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Emittance dilution due to the betatron mismatch in high-intensity hadron accelerators
ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 595 (2008) 561–567
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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Emittance dilution due to the betatron mismatch in high-intensity hadron accelerators J.Y. Tang Institute of High Energy Physics, CAS, Yuquan Road 19B, Beijing 100049, China
a r t i c l e in fo
abstract
Article history: Received 23 June 2008 Received in revised form 6 August 2008 Accepted 6 August 2008 Available online 20 August 2008
The importance of the betatron mismatch between two sections of an accelerator complex was recognized a long time ago. The mismatch together with the filamentation effect will result in a larger diluted beam emittance. However, the influence of the mismatch was underestimated largely in high-intensity hadron synchrotrons in the literature, as the rms emittance is used to represent the dilution. The dilution on the total emittance is significantly larger, which is of more importance in representing the beam loss in proton synchrotrons. This article points out the necessity of using the dilution factor of the total emittance together with the rms emittance in high-intensity hadron accelerators. A proof is given to show that the dilution factor of the rms emittance is the same for any initial beam distribution. The detailed beam distribution after the dilution is discussed for the initial uniform, Gaussian and parabolic distributions, and the numerical simulation results are presented. Examples and discussions about the betatron mismatch in the cases of injection, FODO channel transporting mixed H and proton beams, fast-changing periodic focusing systems and proton linacs of a large tune depression are also presented. & 2008 Elsevier B.V. All rights reserved.
Keywords: Betatron mismatch Filamentation Emittance dilution Beam distribution Beam loss
1. Introduction It is known that the betatron mismatch between the injected beam and the periodic focusing system will result in an emittance dilution. For the case with no nonlinear components in the periodic system, there will be only betatron modulation and the beam emittance will rotate within the boundary of a larger ellipse; in case of the presence of nonlinear components, the filamentation effect in the phase space will lead to a real emittance dilution although Liouville theorem still holds. The emittance increase due to the mismatch is very important in many accelerators where the beam loss or the collision luminosity is concerned, and the betatron modulation has the similar impact. The study on the emittance dilution due to the betatron mismatch can be traced early in 1960s [1]. The following expression for the increase in emittance that is widely used [2–11] ! 2 diluted 1 b1 b1 b2 b2 ¼ þ a1 a þ (1) 2 b2 0 b2 2 b1 b1 where suffixes 1 and 2 are for the acceptance and the injected beam, a and b are the Courant–Snyder parameters. This is true for the rms emittance whatever the beam distribution is (see Section 3.1). However, when the total diluted Tel./fax: +86 10 8823 5908.
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emittance is concerned, the dilution factor is significantly larger. In high-intensity proton synchrotrons where beam losses are of very important concern, the total emittance or the emittance containing 99% particles are often used. In proton linacs, where the beam focusing is space-charge dominant, the betatron mismatch becomes an important factor for the halo formation. In proton synchrotrons with multi-turn injection using H stripping, the phase space is filled by painting method, thus the injection betatron mismatch is less important by affecting only the painted particle distribution. However, in proton synchrotrons with single-turn injection from a booster, the emittance dilution due to the betatron mismatch is a very important factor leading to large beam losses. In this paper, the dilution in the means of the total emittance instead of the rms emittance is stressed for high-intensity hadron accelerators, and the diluted beam distributions for different initial distribution types are studied.
2. Why total emittance in high-intensity hadron accelerators? The betatron mismatch happens in many high-intensity hadron accelerators, such as the proton linacs with spacecharge-dominated focusing, the single-turn injection from a proton booster to a main ring, the synchrotrons or accumulator rings with a large tune shift due to the space-charge force and without focusing compensation. In the case of the multi-turn injection using an H stripping method in a proton synchrotron,
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the emittance dilution due to a betatron mismatch at the injection is less important as it affects mainly the painted particle distribution and has little effect on the total emittance. The rms emittance or sometimes its integer multiple, is widely used in electron accelerators or low-intensity hadron accelerators, as it describes well the property of the beam as a whole. Some halo particles having large Courant–Snyder invariants can be ignored there. However, in high-intensity proton accelerators where the beam loss level is of great concern, the halo particles are no longer to be ignored. In this case, the total emittance or sometimes the emittance including 99% particles is better used along with the rms emittance. The dilution of the total emittance due to a betatron mismatch is depicted by three ellipses in Fig. 1, where E1, E2 and E3 denote the matched beam, the mismatched beam and the diluted emittance, respectively. The betatron modulation along the longitudinal direction created by the mismatch will result in a real larger beam emittance after some periods due to the filamentation effect (see Fig. 2), namely the beam fills the whole E3 ellipse. In Appendix A, the derivations to obtain the total emittance dilution factor are given. There the dilution factor of the total emittance is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ¼ 3 ¼ x þ x2 1 (2)
2
y’
1.5 1 0.5 0 -15
-10
-5.0
0 -0.5
5
10
15 y
-1 -1.5 -2 Fig. 2. Filamentation effect in the early stage in the presence of nonlinear forces. Dashed line outlines the filamented part of the distribution; the outer and inner ellipses are E3 and E4 in Fig. 1.
2
where
x¼
1 b1 b2 b1 þ þ 2 b2 b1 b2
2 ! b a2 2 a1 .
(3)
b1
When the total emittance is used, the emittance dilution due to a betatron mismatch becomes much more significant, quite differently from the case that the rms emittance is used. Especially in the case with a small mismatch factor (x151), the emittance dilution for the total emittance is significant whereas the one for the rms emittance is negligible. We can also express the emittance dilution by the relative change in the emittance, qffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 2 ¼ x 1 þ x 1. (4)
If the mismatch is small, x ¼ 1+D, thus
D
Dþ
pffiffiffiffiffiffiffi 2D.
(5)
As will be seen in Section 3.2, the beam tail with a large action after the dilution is significant in the cases of initial uniform and parabolic distributions, but to a much less degree in the case of an
1.5
E2
E1
0.5
-10
-5 E4
5
3. Diluted beam distribution Not like in an electron synchrotron where the beam distribution is usually a Gaussian like, the beam distribution in hadron synchrotrons is more difficult to predict. Here three different types of distributions: Gaussian, uniform and parabolic are used to show the dilution effect in case of a betatron mismatch. Because the filamentation effect will dilute the particle distribution when a betatron mismatch happens, the beam distribution becomes sparse at the outer part of the diluted emittance [1]. For an initial uniform distribution, the inner part (E4) of the emittance as shown in Fig. 1 will remain uniform. The emittance of E4 is defined by Eq. (A.15) with the negative sign, namely qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z0 ¼ 4 ¼ x x2 1. (6)
2
If the initial distribution is Gaussian or parabolic, the inner part of the beam will also be diluted. In Ref. [4], it was shown that for an initially Gaussian distribution the rms emittance of the diluted beam follows Eq. (1). However, we do not know what the situation is for any kind of distribution. In the following section, the diluted rms emittance in the case of a betatron mismatch will be proven still following Eq. (1) whatever the distribution is.
y’
1.0
initial Gaussian distribution. However, it is usually not easy to predict the beam distribution in hadron accelerators as in electron accelerators.
10
y
3.1. RMS emittance of the diluted beam
-0.5 -1.0 E3 -1.5
Fig. 1. Emittance dilution due to the betatron mismatch. E1: ellipse for the matched beam; E2: ellipse for the mismatched beam; E3: ellipse for the diluted emittance; E4: ellipse for the non-diluted part when the distribution is uniform.
It is often convenient to use the action–angle variables; here the definition of the action variable (also called the Courant–Snyder invariant) has a factor of two in difference from the usual one. If the initial particle distribution is not uniform in the angle, the filamentation effect will smooth out the difference. Therefore, it is convenient to assume that the initial particle distribution for the action–angle variables is uniform in the angle variable and then use various types of distributions for the action variable.
ARTICLE IN PRESS J.Y. Tang / Nuclear Instruments and Methods in Physics Research A 595 (2008) 561–567
Assume the initial distribution g(x,x0 ), the action function is 2
0
0
02
Iðx; x Þ ¼ gx þ 2axx þ bx .
(7)
Here the derivations are given to show the rms emittance of a distribution is related to the mean value of the action. Taking the average to Eq. (7) ZZ I¯ ¼ gðx; x0 ÞIðx; x0 Þ dx dx0 ZZ ¼ gðx; x0 Þðgx2 þ 2axx0 þ bx0 2 Þ dx dx0 ¼ gx2 þ 2axx0 þ bx0 2 .
(8)
Using the statistical definitions of the rms emittance 8 2 > > < x ¼ brms x0 2 ¼ grms > > : xx0 ¼ a
563
Thus, by substituting the averaged action in Eq. (10) the new rms emittance after the dilution is ZZ 1 1 gðrÞr 2 0 rms ¼ cos2 y þ Z sin2 y r dr dy. (13) 2 Z 2bð1 þ 2a Þ With the integration in separate variables, the change in the rms emittance before and after the emittance dilution can be expressed as 0 rms 1 1 ¼ þ Z ¼ x. (14) rms 2 Z This is just the same as Eq. (1). 3.2. Distribution of the diluted beam
(9)
rms
Three different initial distributions: uniform, Gaussian and parabolic, which are often used in hadron accelerators, are considered here to show the dilution effect.
and bg–a2 ¼ 1, it is straightforward to obtain I¯ ¼ 2ð1 þ 2a2 Þrms .
(10)
Eq. (10) shows an important intrinsic relationship between the averaged action and the rms emittance. In order to better describe the betatron mismatch, the transformed coordinates (x,z) instead of (x,x0 ) will be used as in the literature, where z ¼ ax+bx0 is used. In the new coordinates, the distribution properties remain but the distribution now is over a round area. The initial distribution becomes axisymmetric and will be deformed into an ellipse with a mismatch, but will become axisymmetric again due to the filamentation effect. With the initial distribution, one has 2
x2 ¼ z ¼ 12r 2 .
(11)
With the coordinate transformation, there is I(x,x0 ) ) (1/b)r2. Following the depiction in Fig. 3, for the particles in the area element r dr dy, the change in the contribution to the action integral is 1 gðrÞr 2 r dr dy ) gðrÞr 2 2 cos2 y þ a2 sin2 y r dr dy (12) a where a is the elongation factor at the mismatch and a ¼ OZ in Eq. (2). This is to say that the action variable being usually an invariant will change during a mismatch.
r/a r
Matched
x
ar Mismatched
Fig. 3. Emittance mismatch in the normalized coordinates. Circle for the matched emittance and ellipse for the mismatched one.
(1) Uniform distribution (KV distribution) gðx; x0 Þ ¼
b 4ps2
;
x 2 ð2s; 2sÞ
(15)
or gðIÞ ¼
1 2s2 ; I 2 ð0; 2I0 Þ; I0 ¼ 2I0 b
(16)
where s is the rms beam size (the same thereafter). (2) Gaussian distribution (truncated) gðx; x0 Þ ¼
b 2ps2
2
eðx
þðaxþbx0 Þ2 Þ=ð2s2 Þ
;
x 2 ð3s; 3sÞ
(17)
or gðIÞ ¼
1 I=I0 2s2 e ; I 2 ð0; 4:5I0 Þ; I0 ¼ . I0 b
(18)
(3) Parabolic distribution gðx; x0 Þ ¼
b 2ps2
1
! x2 þ ðax þ bx0 Þ2 ; 4s2
x 2 ð2s; 2sÞ
(19)
or gðIÞ ¼
1 I 2s2 1 . ; I 2 ð0; 2I0 Þ; I0 ¼ I0 2I0 b
(20)
Fig. 4 shows the diluted distributions according to different mismatch degrees, which are obtained using multi-particles simulations (a self-made code). The validity of the code has been checked with the theoretical predictions on the dilution factors of the total emittance and the rms emittance. The filamentation process is simulated through the tune spread induced by a sextupole component in a periodic system (see Fig. 2). To simulate the redistribution of particles in the phase plane during a betatron mismatch, the initially populated particles of a given distribution will change their action parameter by a factor of ((1/Z)cos2 y+Z sin2 y) as depicted in Fig. 3, where Z and y are defined in Section 3.1. The relative numbers of the particles situated in different action intervals give the diluted distributions. A number of 5 105 particles and 50 action intervals are used in Fig. 4, which gives statistical errors of about 1%. The initial Courant–Snyder parameters and emittance are of no importance here, and only the relative change in the shape of the phase ellipse is important which is represented by the parameter x. The beam distributions after the dilution in the cases of uniform and parabolic distributions is no longer the same, whereas in the case of an initial Gaussian distribution and a relatively small mismatch factor (xp2) after the dilution it is still
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1.2
Uniform Distribution
1
Probability density
Probability density
1.2
0.8 = 1.5 = 1.1
0.6 0.4
= 2.0
0.2 0
= 1.0
0
1
Gaussian Distribution (Truncated)
1 0.8 = 1.0
0.6
= 1.1 = 1.5 = 2.0
0.4 0.2
3
2 I/I0
Probability density
1.2
0
4
0
1
2 I/I0
3
4
Parabolic Distribution
1 = 1.0
0.8
= 1.1
0.6
= 1.5 = 2.0
0.4 0.2 0
0
1
2 I/I0
3
4
Fig. 4. Diluted beam distribution over the action variable for initial Gaussian, uniform and parabolic distribution. Non-smooth curves are due to the statistical errors when using limited particles in the simulations.
close to a Gaussian distribution but with a different variance. The results with Gaussian distribution are consistent to the one in Ref. [4] (Fig. 8). For the uniform and parabolic distributions, the particle population in the diluted part appears much more important than for the Gaussian distribution. The unperturbed and unmodified core in the case of a uniform distribution is consistent with Eq. (6).
4. Different betatron mismatch scenarios in high-intensity hadron accelerators 4.1. An examples of betatron mismatch at the J-PARC/MR injection To give a feeling of the quantitative magnitude, an example for the betatron mismatch is given here for the injection from a booster into a main ring such as in the J-PARC where the beam extracted from the 3 GeV rapid cycling synchrotron (RCS) will be injected into the 50 GeV main ring (MR) by a multiple single-turn injection method [12]. Because the beam power at the injection already reaches 45 kW, beam loss is a very critical issue in the MR design. As the beam emittance coming from the RCS, which is shaped by a collimation system in the beam transport line, almost equals the acceptance of the MR collimators at its lowest setting, any emittance dilution due to the betatron mismatch may result in important beam loss. The injection beam emittance, MR collimation acceptance and physical acceptance are defined as 54, 54–81 and 81 pmm mrad, respectively. A gap between the collimation acceptance and the physical acceptance is important for the collimation efficiency, which will be determined to
Table 1 Emittance dilution factors due to betatron mismatch for different cases at the J-PARC/MR injection Case
Mismatch conditions
x
Z
etot (pmm mrad)
1 2 3 4 5 6
Dbx/bx ¼ 0.2, Dax/ax ¼ 0. Dbx/bx ¼ 0., Dax/ax ¼ 0.2 Dxc ¼ 1.0 Dby/by ¼ 0.2, Day/ay ¼ 0. Dby/by ¼ 0., Day/ay ¼ 0.2 Dyc ¼ 1.0
1.054 1.045 1.009 1.048 1.037 1.008
1.386 1.346 1.140 1.360 1.312 1.136
74.84 72.68 61.54 73.44 70.85 61.37
Note: The differences in the horizontal and vertical planes are due to the original Courant–Snyder parameters as indicated in Eqs. (2) and (3).
minimize the total uncontrolled beam loss in the ring during the commissioning. With the nominated Courant–Snyder parameters at the injection point, bx ¼ 13.47, ax ¼ 1.493, by ¼ 12.42, ay ¼ 1.361, different betatron mismatch cases are shown in Table 1, where both the total emittance dilution factor and the rms emittance dilution factor are shown. Cases 1 and 4 are for a betatron mismatch of 20% increase in b, and this is equivalent to a mismatch of 10% in the beam size that can be measured by a profile monitor. Cases 2 and 5 are for an increase of 20% in a, and this is not easy to be measured directly. Case 3 and 6 are for a beam center displacement of 1 mm that is equivalent to a mismatch in b, and this usually can be corrected with the help of a beam profile monitor. From the results, one can see that a betatron mismatch of 20% results in a very important increase in
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the total emittance, whereas a beam center displacement of 1 mm gives a modest increase; the emittance dilution is similar in the two transverse planes. Thus, the effort to produce a good betatron matching with the help of intensive beam diagnostics is very important to minimize the beam loss in the MR. In addition, the distribution of the injection beam after the shaping in the beam transport line is close to a parabolic one, the diluted distribution will follow the analysis in Section 3.2. 4.2. An FODO channel transporting mixed H and proton beams Another example of the betatron mismatch is about a wrongtilt mismatching (a1 ¼ –a2), where an FODO lattice in a part of the beam line from the linac injector to the ring is used to transport both an H beam and a proton beam simultaneously [12–14]. The proton beam is converted from the H beam by foil scrapers in order to shape the H beam emittance before the H beam is injected into the 3 GeV synchrotron. As the focusing structure of the beam line is set to match the H beam, the accompanying proton beam is mismatched in the worst manner. Although there is no real emittance dilution in this short transport section, the required acceptance due to the betatron mismatch to avoid large beam losses is similar to the case with an emittance dilution. With the given bx ¼ 10.5, ax ¼ 1.10, by ¼ 10.5, ay ¼ 1.10, the calculated emittance increase factors for the total emittance and the rms emittance for the proton beam are 6.69 and 3.42, respectively. It is evident that the emittance increase factor for the total emittance is significantly larger than the one for the rms emittance, and the acceptance of the beam transport line has to be designed according to the total emittance.
565
where T2 ms denotes the revolution period. By assuming a constant changing rate of Qx (Qx0 ¼ 5.82) during a quarter synchrotron period Ts (Ts125 T or Qs0.008 at the time of 2 ms) and the maximum tune shift DQx 0.3, we have
T DQ x ¼ 0:0017 T s =4 Q x
Therefore, we can say that the adiabatic condition is satisfied in this case. This means that we need not worry about the emittance increase due to the change in the betatron frequency. The beam is always matched to the Courant–Snyder parameters. The physical image of a high-intensity bunched beam circulating in a high-intensity hadron synchrotron becomes clear: The Courant–Snyder parameters or beam envelope is different along the bunch, but there is no betatron mismatch as the beam is automatically matched to the Courant–Snyder parameters modified by the space charge. When particles move back and forth inside the bunch due to the synchrotron motion, they adapt themselves to the local Courant–Snyder parameters. However, there are still two bad things due to the space-charge forces: a large tune spread and the difference in betatron functions. The former will result in the difficulty to avoid dangerous resonances for all particles, and the latter will result in different mistuned lattices. Both might lead to beam losses. A similar situation happen in a non-scaling FFAG designed for a neutrino factory [16], in which the acceleration is so fast that the tunes change quite rapidly and the adiabatic condition is in doubt, although the beam current is weak here. In fact, it is easy to check that the adiabatic condition still holds here even with a large tune excursion during the acceleration, because the tunes are also large (usually larger than 20). Therefore, the beam is also automatically matched to the instant betatron functions during the acceleration.
4.3. Fast-changing periodic focusing systems 4.4. Emittance mismatch in the case of a large tune depression Betatron mismatch usually happens at the injection to a periodic system such as a linac or a ring. It might be also interesting to see what happens if the periodic system has changing focusing properties. If the Courant–Snyder parameters change slowly or adiabatically, the initially matched beam will be sustained. This is the case in a synchrocyclotron where the tunes decrease slowly with the acceleration. However, in some cases where the tunes vary in a non-adiabatic mode, the betatron mismatch will happen. For example, in a high-intensity synchrotron where the tune shifts due to the space-charge force vary rapidly in the early acceleration phase, thus the emittance blowup due to the betatron mismatch is of concern. This situation will be studied in detail in the following. We take the example of the RCS [15] of the China Spallation Neutron Source (CSNS), which has a repetition rate of 25 Hz, the injected particles of 1.8 1013, the injection energy of 80 MeV and the extraction energy of 1.6 GeV. After the beam injection and the initial RF capture in the RCS, the longitudinal bunch of a parabolic shape is formed. Due to the non-uniformity of the longitudinal distribution, the particles along the bunch suffer different space-charge forces thus the betatron functions modified by the space charge are also different. The largest change of about 10% in betatron function is at the bunch center, whereas there is no change at the two ends. During a synchrotron period, the particles with a large action in the longitudinal phase plane will experience an end-center-end process twice. It is important to see if the betatron functions change adiabatically or not during the process. The condition for the adiabatic motion is T dQ x 51 ¼ (21) Q x dt
The emittance mismatch of a bunched beam due to the space charge also happens in proton linacs where the betatron mismatch is even more important as the tune depression can be as high as 0.5 or so (space-charge dominant). The tune depression parameter in linacs is defined as the ratio of the betatron frequency in the presence of space charge to the betatron frequency at zero current. Some proton linacs use quadrupoles in permanent magnet type and the transverse focusing cannot be adjusted to the beam current, and the others have the limited tunability on the focusing by powering some quadrupoles in series. If the beam distribution is not a uniform ellipsoid, there will be transverse tune spread, and the betatron mismatch is complicated in the presence of strong synchro-betatron coupling effect. In addition, for the emittance blow-up due to a betatron mismatch in case of a very strong tune depression, e.g. a tune depression of less than 0.5, the emittance dilution factor by the geometrical effect as described above will be overtaken by the free energy effect [17,18]. To study the beam behavior including space-charge effects in detail, numerical simulations using sophisticated three-dimensional macro-particles codes such as PARMILA [19], IMPACT [20], etc. are required.
5. Conclusions For high-intensity hadron accelerators, the total emittance should be used when the beam loss level is concerned. The emittance dilution due to the betatron mismatch is much more important for the total emittance than for the rms emittance. The beam distribution after the dilution is different from the original one, except an initial Gaussian distribution will still be close to a Gaussian distribution but with different variance. The emittance
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dilution factor for whether the total emittance or the rms emittance is not related to the initial distribution type. In some applications such as high-intensity proton accelerators where the emittance increase is critical to beam losses, one should make efforts to minimize the mismatch. The adiabatic variation condition still holds when the betatron functions are changing either due to the varying space charge in a RCS or in a non-scaling FFAG with fast acceleration. In the case that the tune depression is large but the focusing is still emittance dominant such as in highintensity proton linacs, the betatron mismatch due to the space charge is very important but complicated. As mentioned in Ref. [6], the dispersion mismatch can be treated by the dispersion-dependent effective b-function.
Acknowledgements This work was supported by the National Natural Science Foundation of China (10775153) and the CAS Knowledge Innovation Program—‘‘CSNS R&D Studies’’. The author would like to cordially thank A.W. Chao of SLAC for giving very useful comments, the CSNS colleagues for the discussions, and K.J. Fan of KEK for providing the J-PARC parameters.
Introducing a pair of dimensionless parameters (Y, Y0 ) rffiffiffiffiffi 3 y¼Y g1 rffiffiffiffiffiffi 3 y0 ¼ Y 0 .
(A.3)
b1
Eq. (3) becomes 2
a1 ffi YY 0 þ Y 0 ¼ 1 Y 2 þ p2ffiffiffiffiffiffiffi g1 b1
(A.4)
3 g2 2 3 b2 0 2 0 2p a2ffiffiffiffiffiffiffi 3 ffi 2 g1 Y þ 2 g1 b1 YY þ 2 b1 Y ¼ 1:
The tangent line for a planar curve F(x, y) at the point (x0,y0) can be expressed by qF qF ðx x Þ þ ðy y0 Þ ¼ 0. (A.5) o qxx0 ;y0 qyx0 ;y0 Thus, the tangent line for the ellipse E3 at the joint point (Y0,Y0 0) can be expressed by ! ! 2a1 2a1 2Y 0 þ pffiffiffiffiffiffiffiffiffiffi Y 0 0 ðY Y 0 Þ þ pffiffiffiffiffiffiffiffiffiffi Y 0 þ 2Y 0 0
g1 b1
g 1 b1
0 Y Y 00 ¼ 0
(A.6)
or Appendix A. Total emittance dilution due to a betatron mismatch
a1 g1 b1
Assuming the acceptance of a periodic system is defined by the Courant–Snyder parameters a0, b0, g0, e0; the injected beam has the parameters a1, b1, g1, e1. In a perfect matching, the two sets of parameters should have the following relations:
a1 ¼ a0 ;
b1 ¼ b0 ;
g1 ¼ g0 ;
E1 : g1 y2 þ 2a1 yy0 þ b1 y0 2 ¼ 1 (A.1)
E3 : g3 y2 þ 2a3 yy0 þ b3 y0 2 ¼ 3 where b3 ¼ b1, a3 ¼ a1, g3 ¼ g1, e34e1. As E2 and E3 are tangibly crossed at two points, the two linked equations can be solved for the joint points:
(A.7)
3 g2 a2 3 3 b2 0 a2 3 Y þ pffiffiffiffiffiffiffiffiffiffi Y 0 Y þ Y Y þ pffiffiffiffiffiffiffiffiffiffi Y 2 g1 0 2 g1 b1 0 2 b1 0 0 2 g1 b1 0 Y 0 ¼ 1.
One can omit the relation with g as there is a relation bga ¼ 1. In the case of a betatron mismatch, the injected beam has the parameters a2, b2 that are different from a1, b1 but the emittances are the same. This is depicted by three ellipses in Fig. 1, where E1, E2 and E3 denote the matched beam, the mismatched beam and the diluted emittance, respectively. This will introduce a betatron modulation along the longitudinal direction, which is equivalent to have a larger emittance (ellipse E3). E3 has the same a and b parameters as E1 but it circumscribes with the ellipse E2. In the presence of nonlinear forces, there will be a tune spread according to the action parameter. After some periods the filamentation effect (see Fig. 2) will result in a real larger beam emittance, namely the beam fills the whole E3 ellipse. As depicted in Figs. 1 and 2, the filamentation effect will result in a real larger beam emittance. Here, we use a geometrical method to obtain the emittance dilution factor, which was perhaps also used by other people [21]. However, the negative solution (A.15) that is not shown elsewhere is needed in this article, thus the derivations are included in this Appendix A. The three ellipses in Fig. 1 can be expressed by the following equations:
g1 y2 þ 2a1 yy0 þ b1 y0 2 ¼ 3 g2 y2 þ 2a2 yy0 þ b2 y0 2 ¼ 2 .
!
a g1 b1
1 pffiffiffiffiffiffiffiffiffiffi Y 0 þ Y 0 0 Y 0 ¼ 1.
It is similar to express the tangent line for the ellipse E2, ! !
1 p0 . 2
E2 : g2 y2 þ 2a2 yy0 þ b2 y0 2 ¼ 2
!
Y 0 þ pffiffiffiffiffiffiffiffiffiffi Y 0 0 Y þ
(A.8)
As E3 circumscribes with E2, Eqs. (8) and (9) should be the same, thus
3 g2 a2 3 Y þ pffiffiffiffiffiffiffiffiffiffi Y 0 2 g1 0 2 g1 b1 0 a1 a2 3 3 b2 0 pffiffiffiffiffiffiffiffiffiffi Y 0 þ Y 0 0 ¼ pffiffiffiffiffiffiffiffiffiffi Y 0 þ Y 2 b1 0 g1 b1 2 g1 b1 a1 g1 b1
Y 0 þ pffiffiffiffiffiffiffiffiffiffi Y 0 0 ¼
(A.9)
or in the form 3 g 2 1 Y0 ¼
a 2 g1 b1
2 g 1
a g1 b1
1 pffiffiffiffiffiffiffiffiffiffi
a g1 b1
!
3 1 p2ffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Y 0 0
a 2 g1 b1
!
3 p2ffiffiffiffiffiffiffiffiffiffi Y0 ¼
3 b2 1 Y 00. 2 b1
(A.10)
Thus 3 g2 3 b2 1 1 ¼
2 g 1
a g1 b1
1 pffiffiffiffiffiffiffiffiffiffi
2 b1
a 2 g1 b1
a 3 a1 p2ffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 g1 b1 g1 b1
!
!
3 p2ffiffiffiffiffiffiffiffiffiffi .
(A-11)
Eq. (12) can be simplified as 2 2 a1 3 g 2 3 b2 a2 3 1 . 1 ¼ 1
2 g 1
2 b1
g 1 b1
a1 2
(A.12)
Introducing the emittance dilution factor Z ¼ e3/e2, Eq. (13) becomes ða22 g2 b2 ÞZ2 þ ðb1 g2 þ b2 g1 2a1 a2 ÞZ þ a21 g1 b1 ¼ 0.
(A.13)
2
Using the relation bg–a ¼ 1, Eq. (14) is simplified to (A.2)
2
Z þ ð2a1 a2 b1 g2 b2 g1 ÞZ þ 1 ¼ 0.
(A.14)
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The solution is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ¼ 12½ðb1 g2 þ b2 g1 2a1 a2 Þ ðb1 g2 þ b2 g1 2a1 a2 Þ2 4. (A.15) Here only the solution with ‘‘+’’ is reasonable as Z should be larger than one. We can define the mismatch factor parameter x: 1 b g þ b2 g1 2a1 a2 2 1 2 2 ! 1 b1 b2 b1 b ¼ þ þ a2 2 a1 . 2 b2 b1 b2 b1
x¼
Then the emittance dilution factor qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ¼ 3 ¼ x þ x2 1.
2
(A.16)
(A.17)
References [1] P.M. Hanney, E. Keil, CERN-ISR-TH/69-32, 1969. [2] T.O. Raumenheimer, M. Syphers, in: A.W. Chao, M. Tigner (Eds.), Handbook of Accelerator Physics and Engineering, World Scientific, Singapore, 1999, pp. 252–276.
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[3] P.J. Bryant, in: Proceedings of the Fifth CERN Accelerator School, CERN 94-01, 1992, p. 219. [4] M.J. Sypheres, Fermilab-FN-458, 1987. [5] M.J. Sypheres,T. Sen, in:Proceedings of the IEEE Particle Accelerator Conference, Washington, DC, 1993, p. 134. [6] L. Merminga, et al., in: Proceedings of the IEEE Particle Accelerator Conference, San Francisco, 1991, p. 461. [7] L. Vos, in: Proceedings of Chamonix IX, CERN-SL-99-007 DI, January 1999, p. 99. [8] T.O. Raubenheimer, Phys. Rev. Spec. Top.–Accel. Beams 3 (2000) 121002. [9] F.J. Decker, et al., in: Proceedings of the IEEE Particle Accelerator Conference, San Francisco, 1993, p. 905. [10] D.A. Edwards, M.J. Syphers, An Introduction to the Physics of High Energy Accelerators, Wiley, New York, 1993. [11] R. Cappi, M. Giovannozzi, CERN/PS 2002-083 (AE). [12] Accelerator Technical Design Report for J-PARC, KEK Report 2002-13/JAERITech 2003-044, March 2003. [13] J.Y. Tang, G.H. Wei, C. Zhang, Nucl. Instr. and Meth. A 572 (2007) 601. [14] T. Ohakawa, M. Ikegami, J. Qiang, P. Saha, Nucl. Instr. and Meth. A 589 (2008) 1. [15] S. Wang, S.X. Fang, Q. Qin, J.Y. Tang, J. Wei, in: Proceedings of EPAC06, Edinburgh, 2006, p. 1396. [16] C. Johnstone, S. Koscielniak, Nucl. Instr. and Meth. A 503 (2003) 445. [17] M. Reiser, J. Appl. Phys. 70 (1991) 1919; M. Reiser, Theory and Design of Charged Particle Beams, Wiley, New York, 1994, pp. 470–477. [18] G. Franchetti, I. Hofmann, D. Jeon, Phys. Rev. Lett. 88 (2002) 254802. [19] K.R. Crandall, LANL, Report LA-11054-MS, 1987; G. Boicourt, J. Merson, LANL Report LA-UR-90-127, 1990. [20] J. Qiang, R. Ryne, S. Habib, V. Decyk, J. Comput. Phys. 163 (2000) 434. [21] Bovet, Claude, et al., A selection of formulae and data useful for the design of A.G. synchrotrons, CERN-MPS-SI-INT-DL-68-3-Rev-1.-Geneva: CERN, 1970, p. 49.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Emittance dilution due to the betatron mismatch in high-intensity hadron accelerators J.Y. Tang Institute of High Energy Physics, CAS, Yuquan Road 19B, Beijing 100049, China
a r t i c l e in fo
abstract
Article history: Received 23 June 2008 Received in revised form 6 August 2008 Accepted 6 August 2008 Available online 20 August 2008
The importance of the betatron mismatch between two sections of an accelerator complex was recognized a long time ago. The mismatch together with the filamentation effect will result in a larger diluted beam emittance. However, the influence of the mismatch was underestimated largely in high-intensity hadron synchrotrons in the literature, as the rms emittance is used to represent the dilution. The dilution on the total emittance is significantly larger, which is of more importance in representing the beam loss in proton synchrotrons. This article points out the necessity of using the dilution factor of the total emittance together with the rms emittance in high-intensity hadron accelerators. A proof is given to show that the dilution factor of the rms emittance is the same for any initial beam distribution. The detailed beam distribution after the dilution is discussed for the initial uniform, Gaussian and parabolic distributions, and the numerical simulation results are presented. Examples and discussions about the betatron mismatch in the cases of injection, FODO channel transporting mixed H and proton beams, fast-changing periodic focusing systems and proton linacs of a large tune depression are also presented. & 2008 Elsevier B.V. All rights reserved.
Keywords: Betatron mismatch Filamentation Emittance dilution Beam distribution Beam loss
1. Introduction It is known that the betatron mismatch between the injected beam and the periodic focusing system will result in an emittance dilution. For the case with no nonlinear components in the periodic system, there will be only betatron modulation and the beam emittance will rotate within the boundary of a larger ellipse; in case of the presence of nonlinear components, the filamentation effect in the phase space will lead to a real emittance dilution although Liouville theorem still holds. The emittance increase due to the mismatch is very important in many accelerators where the beam loss or the collision luminosity is concerned, and the betatron modulation has the similar impact. The study on the emittance dilution due to the betatron mismatch can be traced early in 1960s [1]. The following expression for the increase in emittance that is widely used [2–11] ! 2 diluted 1 b1 b1 b2 b2 ¼ þ a1 a þ (1) 2 b2 0 b2 2 b1 b1 where suffixes 1 and 2 are for the acceptance and the injected beam, a and b are the Courant–Snyder parameters. This is true for the rms emittance whatever the beam distribution is (see Section 3.1). However, when the total diluted Tel./fax: +86 10 8823 5908.
E-mail address: [email protected] 0168-9002/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2008.08.030
emittance is concerned, the dilution factor is significantly larger. In high-intensity proton synchrotrons where beam losses are of very important concern, the total emittance or the emittance containing 99% particles are often used. In proton linacs, where the beam focusing is space-charge dominant, the betatron mismatch becomes an important factor for the halo formation. In proton synchrotrons with multi-turn injection using H stripping, the phase space is filled by painting method, thus the injection betatron mismatch is less important by affecting only the painted particle distribution. However, in proton synchrotrons with single-turn injection from a booster, the emittance dilution due to the betatron mismatch is a very important factor leading to large beam losses. In this paper, the dilution in the means of the total emittance instead of the rms emittance is stressed for high-intensity hadron accelerators, and the diluted beam distributions for different initial distribution types are studied.
2. Why total emittance in high-intensity hadron accelerators? The betatron mismatch happens in many high-intensity hadron accelerators, such as the proton linacs with spacecharge-dominated focusing, the single-turn injection from a proton booster to a main ring, the synchrotrons or accumulator rings with a large tune shift due to the space-charge force and without focusing compensation. In the case of the multi-turn injection using an H stripping method in a proton synchrotron,
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the emittance dilution due to a betatron mismatch at the injection is less important as it affects mainly the painted particle distribution and has little effect on the total emittance. The rms emittance or sometimes its integer multiple, is widely used in electron accelerators or low-intensity hadron accelerators, as it describes well the property of the beam as a whole. Some halo particles having large Courant–Snyder invariants can be ignored there. However, in high-intensity proton accelerators where the beam loss level is of great concern, the halo particles are no longer to be ignored. In this case, the total emittance or sometimes the emittance including 99% particles is better used along with the rms emittance. The dilution of the total emittance due to a betatron mismatch is depicted by three ellipses in Fig. 1, where E1, E2 and E3 denote the matched beam, the mismatched beam and the diluted emittance, respectively. The betatron modulation along the longitudinal direction created by the mismatch will result in a real larger beam emittance after some periods due to the filamentation effect (see Fig. 2), namely the beam fills the whole E3 ellipse. In Appendix A, the derivations to obtain the total emittance dilution factor are given. There the dilution factor of the total emittance is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ¼ 3 ¼ x þ x2 1 (2)
2
y’
1.5 1 0.5 0 -15
-10
-5.0
0 -0.5
5
10
15 y
-1 -1.5 -2 Fig. 2. Filamentation effect in the early stage in the presence of nonlinear forces. Dashed line outlines the filamented part of the distribution; the outer and inner ellipses are E3 and E4 in Fig. 1.
2
where
x¼
1 b1 b2 b1 þ þ 2 b2 b1 b2
2 ! b a2 2 a1 .
(3)
b1
When the total emittance is used, the emittance dilution due to a betatron mismatch becomes much more significant, quite differently from the case that the rms emittance is used. Especially in the case with a small mismatch factor (x151), the emittance dilution for the total emittance is significant whereas the one for the rms emittance is negligible. We can also express the emittance dilution by the relative change in the emittance, qffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 2 ¼ x 1 þ x 1. (4)
If the mismatch is small, x ¼ 1+D, thus
D
Dþ
pffiffiffiffiffiffiffi 2D.
(5)
As will be seen in Section 3.2, the beam tail with a large action after the dilution is significant in the cases of initial uniform and parabolic distributions, but to a much less degree in the case of an
1.5
E2
E1
0.5
-10
-5 E4
5
3. Diluted beam distribution Not like in an electron synchrotron where the beam distribution is usually a Gaussian like, the beam distribution in hadron synchrotrons is more difficult to predict. Here three different types of distributions: Gaussian, uniform and parabolic are used to show the dilution effect in case of a betatron mismatch. Because the filamentation effect will dilute the particle distribution when a betatron mismatch happens, the beam distribution becomes sparse at the outer part of the diluted emittance [1]. For an initial uniform distribution, the inner part (E4) of the emittance as shown in Fig. 1 will remain uniform. The emittance of E4 is defined by Eq. (A.15) with the negative sign, namely qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z0 ¼ 4 ¼ x x2 1. (6)
2
If the initial distribution is Gaussian or parabolic, the inner part of the beam will also be diluted. In Ref. [4], it was shown that for an initially Gaussian distribution the rms emittance of the diluted beam follows Eq. (1). However, we do not know what the situation is for any kind of distribution. In the following section, the diluted rms emittance in the case of a betatron mismatch will be proven still following Eq. (1) whatever the distribution is.
y’
1.0
initial Gaussian distribution. However, it is usually not easy to predict the beam distribution in hadron accelerators as in electron accelerators.
10
y
3.1. RMS emittance of the diluted beam
-0.5 -1.0 E3 -1.5
Fig. 1. Emittance dilution due to the betatron mismatch. E1: ellipse for the matched beam; E2: ellipse for the mismatched beam; E3: ellipse for the diluted emittance; E4: ellipse for the non-diluted part when the distribution is uniform.
It is often convenient to use the action–angle variables; here the definition of the action variable (also called the Courant–Snyder invariant) has a factor of two in difference from the usual one. If the initial particle distribution is not uniform in the angle, the filamentation effect will smooth out the difference. Therefore, it is convenient to assume that the initial particle distribution for the action–angle variables is uniform in the angle variable and then use various types of distributions for the action variable.
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Assume the initial distribution g(x,x0 ), the action function is 2
0
0
02
Iðx; x Þ ¼ gx þ 2axx þ bx .
(7)
Here the derivations are given to show the rms emittance of a distribution is related to the mean value of the action. Taking the average to Eq. (7) ZZ I¯ ¼ gðx; x0 ÞIðx; x0 Þ dx dx0 ZZ ¼ gðx; x0 Þðgx2 þ 2axx0 þ bx0 2 Þ dx dx0 ¼ gx2 þ 2axx0 þ bx0 2 .
(8)
Using the statistical definitions of the rms emittance 8 2 > > < x ¼ brms x0 2 ¼ grms > > : xx0 ¼ a
563
Thus, by substituting the averaged action in Eq. (10) the new rms emittance after the dilution is ZZ 1 1 gðrÞr 2 0 rms ¼ cos2 y þ Z sin2 y r dr dy. (13) 2 Z 2bð1 þ 2a Þ With the integration in separate variables, the change in the rms emittance before and after the emittance dilution can be expressed as 0 rms 1 1 ¼ þ Z ¼ x. (14) rms 2 Z This is just the same as Eq. (1). 3.2. Distribution of the diluted beam
(9)
rms
Three different initial distributions: uniform, Gaussian and parabolic, which are often used in hadron accelerators, are considered here to show the dilution effect.
and bg–a2 ¼ 1, it is straightforward to obtain I¯ ¼ 2ð1 þ 2a2 Þrms .
(10)
Eq. (10) shows an important intrinsic relationship between the averaged action and the rms emittance. In order to better describe the betatron mismatch, the transformed coordinates (x,z) instead of (x,x0 ) will be used as in the literature, where z ¼ ax+bx0 is used. In the new coordinates, the distribution properties remain but the distribution now is over a round area. The initial distribution becomes axisymmetric and will be deformed into an ellipse with a mismatch, but will become axisymmetric again due to the filamentation effect. With the initial distribution, one has 2
x2 ¼ z ¼ 12r 2 .
(11)
With the coordinate transformation, there is I(x,x0 ) ) (1/b)r2. Following the depiction in Fig. 3, for the particles in the area element r dr dy, the change in the contribution to the action integral is 1 gðrÞr 2 r dr dy ) gðrÞr 2 2 cos2 y þ a2 sin2 y r dr dy (12) a where a is the elongation factor at the mismatch and a ¼ OZ in Eq. (2). This is to say that the action variable being usually an invariant will change during a mismatch.
r/a r
Matched
x
ar Mismatched
Fig. 3. Emittance mismatch in the normalized coordinates. Circle for the matched emittance and ellipse for the mismatched one.
(1) Uniform distribution (KV distribution) gðx; x0 Þ ¼
b 4ps2
;
x 2 ð2s; 2sÞ
(15)
or gðIÞ ¼
1 2s2 ; I 2 ð0; 2I0 Þ; I0 ¼ 2I0 b
(16)
where s is the rms beam size (the same thereafter). (2) Gaussian distribution (truncated) gðx; x0 Þ ¼
b 2ps2
2
eðx
þðaxþbx0 Þ2 Þ=ð2s2 Þ
;
x 2 ð3s; 3sÞ
(17)
or gðIÞ ¼
1 I=I0 2s2 e ; I 2 ð0; 4:5I0 Þ; I0 ¼ . I0 b
(18)
(3) Parabolic distribution gðx; x0 Þ ¼
b 2ps2
1
! x2 þ ðax þ bx0 Þ2 ; 4s2
x 2 ð2s; 2sÞ
(19)
or gðIÞ ¼
1 I 2s2 1 . ; I 2 ð0; 2I0 Þ; I0 ¼ I0 2I0 b
(20)
Fig. 4 shows the diluted distributions according to different mismatch degrees, which are obtained using multi-particles simulations (a self-made code). The validity of the code has been checked with the theoretical predictions on the dilution factors of the total emittance and the rms emittance. The filamentation process is simulated through the tune spread induced by a sextupole component in a periodic system (see Fig. 2). To simulate the redistribution of particles in the phase plane during a betatron mismatch, the initially populated particles of a given distribution will change their action parameter by a factor of ((1/Z)cos2 y+Z sin2 y) as depicted in Fig. 3, where Z and y are defined in Section 3.1. The relative numbers of the particles situated in different action intervals give the diluted distributions. A number of 5 105 particles and 50 action intervals are used in Fig. 4, which gives statistical errors of about 1%. The initial Courant–Snyder parameters and emittance are of no importance here, and only the relative change in the shape of the phase ellipse is important which is represented by the parameter x. The beam distributions after the dilution in the cases of uniform and parabolic distributions is no longer the same, whereas in the case of an initial Gaussian distribution and a relatively small mismatch factor (xp2) after the dilution it is still
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1.2
Uniform Distribution
1
Probability density
Probability density
1.2
0.8 = 1.5 = 1.1
0.6 0.4
= 2.0
0.2 0
= 1.0
0
1
Gaussian Distribution (Truncated)
1 0.8 = 1.0
0.6
= 1.1 = 1.5 = 2.0
0.4 0.2
3
2 I/I0
Probability density
1.2
0
4
0
1
2 I/I0
3
4
Parabolic Distribution
1 = 1.0
0.8
= 1.1
0.6
= 1.5 = 2.0
0.4 0.2 0
0
1
2 I/I0
3
4
Fig. 4. Diluted beam distribution over the action variable for initial Gaussian, uniform and parabolic distribution. Non-smooth curves are due to the statistical errors when using limited particles in the simulations.
close to a Gaussian distribution but with a different variance. The results with Gaussian distribution are consistent to the one in Ref. [4] (Fig. 8). For the uniform and parabolic distributions, the particle population in the diluted part appears much more important than for the Gaussian distribution. The unperturbed and unmodified core in the case of a uniform distribution is consistent with Eq. (6).
4. Different betatron mismatch scenarios in high-intensity hadron accelerators 4.1. An examples of betatron mismatch at the J-PARC/MR injection To give a feeling of the quantitative magnitude, an example for the betatron mismatch is given here for the injection from a booster into a main ring such as in the J-PARC where the beam extracted from the 3 GeV rapid cycling synchrotron (RCS) will be injected into the 50 GeV main ring (MR) by a multiple single-turn injection method [12]. Because the beam power at the injection already reaches 45 kW, beam loss is a very critical issue in the MR design. As the beam emittance coming from the RCS, which is shaped by a collimation system in the beam transport line, almost equals the acceptance of the MR collimators at its lowest setting, any emittance dilution due to the betatron mismatch may result in important beam loss. The injection beam emittance, MR collimation acceptance and physical acceptance are defined as 54, 54–81 and 81 pmm mrad, respectively. A gap between the collimation acceptance and the physical acceptance is important for the collimation efficiency, which will be determined to
Table 1 Emittance dilution factors due to betatron mismatch for different cases at the J-PARC/MR injection Case
Mismatch conditions
x
Z
etot (pmm mrad)
1 2 3 4 5 6
Dbx/bx ¼ 0.2, Dax/ax ¼ 0. Dbx/bx ¼ 0., Dax/ax ¼ 0.2 Dxc ¼ 1.0 Dby/by ¼ 0.2, Day/ay ¼ 0. Dby/by ¼ 0., Day/ay ¼ 0.2 Dyc ¼ 1.0
1.054 1.045 1.009 1.048 1.037 1.008
1.386 1.346 1.140 1.360 1.312 1.136
74.84 72.68 61.54 73.44 70.85 61.37
Note: The differences in the horizontal and vertical planes are due to the original Courant–Snyder parameters as indicated in Eqs. (2) and (3).
minimize the total uncontrolled beam loss in the ring during the commissioning. With the nominated Courant–Snyder parameters at the injection point, bx ¼ 13.47, ax ¼ 1.493, by ¼ 12.42, ay ¼ 1.361, different betatron mismatch cases are shown in Table 1, where both the total emittance dilution factor and the rms emittance dilution factor are shown. Cases 1 and 4 are for a betatron mismatch of 20% increase in b, and this is equivalent to a mismatch of 10% in the beam size that can be measured by a profile monitor. Cases 2 and 5 are for an increase of 20% in a, and this is not easy to be measured directly. Case 3 and 6 are for a beam center displacement of 1 mm that is equivalent to a mismatch in b, and this usually can be corrected with the help of a beam profile monitor. From the results, one can see that a betatron mismatch of 20% results in a very important increase in
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the total emittance, whereas a beam center displacement of 1 mm gives a modest increase; the emittance dilution is similar in the two transverse planes. Thus, the effort to produce a good betatron matching with the help of intensive beam diagnostics is very important to minimize the beam loss in the MR. In addition, the distribution of the injection beam after the shaping in the beam transport line is close to a parabolic one, the diluted distribution will follow the analysis in Section 3.2. 4.2. An FODO channel transporting mixed H and proton beams Another example of the betatron mismatch is about a wrongtilt mismatching (a1 ¼ –a2), where an FODO lattice in a part of the beam line from the linac injector to the ring is used to transport both an H beam and a proton beam simultaneously [12–14]. The proton beam is converted from the H beam by foil scrapers in order to shape the H beam emittance before the H beam is injected into the 3 GeV synchrotron. As the focusing structure of the beam line is set to match the H beam, the accompanying proton beam is mismatched in the worst manner. Although there is no real emittance dilution in this short transport section, the required acceptance due to the betatron mismatch to avoid large beam losses is similar to the case with an emittance dilution. With the given bx ¼ 10.5, ax ¼ 1.10, by ¼ 10.5, ay ¼ 1.10, the calculated emittance increase factors for the total emittance and the rms emittance for the proton beam are 6.69 and 3.42, respectively. It is evident that the emittance increase factor for the total emittance is significantly larger than the one for the rms emittance, and the acceptance of the beam transport line has to be designed according to the total emittance.
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where T2 ms denotes the revolution period. By assuming a constant changing rate of Qx (Qx0 ¼ 5.82) during a quarter synchrotron period Ts (Ts125 T or Qs0.008 at the time of 2 ms) and the maximum tune shift DQx 0.3, we have
T DQ x ¼ 0:0017 T s =4 Q x
Therefore, we can say that the adiabatic condition is satisfied in this case. This means that we need not worry about the emittance increase due to the change in the betatron frequency. The beam is always matched to the Courant–Snyder parameters. The physical image of a high-intensity bunched beam circulating in a high-intensity hadron synchrotron becomes clear: The Courant–Snyder parameters or beam envelope is different along the bunch, but there is no betatron mismatch as the beam is automatically matched to the Courant–Snyder parameters modified by the space charge. When particles move back and forth inside the bunch due to the synchrotron motion, they adapt themselves to the local Courant–Snyder parameters. However, there are still two bad things due to the space-charge forces: a large tune spread and the difference in betatron functions. The former will result in the difficulty to avoid dangerous resonances for all particles, and the latter will result in different mistuned lattices. Both might lead to beam losses. A similar situation happen in a non-scaling FFAG designed for a neutrino factory [16], in which the acceleration is so fast that the tunes change quite rapidly and the adiabatic condition is in doubt, although the beam current is weak here. In fact, it is easy to check that the adiabatic condition still holds here even with a large tune excursion during the acceleration, because the tunes are also large (usually larger than 20). Therefore, the beam is also automatically matched to the instant betatron functions during the acceleration.
4.3. Fast-changing periodic focusing systems 4.4. Emittance mismatch in the case of a large tune depression Betatron mismatch usually happens at the injection to a periodic system such as a linac or a ring. It might be also interesting to see what happens if the periodic system has changing focusing properties. If the Courant–Snyder parameters change slowly or adiabatically, the initially matched beam will be sustained. This is the case in a synchrocyclotron where the tunes decrease slowly with the acceleration. However, in some cases where the tunes vary in a non-adiabatic mode, the betatron mismatch will happen. For example, in a high-intensity synchrotron where the tune shifts due to the space-charge force vary rapidly in the early acceleration phase, thus the emittance blowup due to the betatron mismatch is of concern. This situation will be studied in detail in the following. We take the example of the RCS [15] of the China Spallation Neutron Source (CSNS), which has a repetition rate of 25 Hz, the injected particles of 1.8 1013, the injection energy of 80 MeV and the extraction energy of 1.6 GeV. After the beam injection and the initial RF capture in the RCS, the longitudinal bunch of a parabolic shape is formed. Due to the non-uniformity of the longitudinal distribution, the particles along the bunch suffer different space-charge forces thus the betatron functions modified by the space charge are also different. The largest change of about 10% in betatron function is at the bunch center, whereas there is no change at the two ends. During a synchrotron period, the particles with a large action in the longitudinal phase plane will experience an end-center-end process twice. It is important to see if the betatron functions change adiabatically or not during the process. The condition for the adiabatic motion is T dQ x 51 ¼ (21) Q x dt
The emittance mismatch of a bunched beam due to the space charge also happens in proton linacs where the betatron mismatch is even more important as the tune depression can be as high as 0.5 or so (space-charge dominant). The tune depression parameter in linacs is defined as the ratio of the betatron frequency in the presence of space charge to the betatron frequency at zero current. Some proton linacs use quadrupoles in permanent magnet type and the transverse focusing cannot be adjusted to the beam current, and the others have the limited tunability on the focusing by powering some quadrupoles in series. If the beam distribution is not a uniform ellipsoid, there will be transverse tune spread, and the betatron mismatch is complicated in the presence of strong synchro-betatron coupling effect. In addition, for the emittance blow-up due to a betatron mismatch in case of a very strong tune depression, e.g. a tune depression of less than 0.5, the emittance dilution factor by the geometrical effect as described above will be overtaken by the free energy effect [17,18]. To study the beam behavior including space-charge effects in detail, numerical simulations using sophisticated three-dimensional macro-particles codes such as PARMILA [19], IMPACT [20], etc. are required.
5. Conclusions For high-intensity hadron accelerators, the total emittance should be used when the beam loss level is concerned. The emittance dilution due to the betatron mismatch is much more important for the total emittance than for the rms emittance. The beam distribution after the dilution is different from the original one, except an initial Gaussian distribution will still be close to a Gaussian distribution but with different variance. The emittance
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dilution factor for whether the total emittance or the rms emittance is not related to the initial distribution type. In some applications such as high-intensity proton accelerators where the emittance increase is critical to beam losses, one should make efforts to minimize the mismatch. The adiabatic variation condition still holds when the betatron functions are changing either due to the varying space charge in a RCS or in a non-scaling FFAG with fast acceleration. In the case that the tune depression is large but the focusing is still emittance dominant such as in highintensity proton linacs, the betatron mismatch due to the space charge is very important but complicated. As mentioned in Ref. [6], the dispersion mismatch can be treated by the dispersion-dependent effective b-function.
Acknowledgements This work was supported by the National Natural Science Foundation of China (10775153) and the CAS Knowledge Innovation Program—‘‘CSNS R&D Studies’’. The author would like to cordially thank A.W. Chao of SLAC for giving very useful comments, the CSNS colleagues for the discussions, and K.J. Fan of KEK for providing the J-PARC parameters.
Introducing a pair of dimensionless parameters (Y, Y0 ) rffiffiffiffiffi 3 y¼Y g1 rffiffiffiffiffiffi 3 y0 ¼ Y 0 .
(A.3)
b1
Eq. (3) becomes 2
a1 ffi YY 0 þ Y 0 ¼ 1 Y 2 þ p2ffiffiffiffiffiffiffi g1 b1
(A.4)
3 g2 2 3 b2 0 2 0 2p a2ffiffiffiffiffiffiffi 3 ffi 2 g1 Y þ 2 g1 b1 YY þ 2 b1 Y ¼ 1:
The tangent line for a planar curve F(x, y) at the point (x0,y0) can be expressed by qF qF ðx x Þ þ ðy y0 Þ ¼ 0. (A.5) o qxx0 ;y0 qyx0 ;y0 Thus, the tangent line for the ellipse E3 at the joint point (Y0,Y0 0) can be expressed by ! ! 2a1 2a1 2Y 0 þ pffiffiffiffiffiffiffiffiffiffi Y 0 0 ðY Y 0 Þ þ pffiffiffiffiffiffiffiffiffiffi Y 0 þ 2Y 0 0
g1 b1
g 1 b1
0 Y Y 00 ¼ 0
(A.6)
or Appendix A. Total emittance dilution due to a betatron mismatch
a1 g1 b1
Assuming the acceptance of a periodic system is defined by the Courant–Snyder parameters a0, b0, g0, e0; the injected beam has the parameters a1, b1, g1, e1. In a perfect matching, the two sets of parameters should have the following relations:
a1 ¼ a0 ;
b1 ¼ b0 ;
g1 ¼ g0 ;
E1 : g1 y2 þ 2a1 yy0 þ b1 y0 2 ¼ 1 (A.1)
E3 : g3 y2 þ 2a3 yy0 þ b3 y0 2 ¼ 3 where b3 ¼ b1, a3 ¼ a1, g3 ¼ g1, e34e1. As E2 and E3 are tangibly crossed at two points, the two linked equations can be solved for the joint points:
(A.7)
3 g2 a2 3 3 b2 0 a2 3 Y þ pffiffiffiffiffiffiffiffiffiffi Y 0 Y þ Y Y þ pffiffiffiffiffiffiffiffiffiffi Y 2 g1 0 2 g1 b1 0 2 b1 0 0 2 g1 b1 0 Y 0 ¼ 1.
One can omit the relation with g as there is a relation bga ¼ 1. In the case of a betatron mismatch, the injected beam has the parameters a2, b2 that are different from a1, b1 but the emittances are the same. This is depicted by three ellipses in Fig. 1, where E1, E2 and E3 denote the matched beam, the mismatched beam and the diluted emittance, respectively. This will introduce a betatron modulation along the longitudinal direction, which is equivalent to have a larger emittance (ellipse E3). E3 has the same a and b parameters as E1 but it circumscribes with the ellipse E2. In the presence of nonlinear forces, there will be a tune spread according to the action parameter. After some periods the filamentation effect (see Fig. 2) will result in a real larger beam emittance, namely the beam fills the whole E3 ellipse. As depicted in Figs. 1 and 2, the filamentation effect will result in a real larger beam emittance. Here, we use a geometrical method to obtain the emittance dilution factor, which was perhaps also used by other people [21]. However, the negative solution (A.15) that is not shown elsewhere is needed in this article, thus the derivations are included in this Appendix A. The three ellipses in Fig. 1 can be expressed by the following equations:
g1 y2 þ 2a1 yy0 þ b1 y0 2 ¼ 3 g2 y2 þ 2a2 yy0 þ b2 y0 2 ¼ 2 .
!
a g1 b1
1 pffiffiffiffiffiffiffiffiffiffi Y 0 þ Y 0 0 Y 0 ¼ 1.
It is similar to express the tangent line for the ellipse E2, ! !
1 p0 . 2
E2 : g2 y2 þ 2a2 yy0 þ b2 y0 2 ¼ 2
!
Y 0 þ pffiffiffiffiffiffiffiffiffiffi Y 0 0 Y þ
(A.8)
As E3 circumscribes with E2, Eqs. (8) and (9) should be the same, thus
3 g2 a2 3 Y þ pffiffiffiffiffiffiffiffiffiffi Y 0 2 g1 0 2 g1 b1 0 a1 a2 3 3 b2 0 pffiffiffiffiffiffiffiffiffiffi Y 0 þ Y 0 0 ¼ pffiffiffiffiffiffiffiffiffiffi Y 0 þ Y 2 b1 0 g1 b1 2 g1 b1 a1 g1 b1
Y 0 þ pffiffiffiffiffiffiffiffiffiffi Y 0 0 ¼
(A.9)
or in the form 3 g 2 1 Y0 ¼
a 2 g1 b1
2 g 1
a g1 b1
1 pffiffiffiffiffiffiffiffiffiffi
a g1 b1
!
3 1 p2ffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Y 0 0
a 2 g1 b1
!
3 p2ffiffiffiffiffiffiffiffiffiffi Y0 ¼
3 b2 1 Y 00. 2 b1
(A.10)
Thus 3 g2 3 b2 1 1 ¼
2 g 1
a g1 b1
1 pffiffiffiffiffiffiffiffiffiffi
2 b1
a 2 g1 b1
a 3 a1 p2ffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 g1 b1 g1 b1
!
!
3 p2ffiffiffiffiffiffiffiffiffiffi .
(A-11)
Eq. (12) can be simplified as 2 2 a1 3 g 2 3 b2 a2 3 1 . 1 ¼ 1
2 g 1
2 b1
g 1 b1
a1 2
(A.12)
Introducing the emittance dilution factor Z ¼ e3/e2, Eq. (13) becomes ða22 g2 b2 ÞZ2 þ ðb1 g2 þ b2 g1 2a1 a2 ÞZ þ a21 g1 b1 ¼ 0.
(A.13)
2
Using the relation bg–a ¼ 1, Eq. (14) is simplified to (A.2)
2
Z þ ð2a1 a2 b1 g2 b2 g1 ÞZ þ 1 ¼ 0.
(A.14)
ARTICLE IN PRESS J.Y. Tang / Nuclear Instruments and Methods in Physics Research A 595 (2008) 561–567
The solution is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ¼ 12½ðb1 g2 þ b2 g1 2a1 a2 Þ ðb1 g2 þ b2 g1 2a1 a2 Þ2 4. (A.15) Here only the solution with ‘‘+’’ is reasonable as Z should be larger than one. We can define the mismatch factor parameter x: 1 b g þ b2 g1 2a1 a2 2 1 2 2 ! 1 b1 b2 b1 b ¼ þ þ a2 2 a1 . 2 b2 b1 b2 b1
x¼
Then the emittance dilution factor qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ¼ 3 ¼ x þ x2 1.
2
(A.16)
(A.17)
References [1] P.M. Hanney, E. Keil, CERN-ISR-TH/69-32, 1969. [2] T.O. Raumenheimer, M. Syphers, in: A.W. Chao, M. Tigner (Eds.), Handbook of Accelerator Physics and Engineering, World Scientific, Singapore, 1999, pp. 252–276.
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