik, Table of Integrals, Series, and Products, fifth ed., Academic Press, 1994. I.M. Kapchinsky, Theory of Resonance Linear Accelerators, Harwood, 1985. M. Reiser, Part. Accel. 8 (1978) 167. M. Reiser, J.Appl. Phys. 52 (2) (1981) 555. T.P. Wangler, RF Linear Accelerators, second ed., Wiley-VCH, 2008. R. Hutter, Beams with Space Charge, in: A. Septier (Ed.) (Ed.), Focusing of charged particles, Vol. 2, Academic Press, 1967. P.W. Hawkes, Lens Aberrations, in: A. Septier (Ed.,) (Ed.), Focusing of charged particles, Vol. 1, Academic Press, 1967. B. Biswas, Rev. Sci. Instrum. 84 (2013) 103301. W. Glazer, Grundlagen der Electronoptik, Springer, Wien, 1952. O..Klemperer, Electron Optics, Cambridge, 1971. Y.K. Batygin, NIM-A 539 (2005) 455. T.P.Wangler, P.Lapostolle, A.Lombardi, Proceeding of PAC, 1993, p. 3606.