Beam matching with space charge in energy recovery linacs

Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162822

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Beam matching with space charge in energy recovery linacs A. Khan a ,∗, O. Boine-Frankenheim a , F. Hug b , C. Stoll b a b

Technische Universität Darmstadt, Schlossgartenstraße 8, 64289 Darmstadt, Germany Johannes Gutenberg-Universität Mainz, Becher-Weg 45, D-55128 Mainz, Germany

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Keywords: Dispersion ERL Momentum compaction Space charge

ABSTRACT Matching with space charge of an Energy-Recovery Linac (ERL) arc into the subsequent RF structure is essential to preserve beam quality. We show how to match beam envelopes and dispersion along the bends and recirculation arcs of an ERL, including space charge forces, in order to adjust the beam to the parameters of the subsequent RF structure. For a qualitative analysis, we show that one can use a beam matrix approach together with the smooth focusing approximation but with longitudinal–transverse coupling. It is also shown that the space-charge-modified dispersion plays a key role for the adjustment of the momentum compaction 𝑅56 required for both the isochronous and the non-isochronous recirculation mode of an ERL. In this work, a simple coupled transverse–longitudinal beam matrix approach for matching with space charge is employed and compared with particle tracking simulations using ELEGANT. As an example case, we use the 5 MeV ◦ low-energy, 180 injection arc, which also works as a bunch compressor, and matched to the subsequent first RF structure of the projected multi-turn Mainz Energy-recovering Superconducting Accelerator (MESA).

1. Introduction Energy-recovery-linacs (ERLs) are an emerging generation of electron accelerators with the potential of high beam power in continuous wave (CW) operation at moderate operational costs. They are set up as recirculating linacs, allowing for multiple passes through the superconducting accelerating cavities. After the fully accelerated beam completed its interaction, for example with an internal target, the electrons are decelerated in the linac, transferring their energy back to the cavity radio-frequency (RF) fields. This recovery of beam energy results in an enhanced energy efficiency of ERLs. Several high-current, high-brilliance ERL facilities with short pulses have been recently proposed for various applications, for example, to achieve high precision in internal target experiments; to achieve high optical beam power in free-electron laser sources; advancement of photon brilliance in light sources; electron cooling devices that would benefit from both high average current and good beam quality; or, possibly, as the electron accelerator in an electron–ion collider intended to achieve an operating luminosity beyond that provided by existing, storage-ring-based-colliders [1–3]. The goal of an ERL machine is to preserve the high beam quality and brilliance delivered from the particle source throughout the machine. Therefore emittance growth and beam loss should be well controlled to achieve the required efficiency of the energy recovery process [4,5]. For intense electron bunches at low to medium energy traversing through bends, it is essential to understand the details of space-chargeinduced effects to maintain beam quality throughout the ERL operation.

Particularly, current dependent matching of an arc into the subsequent RF structure has been found to be essential to preserve the beam quality. Beam matching with space charge has been discussed mostly in the context of high intensity beams in conventional linacs and synchrotrons [6,7]. For example, the effect of transverse space charge on the dispersion function has been a topic of very active research for many years [8–10]. Current dependent non-zero dispersion might lead to emittance growth in the RF structure because of longitudinal– transverse phase space coupling. Space-charge-induced distortions of beam envelopes in ERLs have been discussed in [11,12]. Dispersion together with transverse space charge has been studied by Venturini–Reiser (V–R) [9] and Lee–Okamoto (L–O) [10]. They independently derived equations, describing the beam envelopes together with the dispersion function under the influence of a linear space charge forces. Theoretical aspects of beam matching with space charge and dispersion have been discussed in [13–16]. For example, [15,16] outlined the concept of two different dispersion functions, one for the beam center, which is not affected by space charge, and one for the off-center particles. Experiments related to space charge and dispersion with low energy proton beams were performed in the CERN PS Booster, matching the beam from the linac into the synchrotron. Although the space charge was found to be relevant, it was sufficient to use the zerointensity dispersion for the matching of the beam center, in order to improve the injection efficiency [17]. An important role of the recirculation arc in an ERL is to provide path length adjustment options to set the accurate required RF phase

∗ Corresponding author. E-mail address: [email protected] (A. Khan).

https://doi.org/10.1016/j.nima.2019.162822 Received 23 July 2019; Received in revised form 12 August 2019; Accepted 19 September 2019 Available online 23 September 2019 0168-9002/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

A. Khan, O. Boine-Frankenheim, F. Hug et al.

Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162822

of 0◦ to 180◦ for acceleration and deceleration. Transverse space charge modifies the dispersion function along the arc and so the momentum compaction which is the transport matrix element 𝑅56 for the individual particles. In case the arc settings are chosen for zero-intensity, one would end up with a dispersion and bunch length different from the design values at a subsequent RF structure. It is therefore necessary to understand the modification of dispersion due to space charge along the arc in order to do proper matching into the next lattice section. Longitudinal space charge also plays an important role, especially for small momentum deviations. Longitudinal space-charge-induced variations in the bunch length or momentum deviation also affect the transverse space charge force by varying the local current density and the transverse beam size through the dispersion. Mainz Energy-recovering Superconducting Accelerator (MESA) is a small-scale, multi-turn, double-sided recirculating linac with vertical stacking of the return arcs operating in CW mode under construction at the Johannes Gutenberg-Universität Mainz Appendix A. Because internal target experiments at the proposed multi-turn MESA demand small momentum deviation (≈ 10−4 ), and moderate 𝛽 functions (≈ 0.5 m) with very small transverse emittance at the interaction point (IP), we need a coupled transverse–longitudinal beam matrix approach to optimize the lattice settings for the given intensity parameters. In this work, we focus on the optimized settings for the MESA low energy (5 MeV) injection arc (MARC0), used also as a bunch compressor, which will be crucial to achieve the desired beam parameters at the IP. We optimized the injection arc for two different cases, first case with small initial momentum deviation 1 × 10−4 , and second case with an initial momentum chirp −7% and the much larger initial momentum deviation of 4 × 10−3 [18]. In subsequent recirculation arcs of MESA, the beam energy is much higher than the injection energy. Therefore, space charge defocusing reduces accordingly. Thus, we will limit our discussions in this paper to the MARC0 optimization only. In this work, we show that the space-charge-modified dispersion plays a key role for the adjustment of the 𝑅56 required for both the isochronous and the non-isochronous recirculation mode of an ERL. A longitudinal bunch envelope equation has been derived by Neuffer [19]. In Refs. [6,20,21] this equation has been used to predict bunch lengths with space charge and momentum compaction in synchrotrons. For simplified estimates we use this approach to predict the momentum deviation evolution along the arc, which in turn affects the transverse beam size, dispersion, and 𝑅56 . The paper is organized as follows: the basic relations for the beam centroid, rms beam size, and path length in the presence of dispersion are explained with and without space charge in Section 2. Section 3 outlines the transverse envelope-dispersion equations and transverse smooth approximation applied to the different operation modes of MESA, respectively, followed by a longitudinal analytical model to obtain an analytical expression for the final momentum deviation in terms of initial beam and machine parameters in Section 4. A beam matrix model with space charge is implemented [15,22,23], referred to as ‘‘beam matrix approach" throughout the paper, and is briefly discussed in Section 5. Results and parameter adjustments for beam matching with space charge from the MESA injection arc (MARC0) into the first RF structure Appendix A are discussed in Sections 6–7 with the presented beam matrix approach and ELEGANT particle tracking [24]. Finally, the ideas are briefly summarized in Section 8.

linear dispersion function without space charge 𝐷0 (𝑠) is the solution of the equation [6]: ′′ 1 𝐷0 (𝑠) + 𝜅𝑥 (𝑠)𝐷0 (𝑠) = , (2) 𝜌(𝑠) which gives the local sensitivity of the particle trajectory to the fractional momentum deviation 𝛿, and the prime denotes derivative with respect to distance 𝑠 along the beamline. 𝜌 is the bending radius and 𝜅𝑥 is the linearized horizontal external focusing gradient. With space charge, we can write: 𝑥sc (𝑠) = 𝑥sc,𝛽 (𝑠) + 𝐷 (𝑠) 𝛿,

where 𝐷 ≡ 𝐷 (𝑠) is the dispersion function with space charge. Taking the average of Eq. (1) and Eq. (3) and subtracting them over the symmetrical phase space distribution, such that ⟨𝑥𝛽 ⟩ = ⟨𝑥sc,𝛽 ⟩ = 0, we obtain (see also [16]): ⟨𝑥sc ⟩ − ⟨𝑥⟩ = (𝐷 − 𝐷0 )⟨𝛿⟩.

(4)

For a beam with a momentum distribution centered at the design momentum the dispersion describing the position of the beam centroid is space charge independent. This also explains the experimental results obtained in the CERN PS Booster [17]. There the dispersion was measured and matched by changing the beam momentum and recording the displacement of the beam center. To observe the effect of space charge on the individual particle dispersion, we have to compute the second moments of the beam distribution. Using the assumption that the momentum deviation is uncorrelated to the betatron oscillations, such that ⟨𝑥𝛽 𝛿⟩ = ⟨𝑥sc,𝛽 𝛿⟩ = 0, we get the expression for 𝐷 by multiplying Eq. (3) by 𝛿 and taking the average over the phase space. Following a similar procedure for 𝐷′ : 𝐷=

⟨𝑥sc 𝛿⟩ ⟨𝛿 2 ⟩

𝐷′ =

,

⟨𝑥′sc 𝛿⟩ ⟨𝛿 2 ⟩

.

(5)

The path length variation of particles with different momenta along a bend is needed to obtain the bunch length at a subsequent RF structure or IP. The deviation of the test particle trajectory from a reference particle trajectory can be described as: 𝑑𝑠1 𝜌 + 𝐷𝛿 = , 𝑑𝑠 𝜌

(6)

where 𝜌 is the bending radius, 𝑑𝑠 and 𝑑𝑠1 are the distances traveled by the reference particle and test particle, respectively. The local longitudinal bunch coordinate with respect to the reference particle is 𝑧. We use the notation that a particle ahead of the reference particle has 𝑧 > 0 and a particle behind the reference particle has 𝑧 < 0. The path length difference between the reference particle and the test particle is obtained by integrating Eq. (6) along the beamline: 𝛥𝐿 = 𝛿

∫

𝐷 d𝑠. 𝜌

(7)

The momentum compaction 𝑅56 is the variation of the path length with momentum deviation for a relativistic beam: 𝐿

𝑅56

𝛥𝐿 𝐷 = = d𝑠, ∫ 𝜌 𝛿

(8)

0

which also is the (5,6)th component of the 6 × 6 transport matrix of the beamline elements. Where 𝐿 is the total path length of the beamline. By definition, 𝑧(𝑠) = 𝑅56 (𝑠)𝛿. An additional expression for the individual particle momentum compaction is obtained by multiplying the last expression by 𝛿 and taking the average over the phase space distribution: ⟨𝑧𝛿⟩ 𝑅56 = . (9) ⟨𝛿 2 ⟩ The above result can be used to obtain 𝑅56 from the beam matrix and tracking codes (see Section 5).

2. Dispersion and momentum compaction In the presence of bending magnets, the horizontal displacement 𝑥 of a particle from the reference particle is written as 𝑥 (𝑠) = 𝑥𝛽 (𝑠) + 𝐷0 (𝑠) 𝛿,

(3)

(1)

where 𝑥𝛽 is the betatron oscillation amplitude, 𝐷0 ≡ 𝐷0 (𝑠) is the dispersion function, and 𝛿 = 𝛥𝑝∕𝑝0 is the fractional momentum deviation. The 2

A. Khan, O. Boine-Frankenheim, F. Hug et al.

Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162822 Table 1 Example current, normalized emittance and space charge parameter after the MESA injector at 5 MeV together with the beam radii and momentum compaction from Eq. (17b) and Eq. (18) for different momentum deviations.

In a beamline with dispersion D, space charge modifies the path length or time of flight of individual particles following the space charge dependent phase slip factor 𝜂: 𝜂=

𝑅56 || 𝐿 ||𝑠=𝐿

(10)

The above expressions define the modification of dispersion and path length for individual particles in a beam affected by space charge. In this work, we use a beam matrix approach as well as particle tracking to obtain the optical functions along a beamline with bends, including the self-consistent space charge force. However, for a qualitative analysis and to obtain approximate scaling laws with intensity, it is convenient to derive simplified expressions from the envelope equations with space charge.

We use the envelope-dispersion equations written in symmetric form by Okamoto and Machida [14]: ) 𝐾sc 1 𝐷𝑥 − = 0, 2𝑋(𝑋 + 𝑌 ) 𝜌 ( ) 𝜖2 𝐾sc 𝜎𝑥′′ + 𝜅𝑥 (𝑠) − 𝜎𝑥 − 𝑑𝑥 = 0, 2𝑋(𝑋 + 𝑌 ) 𝜎𝑥3 ( ) 𝜖𝑦2 𝐾sc 𝜎𝑦′′ + 𝜅𝑦 (𝑠) − 𝜎𝑥 − = 0. 2𝑌 (𝑋 + 𝑌 ) 𝜎𝑦3

(11a) (11b)

( 𝜅0 − ( 𝜅0 −

𝐾sc 2𝑎(𝑎 + 𝑏) 𝐾sc 2𝑏(𝑎 + 𝑏)

) 𝐷− )

R56 (m)

𝑎∕𝑎0 R56 (m) 𝜎𝛿 = 0.4 %

0.0 1.0 5.0 10.0

2.0 2.0 3.5 6.0

1.00 0.95 0.92 0.89

1.0 1.025 1.042 1.059

0.140 0.159 0.165 0.185

2.45 2.58 2.21 1.97

2 𝜒𝑥𝜎 𝜅0 𝐷 − 𝛿

𝐷0 1 =0 ⟹ 𝐷= , 2 𝜌 𝜒𝑥𝜎

2 𝜒𝑥𝜎 𝜎𝑥 0 − 𝛿

𝜖𝑥2 𝜅0 𝜎𝑥3

= 0 ⟹ 𝜎𝑥2 = 0

0

𝑌 = 𝜎𝑦 .

1 = 0, 𝜌

𝜎𝑥 0 −

𝜖𝑥2 𝜎𝑥3

(12) 2 𝜒𝑦𝜎 𝜎𝑦0 − 𝛿

(15b)

𝜎𝑦 0 −

𝜖𝑦2 𝜎𝑦3

= 0,

(15c)

(17a)

𝜅0 𝜎𝑦3

= 0 ⟹ 𝜎𝑦2 = 0

𝑎20 𝜒𝑥𝜎𝛿 𝑎20 𝜒𝑦𝜎𝛿

,

(17b)

,

(17c)

√ √ √ where 𝑎 = 𝜎𝑥2 + (𝐷𝜎𝛿 )2 and 𝑎0 = 𝜖𝑥 ∕ 𝜅0 . Assuming a round beam 0 (a ≈ b) in the limit of small momentum deviation and arbitrary space ( ) charge 𝐷𝜎𝛿 ∕𝜎𝑥0 ≪ 1 results in 𝜒𝑥𝜎𝛿 = 𝜒𝑦𝜎𝛿 = 𝜒. An expression for momentum compaction in the presence of space charge is obtained by substituting the obtained expression for the dispersion from Eq. (17a) into Eq. (8): 𝑅56 =

𝐷0 𝐿𝑑 𝜒 2𝜌

,

(18)

where 𝐿𝑑 is the total length of the dispersive element. The last equation is important for the ERL design, since it gives the scaling of momentum compaction with intensity. We illustrate Eqs. (17) and (18) for parameters relevant to the MESA injection arc. The first three columns in Table 1 list the main beam parameters after the injector at 5 MeV [26]. The last four columns list the obtained horizontal beam radii, normalized to 𝑎0 , and the momentum compaction for two rms momentum deviations 𝜎𝛿 = 0, 0.4 % (see Section 7 for a discussion of the MESA beam parameters). Fig. 1 shows the results for the normalized horizontal beam radius 𝑎∕𝑎0 (using Eq. (17b)) and momentum compaction Eq. (18) as a function of space charge parameter 𝜒 for four values of a dimensionless parameter 𝜉0 = 𝐷0 𝜎𝛿 ∕𝑎0 = 0 (no momentum deviation), 1.59, 2.10 or 2.76 for 0.4 % momentum deviation. Also results shown by dotes on Fig. 1 are obtained using the matrix approach (explained in Section 7) for three different cases of MESA beams (𝜎𝛿 = 0.4 %) referred to in Table 1. The results using the smooth approximation are in good agreement with the matrix approach results for the listed values of momentum deviation. In conclusion, the full solution with discrete focusing agrees with the smooth approximation for the MARC0 optics. If the longitudinal space charge force can be neglected (see next section and Section 7), the half bunch length at the end of the beamline,

(15a)

= 0,

𝜖𝑦2 0

0

)

0.140 0.140 0.138 0.134

𝛿

2 𝜖𝑑𝑥 = (𝜖𝑥2 − 𝜎𝛿2 ⟨(𝑥′ 𝐷𝑥 − 𝑥𝐷𝑥′ )2 ⟩), (14) √ where 𝜎𝛿 = ⟨ 𝛿 2 ⟩ is the rms momentum deviation, and ⟨⋅⟩ angular brackets denote phase space averages. 𝜖𝑦 is the vertical rms emittance. For simplicity, we assume that bending occurs only in the horizontal plane. A numerical analysis is required to solve the rms envelopes equations together with the dispersion function along a beamline, including space charge. In this work we will not solve the envelope equation directly, but will make use of the beam matrix (see Section 5). The envelope equations are used in the following to obtain simplified scaling laws for the beam radii and dispersion as a function of the space charge parameter. First, the system of envelope-dispersion equations is written in the smooth approximation (see also [13,16]). Assuming a matched beam ′′ ′′ 𝜎𝑥,𝑦 , 𝐷𝑥 = 0, the envelope-dispersion equations are:

𝐾sc 2𝑎(𝑎 + 𝑏)

𝑎∕𝑎0 𝜎𝛿 = 0

(11c)

𝐾sc is the space charge perveance, which measures the strength of space charge [6]: 2𝐼 𝐾sc = , (13) 𝛽 3 𝛾 3 𝐼𝐴 where 𝐼 is the beam current, 𝐼𝐴 ≈ 17.045 kA is the Alfvén current, and ( )−1∕2 are the relativistic factors. 𝛽 and 𝛾 = 1 − 𝛽 2 The generalized invariant emittance with dispersion in Eq. (11b) is (see [25])

( 𝜅0 −

𝜒

resulting in values between 0 and 1 for zero intensity and in the space charge dominated limit. Equations (15a)–(15c) are now rewritten in a more compact form by using Eq. (16):

√ Here, 𝜅𝑥,𝑦 are the linearized external focusing gradients, 𝜎𝑥 = ⟨𝑥2 ⟩, √ 𝜎𝑦 = ⟨𝑦2 ⟩ are the rms betatron amplitudes and 𝑋 and 𝑌 are the effective transverse rms beam radii including dispersion: 𝑋 2 = 𝜎𝑥2 + 𝐷𝑥2 𝜎𝛿2 ,

𝜖𝑛𝑥 (μm)

where 𝜅0 is the averaged constant focusing strength, 𝐷 = 𝐷𝑥 is the average horizontal dispersion, 𝜌 is the average bending radius, 𝜎𝑥0 ,𝑦0 are the average rms betatron amplitudes, a, b are the effective rms beam radii, and 𝜖𝑥 is the horizontal emittance which is conserved in the presence of linear dispersion. By defining horizontal and vertical space charge intensity parameters as follows: √ √ 𝐾sc 𝐾sc 𝜒𝑥𝜎𝛿 ≡ 1 − , 𝜒𝑦𝜎𝛿 ≡ 1 − , (16) 2𝑎(𝑎 + 𝑏)𝜅0 2𝑏(𝑎 + 𝑏)𝜅0

3. Transverse envelope-dispersion equations

( 𝐷𝑥′′ + 𝜅𝑥 (𝑠) −

I (mA)

0

3

A. Khan, O. Boine-Frankenheim, F. Hug et al.

Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162822

𝐾𝑙 is the longitudinal perveance [6,19]: −3𝑔𝑁𝑟𝑒 𝜂 𝐾𝑙 = ( (22) ) , 2𝛽 2 𝛾 3 ) ( where 𝑔 = 0.67 + 2 ln 𝑟𝑝 ∕𝑟𝑏 is the geometry factor (𝑟𝑝 and 𝑟𝑏 are the radii of the beam pipe and beam respectively), N is the number of particles in the bunch and 𝑟𝑒 is the classical electron radius. The beam current for a parabolic particle distribution is 𝐼 = (3𝑁𝑞)∕(4𝑧𝑚 ). For a relativistic beam 𝑧𝑚 is replaced by 𝛾𝑧𝑚 [6]. It should be noted that for a non-parabolic distribution, it is necessary to use rms values for the bunch length and the emittance. Equivalent rms half-bunch length and emittance are [20,27]: √ 1∕2 𝜎𝑧 = ⟨𝑧2𝑚 ⟩ = 𝑧𝑚 ∕ 5 , 𝜖𝑙rms = 𝜖𝑙 ∕5. (23) The longitudinal emittance 𝜖𝑙 is the 𝑧𝑧′ phase space area: 𝜖𝑙 = |𝜂| 𝑧𝑚 𝛿0 .

(24)

For an arbitrary ellipse of the longitudinal phase space (𝑧, 𝛿), where 𝛿0 = (𝛿𝑝∕𝑝0 )0 is the maximum momentum deviation at the bunch center, and 𝛿𝑚 = (𝛿𝑝∕𝑝)𝑚 is the momentum deviation at the bunch ends. ( ) By using 𝑧′ = −𝜂 𝛿𝑝∕𝑝0 , the total momentum deviation 𝜎𝛿 = 𝛿𝑝∕𝑝0 is 𝜎𝛿2 = 𝛿02 + 𝛿𝑚2 [20]: √ ( ) ( ′ )2 𝑧𝑚 𝜖𝑙 2 + . (25) 𝜎𝛿 = 𝜂𝑧𝑚 𝜂 In the absence of RF kicks 𝜅𝑧0 = 0 in a beamline, a longitudinal invariant 𝐼𝑙 is obtained by multiplying Eq. (21) by 𝑧′𝑚 and integrating with respect to 𝑠: 𝑧′𝑚 2

𝜖2 𝐾𝑙 + 𝑙 . (26) 2 𝑧𝑚 2𝑧2𝑚 The longitudinal invariant 𝐼𝑙 is now written in terms of total momentum deviation by using Eq. (25): 𝐼𝑙 =

𝐼𝑙 =

2 𝜎𝛿𝑖 +

(19)

where 𝑧𝑓 and 𝑧𝑖 are the initial and final half-bunch length, respectively.

𝛿𝑖

In an ERL beamline one can expect longitudinal–transverse coupling due to the space charge force and dispersion. In our numerical beam matrix and particle tracking approaches coupling effects are taken into account. A simplified approach within a longitudinal envelope model is complicated by the variation of the local momentum compaction 𝜂(𝑠). However, if we use the constant 𝜂 obtained from the smooth focusing approximation, discussed in the previous section, we can employ the well established longitudinal envelope equation (see for example [19]) under an assumption that bunch length is longer than the beam radius with the slowly varying momentum compaction 𝐷0

2𝐾𝑙 𝜂 2 𝑧𝑚,𝑖

2 = 𝜎𝛿𝑓 +

2𝐾𝑙 𝜂 2 𝑧𝑚,𝑓

,

(28)

𝛿𝑖

If we neglect the 𝑧𝑚,𝑖 ∕𝑧𝑚,𝑓 on the right-hand side in Eq. (29) such that 𝑧𝑚,𝑓 ≫ 𝑧𝑚,𝑖 because space charge stretches the bunch length. The final expression for the total momentum deviation in terms of longitudinal space charge perveance, initial bunch length, and phase slip factor is obtained by rearranging Eq. (29): √ 2𝐾𝑙 2 + 𝜎𝛿𝑓 = 𝜎𝛿𝑖 . (30) 𝜂 2 𝑧𝑚,𝑖

4. Longitudinal envelope model

𝜒 2𝜌

(27)

with 𝜎𝛿𝑖 as the initial total momentum deviation at the start of the beamline and 𝜎𝛿𝑓 as the final total momentum deviation at the end of the beamline. Rearranging Eq. (28), we find: ) ( 2 𝜎𝛿𝑓 𝑧𝑚,𝑖 2𝐾𝑙 (29) =1+ 1− 𝑧𝑚,𝑓 𝜎2 𝜂 2 𝑧𝑚,𝑖 𝜎 2

including the transverse space charge, can be obtained using Eq. (18) as

𝜂=

1 2 2 𝐾𝑙 𝜂 𝜎𝛿 + , 2 𝑧𝑚

if 𝑧𝑚,𝑖 and 𝑧𝑚,𝑓 are the bunch lengths at the start and at the end of the beamline, respectively. Then the conservation of 𝐼𝑙 results in:

Fig. 1. (a) Horizontal beam radius, and (b) momentum compaction as a function of the space charge intensity parameter 𝜒 for four values of 𝜉0 = 𝐷0 𝜎𝛿 ∕𝑎0 . 𝜎𝛿 = 0 for first case (yellow solid line) and 𝜎𝛿 = 0.4 % for 𝜉0 = 1.59 (green dotted line), 𝜉0 = 2.10 (red dash-dotted line), and 𝜉0 = 2.79 (blue dashed line) in the presence of space charge.

𝑧𝑓 = 𝑧𝑖 + 𝑅56 𝜎𝛿

+

In Section 6 Eq. (30) is compared to results obtained with beam matrix tracking (see solid blue line in Fig. 4). 5. Beam matrix tracking

(20)

Equation (8) shows that the path length varies with 𝐷𝑥 along the beamline. The variation of current with the bunch length modifies the transverse space charge perveance Eq. (13), and the longitudinal space charge perveance Eq. (22) depends weakly on the beam radius 𝑟𝑏 . The transverse and longitudinal envelopes are therefore coupled. In the previous sections we derived simplified expressions, without

The longitudinal envelope equation for half-bunch length 𝑧𝑚 (𝑠) is then [19]: 𝜖2 𝐾 ′′ 𝑧𝑚 − 𝑙 − 𝑙 = 0. (21) 2 𝑧𝑚 𝑧3𝑚 4

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Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162822

Fig. 2. Layout of MESA.

Fig. 3. (Color Online) Horizontal beam envelopes (a), dispersion (b), and momentum compaction (c) along the MARC0 beamline obtained from the matrix approach (blue dashed line) and ELEGANT (red dash-dotted line) for ‘‘zero" and 10 mA current for 𝜎𝛿 = 10−4 . Here, the blue rectangles illustrate dipole bends, and red rectangles are focusing or defocusing quadrupoles.

5

A. Khan, O. Boine-Frankenheim, F. Hug et al.

Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162822 Table 2 Beam parameters for the injection arc of MESA.

coupling. The beam envelopes and lattice functions including coupling and for the detailed lattice structure we obtain numerically from a tracking scheme for the beam matrix, including linear space charge kicks; details can be found in Appendix B. The beam matrix tracking uses the transport matrices obtained from MAD-X [28] together with space charge kicks as explained in Appendix B. Results are compared to particle tracking using ELEGANT [24] as explained in the next section. The matching constraints of an example ERL beamline with dispersion, first order momentum compaction and space charge are outlined. Matching is required for the different isochronous/non-isochronous recirculating modes of ERLs. The aim of beam matching is to fulfill given constraints for the lattice functions at a subsequent RF structure or an internal target. The Twiss parameters can be expressed in terms of second moments and beam matrix elements: 𝛽𝑥 =

⟨𝑥2𝛽 ⟩ 𝜖𝑥

=

2 𝜎11 − 𝜎16

𝜖𝑥

,

𝛼𝑥 = −

⟨𝑥𝛽 𝑥′𝛽 ⟩ 𝜖𝑥

=−

𝜎12 − 𝜎16 𝜎26 , 𝜖𝑥

Parameters [unit]

Symbol

Value

Kinetic energy [MeV] Bunch charge [pC] RF frequency [GHz] Initial beta functions [m] Initial alpha functions Initial rms momentum deviation [/%] Initial half bunch length [ps] Final beta functions [m] Final alpha functions Normalized emittance [𝜋 mm mrad] Momentum compaction [m]

E𝑘 𝑁𝑞 𝑓𝑟𝑓 𝛽𝑥0 , 𝛽𝑦0 𝛼𝑥0 , 𝛼𝑦0 𝜎𝛿 𝑧𝑚 𝛽𝑥 , 𝛽𝑦 𝛼𝑥 , 𝛼𝑦 𝜖𝑛𝑥,𝑛𝑦 R56

5 0.77/7.7 1.3 1.30, 0.90 −0.17, −0.57 1 × 10−4 / 4 × 10−3 4.2 1.57, 1.57 0.12, 0.12 2/6 0.14

(31)

where 𝜖𝑥 is the horizontal rms emittance. Similar expressions can be written along the vertical plane for 𝛽𝑦 and 𝛼𝑦 . The dispersion and its derivative are written in terms of beam matrix elements by comparing Eq. (B.1) with Eq. (5) [15]: 𝜎 𝜎 𝐷 = 16 , 𝐷′ = 26 . (32) 𝜎66 𝜎66 For an ERL beamline, the optimization parameters are usually the quadrupole strengths. Optimization of a non-periodic lattice with multiple constraints leads to nonlinear and implicit objective functions for which numerical solutions are required. In the case of our example MARC0 arc, the objective function depends on the beam matrix elements corresponding to the Twiss parameters, and dispersion at the end of the arc, Eq. (31) and Eq. (32). With a search algorithm (‘‘random walk"), a first trial solution is computed, and then the solution is subsequently improved based on the corresponding objective function value until convergence to get the matched solution.

Fig. 4. The final total momentum deviation at the end of the MARC0 as a function of the beam current. A reasonable agreement between matrix approach (red circles), ELEGANT (green triangles) and Eq. (30) approximation model (blue solid line) can be observed.

6. Comparison of analytical models, beam matrix and particle tracking At high beam current (I = 10 mA), there is a significant increase in momentum deviation by a factor of 5 at the end of the arc. The spacecharge induced changes in the momentum deviation also affect the dispersion computed from second moments of beam distribution (see Eq. (32)). Therefore, it is important to consider the coupled transverse– longitudinal simulations to match the beam-based dispersion at high beam currents. Fig. 4 also shows reasonable agreement for the evolution of the momentum deviation between our simplified model Eq. (30) and beam matrix tracking Eq. (32). The small deviation between the simplified model and beam matrix tracking can be explained by the geometry factor in 𝐾𝑙 . In our matrix tracking model, the beam radius is 𝑟𝑏 ≡ 1.7(𝜎𝑥 + 𝜎𝑦 )∕2, and in the simplified model an average value computed by smooth approximation Eq. (17) is used for 𝑎. Fig. 5 (a)–(b) show the variation of horizontal and vertical transverse envelopes along 𝑠, for 𝐼 = 1, and 10 mA in MARC0. It can be seen from Fig. 5 (c) that with longitudinal and transverse space charge the dispersion is significantly modified for non-zero currents. The sign reversal of 𝐷𝑥 may contribute to phase mismatch at the entrance of the RF structure due to changes in the time of flight of particles (see Eq. (8)). Thus, it is important to consider both longitudinal and transverse space charges for optimal matching.

In this section, we present the solutions for the beam envelopes, dispersion and momentum compaction with space charge. We compare beam matrix tracking results with particle tracking in ELEGANT. The implementation of transverse space charge in ELEGANT is briefly described in Ref. [16,29]. The MESA injection arc (MARC0) is used as an example case. An overview of the MESA facilities is shown in Fig. 2, for details see Appendix A. MARC0 is a 5 MeV, 180◦ , first-order double bend achromatic with flexible 1st order momentum compaction 𝑅56 , which is required for non-isochronous recirculating schemes and also to support the two different operation modes of MESA. By accelerating off-crest in the injector linac and adjusting 𝑅56 in the injector arc MARC0 is designed as a bunch compressor. Estimation of space charge effects is done for a typical set of beam parameters listed in Table 2. Note that we are using an idealized lattice for the MARC0 ignoring magnet misalignments and multipole errors for the simulations. Additional studies are required to include these effects. First, we compute the horizontal beam envelope, dispersion, and momentum compaction profile along the MARC0 arc for ‘‘zero" and 10 mA current and an rms momentum deviation of 10−4 , accounting for transverse space charge only. Fig. 3 shows a good agreement between the beam matrix tracking results and ELEGANT particle tracking. As expected from Eq. (8), the variation in the dispersion due to space charge leads to a variation in the momentum compaction in Fig. 3(b)–(c). Second, simulations are performed with longitudinal space charge to show the variation of the momentum deviation with current as shown in Fig. 4. The momentum deviation is computed from the beam √ matrix as 𝜎𝛿 = 𝜎66 . It can be seen that longitudinal space charge does not have much impact on the momentum deviation below I = 1 mA.

7. Lattice optimization and discussions In this work, we optimize the lattice parameters of MARC0 to get fixed values of 𝛽𝑥,𝑦 , 𝛼𝑥,𝑦 , 𝐷𝑥 , 𝐷𝑥′ and 𝑅56 at the end. While the transverse dispersion functions 𝐷𝑥 and 𝐷𝑥′ need to be zero at the end of the arc, the longitudinal dispersion is fixed at a finite value of 𝑅56 = 0.14 m in 6

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Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162822

Fig. 5. Evolution of (a) horizontal beam envelope, (b) vertical beam envelope, and (c) dispersion along the MARC0 beamline for I = 0 mA (blue solid line), 1 mA (red dotted line), 10 mA (black dash-dotted line) including transverse and longitudinal space charge. Table 3 Example current, the horizontal and vertical beta functions mismatch respectively, and the horizontal dispersion mismatch from the design value at the end of MARC0 with space charge.

order to use the arc as part of a bunch compressor to achieve a short bunch length at the start of the first superconducting RF structure. A multi-turn ERL with a split linac like MESA, explicitly demands that both the accelerating and the decelerating beams share the individual return arcs. This condition imposes symmetry about the RF structures. The return arcs accept two beams of the same energy – one will be accelerated and the other one will be decelerated – which requires a lattice with its symmetry axes in the middle of the RF structure. Additionally, the beta functions at the position of the RF structure need to be as small as possible to achieve maximum beam current. The main reason for this second requirement on the beam optics is to control the transverse beam break-up. Beam break-up is related to the excitation of unwanted dipole higher-order modes, which can be diminished by minimizing the beta functions and thus the beam size inside the RF structure [30]. For ER operation, the nominal design of the arc should deliver fixed beam parameters with zero transverse dispersion at the start of the RF structure. As can be seen in Fig. 2, only about 3.8 m of space is available for the 180◦ arc. This limits the number of knobs to adjust dispersion and Twiss parameters. As can be seen from Fig. 5, MARC0 consists of two double-bend achromats (DBA) [31]. It controls both transverse beam confinement

I (mA)

𝛥𝛽𝑥 𝛽𝑥

1 5 10 10 (matched)

3.0 17.5 39.7 0

(%)

𝛥𝛽𝑦 𝛽𝑦

(%)

2.6 17.3 41.0 0

𝛥𝐷𝑥 (m) 0.037 −0.017 −0.046 0

and longitudinal phase space to compress the bunch. A set of four quadrupoles at the start of the arc matches the first DBA to the MAMBO injector. The central part of the arc between the two DBAs contains three quadrupoles to again match the Twiss parameters. A total of 15 quadrupole gradients are available knobs to optimize the lattice. Table 3 illustrates the horizontal beta function mismatch 𝛥𝛽𝑥 ∕𝛽𝑥 and the vertical beta function 𝛥𝛽𝑦 ∕𝛽𝑦 mismatch from the design value respectively, and the horizontal dispersion mismatch 𝛥𝐷𝑥 from the design value at the end of MARC0 with space charge for 1, 5 and 10 mA. Here, 𝛥𝛽𝑥 = 𝛽sc,𝑥 − 𝛽𝑥 , 𝛥𝛽𝑦 = 𝛽sc,𝑦 − 𝛽𝑦 , and 𝛥𝐷𝑥 = 𝐷sc,𝑥 − 𝐷𝑥 . We can see 7

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Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162822

dispersion, and momentum compaction. We focused on the example of the MARC0 arc, but the same scheme applies to the MESA recirculation arcs and also to other ERLs for a qualitative analysis. 8. Conclusion The correct matching with space charge is important for the injection arc into the RF structure of an ERL. This was shown using beam matrix tracking and analytic expression within the smooth approximation. We show how space charge modifies the dispersion function along the arc and so the momentum compaction which leads to change in the path length of electrons. Momentum compaction needs to be controlled in an ERL to control the bunch length and to avoid the phase mismatch at the RF structure because of changes in path length. A ‘‘random walk" scheme was employed to get the matched parameters with space charge at the subsequent RF structure of an arc. Further, a simple expression for the momentum deviation with space charge was obtained, under the assumption of slowly varying momentum compaction along an arc . We have applied our models to the 5 MeV low energy injection arc (MARC0) of MESA, which also compress the bunch at the start of the subsequent RF structure, and compared the results of beam matrix, smooth approximation and ELEGANT particle tracking. In subsequent recirculation arcs of MESA, the beam energy is much higher than the injection energy. Thus, space charge effects are weaker. The future application of this work is to predict the longitudinal space-chargeinduced microbunching instability in MESA, which depends on the details of the dispersion function along the arcs [5].

Fig. 6. Momentum compaction with space charge along the MARC0 beamline for a bunch with momentum chirp -7% and momentum deviation of 0.4% at 1 mA and 10 mA.

from the last column of Table 3 that current dependent dispersion is non-zero at the end of MARC0 which might lead to emittance growth in the RF structure because of longitudinal–transverse phase space coupling. A simple ‘‘random walk" routine is used to optimize all lattice parameters to get the matched solution of MARC0 with space charge. Note that 𝐷′ should be considered for optimization of 𝐷 in the beamline with space charge. 𝐷′ must be zero at the central quadrupole of the achromat to achieve zero dispersion at the end of the last dipole. A new set of quadrupole strengths is obtained with corrections of up to 15% in the original quadrupole to get the matched solution. In the second case, the longitudinal phase space is shaped within the MAMBO injector to achieve a linear correlation between the phase and energy of the particles. As discussed in Appendix A, the MESA ER mode is planned to be used for electron scattering experiments at an internal gas target for high-precision experiments. The final momentum deviation of less than 2 × 10−4 can be achieved in either isochronous or non-isochronous operation when the bunch length at injection into the first RF structure is small [2,18,30]. Particle tracking simulations in Ref. [26] show that for a given longitudinal emittance in the presence of space charge, a beam optimized for a shorter bunch length with an increased momentum deviation at the start of the first RF structure yields a better momentum deviation at the internal target experiment energy (105 MeV) than vice versa. Therefore, the target rms values for longitudinal properties of the 5 MeV beam behind MARC0 can be defined to be less than 5 × 10−3 in momentum deviation [18,26]. Thus, we use the bunch distribution with a momentum chirp of −7 % and a momentum deviation of 0.4 % in ELEGANT at 1 mA, and 10 mA at the start of MARC0. Fig. 6 shows the momentum compaction has the designed value of around 0.14 m to get short bunch at the start of the first RF structure with space charge along the MARC0 beamline for a bunch with momentum chirp -7% and momentum deviation of 0.4% at 1 mA and 10 mA. It can be concluded that by accelerating off-crest in the injector linac and adjusting 𝑅56 in the injection arc (MARC0) successfully works as a bunch compressor in the presence of linear space charge forces even at the highest beam current of 10 mA as can be seen in Fig. 7(b)–(c). The nonlinearities outside the rms beam ellipse from matrix tracking (red ellipses) are expected from the fact that the red ellipses are obtained from the matrix approach with a linear space charge, which is assumed to be consistent for all beam particles. Our relatively simple beam matrix tracking scheme, including space charge kicks, provides a valuable understanding of the beam envelopes,

Acknowledgments A. Khan would like to thank Kurt Aulenbacher (JGU Mainz) and Uwe Niedermayer for valuable discussions and feedback. We are grateful for the financial support provided by DFG, Germany through GRK 2128 Accelence project D-3 to study the effect of space charge and wake fields in MESA. Appendix A. MESA An overview of the MESA facilities is shown in Fig. 2. The electron source provides up to 1 mA of a polarized beam at 100 keV. In the next planned stage of MESA, the electron source will provide 10 mA of unpolarized beam current. This electron source is followed by a spin manipulation system containing two Wien filters. A chopper and two buncher cavities prepare the bunches for the normal-conducting milliampere booster (MAMBO), which accelerates them to 5 MeV. All radio-frequency (RF) devices of MESA operate at 1.3 GHz or higher harmonics. In MESA electrons will be accelerated by two ELBE-type cryomodules each housing two superconducting TESLA-type cavities with an accelerating gradient of 12.5 MeV∕m, which results in 25 MeV per pass. There are four spreader sections for separating and recombining the beam and two chicanes for injection and extraction of the 5 MeV beam. MESA is planned to operate in two modes: external beam (EB) mode and energy-recovery (ER) mode. In EB mode, the electron beam will gain 155 MeV with up to 150 μA beam current by circulating thrice around the accelerator. The beam is planned to use for high-precision fixed-target experiments. The main experiment will be the measurement of the electroweak mixing angle at the P2 setup [32]. The second operation mode is a twice-recirculating ER mode at 105 MeV with 1(10) mA beam current for pseudo-internal target (PIT) experiments in a search for dark photons with high luminosity [33]. In ER mode, 100 MeV of the beam energy can be recovered by decelerating the beam in the superconducting cavities and using this recovered energy to accelerate further bunches [2]. Both operational modes set high demands on beam quality and stability. Mainly, a small momentum deviation is required to achieve higher precision of experiments. One main issue for sustaining beam quality is a proper beam transport through the injection arc (MARC0), which is discussed in this article. 8

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Nuclear Inst. and Methods in Physics Research, A 948 (2019) 162822

Fig. 7. (Color online) Longitudinal phase spaces at (a) the start of MARC0, (b) the end of MARC0 for 1 mA of current, and (c) at the end of MARC0 for 10 mA. The red ellipses indicate the rms beam ellipse from matrix tracking.

Appendix B. Beam matrix model

[8] J. Barnard, F. Deadrick, A. Friedman, D. Grote, L. Griffith, H. Kirbie, V. Neil, M. Newton, A. Paul, W. Sharp, et al., Phys. Fluids B 5 (1993) 2698. [9] M. Venturini, M. Reiser, Phys. Rev. E 57 (1998) 4725. [10] S.-Y. Lee, H. Okamoto, Phys. Rev. Lett. 80 (1998) 5133. [11] J.-G. Hwang, E.-S. Kim, T. Miyajima, Nucl. Instrum. Methods Phys. Res. A 684 (2012) 18. [12] B. Muratori, C. Gerth, N. Vinokurov, Proceedings of EPAC04, 2004. [13] M. Ikegami, S. Machida, T. Uesugi, Phys. Rev. Spec. Top. Accel. Beams 2 (1999) 124201. [14] H. Okamoto, S. Machida, Nucl. Instrum. Methods Phys. Res. A 482 (2002) 65. [15] T. Ohkawa, M. Ikegami, Nucl. Instrum. Methods Phys. Res. A 576 (2007) 274. [16] S. Bernal, B. Beaudoin, T. Koeth, P.G. O’Shea, Phys. Rev. Spec. Top. Accel. Beams 14 (2011) 1. [17] K. Hanke, J. Sanchez-Conejo, R. Scrivens, Proceedings of the 2005 Particle Accelerator Conference, 2005, http://dx.doi.org/10.1109/PAC.2005.1591442. [18] F. Hug, Proceedings of the 8th International Particle Accelerator Conference, 2017, p. 873. [19] D. Neuffer, IEEE Trans. Nucl. Sci. 26 (1979) 3031. [20] G. Franchetti, I. Hofmann, G. Rumolo, Phys. Rev. Spec. Top. Accel. Beams 3 (2000) 084201. [21] S. Appel, O. Boine-Frankenheim, Phys. Rev. Spec. Top. Accel. Beams 15 (2012) 054201. [22] K.R. Crandall, Trace 3-d documentation, Tech. Rep., Los Alamos National Lab., 1987. [23] C.K. Allen, N. Pattengale, Los Alamos National Laboratory Internal Report LA-UR-02-4979, 2002. [24] M. Borland, User’s Manual for ELEGANT, 2006. [25] M. Venturini, R.A. Kishek, M. Reiser, Particle Accelerator Conference, 1999. Proceedings of the 1999, vol. 5, IEEE, 1999, pp. 3274–3276. [26] F. Hug, R. Heine, J. Phys. Conf. Ser. 874 (2017) 012012. [27] F.J. Sacherer, IEEE Trans. Nucl. Sci. 18 (1971) 1105. [28] L. Deniau, H. Grote, G. Roy, F. Schmidt, The MAD-X Program (Methodical Accelerator Design) User’s Reference Manual, 2018. [29] A. Xiaot, M. Borland, L. Emery, Y. Wang, K.Y. Ng, 2007 IEEE Particle Accelerator Conference, PAC, IEEE, 2007, pp. 3456–3458. [30] D. Simon, K. Aulenbacher, R. Heine, F. Schlander, Proceedings of the 6th International Particle Accelerator Conference, 2015, p. 220. [31] H. Wiedemann, Particle Accelerator Physics, Springer, 2015, p. 468. [32] D. Becker, R. Bucoveanu, C. Grzesik, et al., Eur. Phys. J. A 54 (2018) 208. [33] A. Denig, AIP Conf. Proc. 1735 (2016) 020006.

Instead of individual particle tracking, we track the second moments defined as [15,22,27]: 𝜎𝑖𝑗 = ⟨𝑢𝑖 𝑢𝑗 ⟩

(B.1)

where 𝑢𝑘 stands for the distance along the 𝑘th coordinate axis. Specifically, ⎛𝑢1 ⎞ ⎛ 𝑥 ⎞ ⎜𝑢2 ⎟ ⎜𝑥′ ⎟ ⎜ ⎟ ⎜ ⎟ 𝑢 𝑦 𝒖 ≡ ⎜ 3⎟ = ⎜ ′ ⎟ (B.2) ⎜𝑢4 ⎟ ⎜𝑦 ⎟ ⎜𝑢 ⎟ ⎜ 𝑧 ⎟ ⎜ 5⎟ ⎜ ⎟ ⎝𝑢6 ⎠ ⎝ 𝛿 ⎠ 𝑥, 𝑦 and 𝑧 are the horizontal, vertical, and longitudinal coordinates respectively, and prime designates the derivative with respect to 𝑠. Here, 𝜎 represents the 6 × 6 matrix, with elements 𝜎𝑖𝑗 . The averages are taken over the phase space variables, and the subscripts i, j run from 1 to 6 representing 𝑥, 𝑥′ , 𝑦, 𝑦′ , 𝑧, 𝑧′ . The time evolution of a beam matrix 𝜎𝑠 from 𝑠0 to 𝑠1 along the longitudinal position 𝑠 is given by 𝜎𝑠1 = 𝑅(𝑠0 → 𝑠1 ) 𝜎𝑠0 𝑅𝑇 (𝑠0 → 𝑠1 ) [15,22,23], where 𝑅 is the transport matrix. The ( ) space charge kick is implemented as 𝑅 𝑠0 , 𝑠0 + 𝛥𝑠 = 𝑅𝛥𝑠 𝑅sc , where 𝑅𝛥𝑠 is a drift of length 𝛥𝑠 and 𝑅sc is the space charge kick as described in Ref. [15,22,23]. References [1] L. Merminga, D.R. Douglas, G.A. Krafft, Annu. Rev. Nucl. Part. Sci. 53 (2003) 387. [2] K. Aulenbacher, Hyperfine Interact. 200 (2011) 3. [3] S.M. Gruner, D.H. Bilderback, Nucl. Instrum. Methods Phys. Res. A 500 (2003) 25. [4] A. Jankowiak, M. Abo-Bakr, A. Matveenko, CERN Yellow Reports: School Proceedings, vol. 1, 2018, p. 439. [5] A. Khan, O. Boine-Frankenheim, C. Stoll, J. Phys. Conf. Ser. (2018). [6] M. Reiser, Theory and Design of Charged Particle Beams, John Wiley & Sons, 2008. [7] T.P. Wangler, RF Linear Accelerators, John Wiley & Sons, 2008.

9