PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS 14, 014401 (2011)

Estimation of orbit change and emittance growth due to random misalignment in long linacs Kiyoshi Kubo KEK, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan (Received 15 September 2010; published 4 January 2011) In linear accelerators, the transverse beam orbit is induced by tilts of accelerating cavities (deviation of accelerating field direction from designed beam direction) and transverse offset of quadrupole magnets. Estimating induced orbit and emittance growth due to such random errors is important for evaluation of performance of linacs, especially where stable and low emittance beam is required, such as linear colliders. Usually, such estimations are performed using tracking simulations, including the Monte Carlo method, which tends to take a long time. Here, a much faster and simpler method of quantitative estimation of beam orbit and emittance growth is reported. This method is valid for very long linacs with many components, where statistical treatment of errors is justified. It is shown that the results from the method agree well with tracking simulations for the ILC (International Linear Collider) main linac. DOI: 10.1103/PhysRevSTAB.14.014401

PACS numbers: 29.27.Bd

magnets. However, its assumption is not valid in the case of the ILC main linac.

I. INTRODUCTION In linear accelerators, the transverse beam orbit is induced by tilts of accelerating cavities (deviation of accelerating field direction from designed beam direction) and transverse offset misalignment of quadrupole magnets. As for linear colliders, in cases where very stable and low emittance beam is required, estimating induced orbit and emittance growth due to such random errors is important for evaluation of performance of the linacs. Usually, such estimation is performed using tracking simulations including the Monte Carlo method, with many different sets of random numbers, which tend to take a long time. In the following sections we derive formulas of beam orbit and emittance growth due to cavity tilt and quadrupole magnet offset. Then, results are compared with tracking simulations for the ILC main linac. For simplicity, we only consider one transverse direction, denoting y. It is straightforward to include the other direction. Also, we do not consider effects of wakefield in this report. There were various past studies on analytic and semianalytic estimations of orbit change due to random misalignments in high energy linacs (for example, [1–6]). Our method for beam orbit is basically the same as in these past works and gives similar formulas. However, our method and formulas for emittance growth are new. Reference [1] extensively studied analytic estimation of emittance growth in high energy linear accelerators. Though it gave a formula of emittance growth induced by cavity tilt, it did not consider beam energy spread, which is dominantly important in the ILC main linac. This reference also gave emittance growth with correlated misalignment of magnets and emittance after orbit corrections, but no formula for random misalignment of quadrupole magnets was given. Reference [3] gives expressions of chromatic dependence of the orbit with random misalignment of quadrupole 1098-4402=11=14(1)=014401(8)

II. EFFECT OF TILT OF ACCELERATING CAVITY A. Transverse kick by one cavity with tilt angle In a linac, if a cavity axis deviates from the designed beam direction, beam particles are kicked transversely by the accelerating field of the cavity. For ultrarelativistic beam, the transverse momentum change in a cavity with voltage Vc and tilt angle can be expressed as py;in ¼

eVc sin: c

(1)

We should also consider edge fields of the cavity. (The effect of the edge field can be derived applying Maxwell’s equations to field of an accelerating structure. See, for example, [7].) Using a hard edge model, transverse momentum change at the entrance and the exit of a cavity can be expressed as py;ent ¼ yent

eVc ; 2Lc

(2)

and py;ext ¼ yext

eVc ; 2Lc

(3)

respectively, where yent and yext are transverse offset of the particle with respect to the cavity center, and L is length of the cavity. Then, py;ent þ py;ext ¼ ðyent yext Þ

eVc : 2Lc

(4)

Assuming the beam orbit angle is negligibly small compared with the cavity tilt angle,

014401-1

yent yext ¼ L sin;

(5)

Ó 2011 The American Physical Society

KIYOSHI KUBO

Phys. Rev. ST Accel. Beams 14, 014401 (2011) TABLE I. List of assumptions and approximations for analytic formulas.

and py;ent þ py;ext ¼

eVc 1 sin ¼ py;in : 2 2c

(6)

It means the edge fields reduce the transverse momentum change by a half of that from the field inside the cavity. As total momentum change, we have py ¼

eVc sin: 2c

(7)

Then, transverse angle change (kick angle) is transverse momentum change divided by E=c for ultrarelativistic particles, ¼

eVc eV sin c ; 2E 2E

(8)

where E is the beam energy at the cavity.

Same mean square of tilt angle for all cavities Same mean square of offset for all quadrupole magnets Energy deviation, E, is constant for each particle Large number of components, for taking averages. Uniform FODO lattice Same energy gain in all cavities Take lowest order of energy spread, 2E , for emittance growth

Here, we derive a more simple expression with some approximations. For a long linac with many cavities with the same accelerating voltage, assuming beta function does not depend on the beam energy, we can take averages of i , Ei , and h2i i separately. We also assume the expected tilt angle square is the same for all cavities. Then, we can have i ! ;

B. Orbit due to random tilt of many cavities Summing up effects of all cavities in a linac, the vertical position and angle at the end of linac can be expressed as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ yf ¼ i f sini Ei =Ef i ; i (9) qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ i =f ðcosi f sini Þ Ei =Ef i ; y0f ¼

h2i i ! h2 i;

(12)

where is the average of the beta function. And Z Ef dE X1 1 ¼ ! logðEf =E0 Þ; eVc E0 eVc E i Ei

(13)

where E0 is the initial beam energy. Then,

i

hJf i

eVc logðEf =E0 Þh2 i: 8Ef

(14)

where the summation is taken for all cavities in the linac, f and f are the alpha and beta function at the end of linac, i is the beta function at the ith cavity, i is the betatron phase advance between the ith cavity to the linac end, Ei is the beam energy at the ith cavity, Ef is the beam energy at the end of linac, and i is the kick angle at the ith qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cavity. The factor Ei =Ef represents adiabatic damping due to acceleration. Assuming i is random and independent for each cavity, expected (average) action at the linac end is

Here, we have obtained a simple formula for expected orbit change due to random tilt angle of accelerating cavities. Assumptions and approximations used in this article are listed in Table I. Some of them will be used later.

hJf i hðf y2f þ 2f yf y0f þ f y2f Þ=2i X ¼ i Ei =Ef h2i i=2

C. Emittance growth due to random tilt of many cavities

i

X ¼ i Ei =Ef hðeVc i =2Ei Þ2 i=2 i

¼

1 X ðeVc Þ2 i h2i i ; 8Ef i Ei

(10)

where h i denote average over many sets of random errors and f ð1 þ 2f Þ=f . We used hi j i ¼ ij h2i i:

It is convenient to multiply this by (energy factor) for comparison with normalized emittance, hJf i

(15)

Here, we consider emittance growth from the dispersive effect. Effects of wakefield are ignored, which are usually not very important for superconducting cavities using relatively low rf frequencies, such as for ILC. Dispersive effect is estimated from orbit difference between particles with different energies. Let us assume that position and angle deviation are proportional to energy deviation, as follow: y ¼ E=E

(11)

This formula is suitable for numerical calculation to estimate expected orbit for a given design of a linac, using a computer.

eVc logðEf =E0 Þh2 i: 8mc2

y0 ¼ 0 E=E;

(16)

where E=E is relative energy deviation of a particle, and and 0 are dispersion and angle dispersion, respectively. The square of emittance with such deviation is expressed as

014401-2

ESTIMATION OF ORBIT CHANGE AND EMITTANCE . . . 2 ðy0 y0 Þ2 ½ðy yÞðy 0 y0 Þ2 2 ¼ ðy yÞ ¼ ðy0 þ y y0 yÞ2 ðy00 þ y0 y00 y0 Þ2 2 0 0 0 0 ðy0 þ y y0 yÞðy0 þ y y0 y Þ ¼ 20 þ 0 ½y ðyÞ2 þ 2y yy0 þ y ðy0 Þ2 ¼ 20 þ 0 ½y ðÞ2 þ 2y 0 þ y ð0 Þ2 ðE=EÞ2 ; (17) where overlines denote average over all particles, y0 , y00 , and 0 are position, angle, and emittance without deviation due to the dispersion, y , y , and y are Twiss parameters. So, for evaluating emittance growth, we calculate ½y ðyÞ2 þ 2y yy0 þ y ðy0 Þ2 or ½y ðÞ2 þ 2y 0 þ y ð0 Þ2 , at the end of the linac. First, angle change due to cavity tilt is proportional to the inverse of particle energy. Therefore, for orbit difference due to energy difference, we replace in Eq. (8) by ðE=EÞ, assuming E < E. Then, qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ yf;1 ¼ i f sini Ei =Ef ðE=Ei Þi ; i

y0f;1

X qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ i =f ðcosi f sini Þ ¼

(18)

i

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ei =Ef ðE=Ei Þi :

Phys. Rev. ST Accel. Beams 14, 014401 (2011)

magnet, and q is the phase advance between each quadrupole magnet to the linac end. From Eq. (9), qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ yq ¼ (22) q i siniq Ei =Eq i ; i

Estimation of orbit change and emittance growth due to random misalignment in long linacs Kiyoshi Kubo KEK, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan (Received 15 September 2010; published 4 January 2011) In linear accelerators, the transverse beam orbit is induced by tilts of accelerating cavities (deviation of accelerating field direction from designed beam direction) and transverse offset of quadrupole magnets. Estimating induced orbit and emittance growth due to such random errors is important for evaluation of performance of linacs, especially where stable and low emittance beam is required, such as linear colliders. Usually, such estimations are performed using tracking simulations, including the Monte Carlo method, which tends to take a long time. Here, a much faster and simpler method of quantitative estimation of beam orbit and emittance growth is reported. This method is valid for very long linacs with many components, where statistical treatment of errors is justified. It is shown that the results from the method agree well with tracking simulations for the ILC (International Linear Collider) main linac. DOI: 10.1103/PhysRevSTAB.14.014401

PACS numbers: 29.27.Bd

magnets. However, its assumption is not valid in the case of the ILC main linac.

I. INTRODUCTION In linear accelerators, the transverse beam orbit is induced by tilts of accelerating cavities (deviation of accelerating field direction from designed beam direction) and transverse offset misalignment of quadrupole magnets. As for linear colliders, in cases where very stable and low emittance beam is required, estimating induced orbit and emittance growth due to such random errors is important for evaluation of performance of the linacs. Usually, such estimation is performed using tracking simulations including the Monte Carlo method, with many different sets of random numbers, which tend to take a long time. In the following sections we derive formulas of beam orbit and emittance growth due to cavity tilt and quadrupole magnet offset. Then, results are compared with tracking simulations for the ILC main linac. For simplicity, we only consider one transverse direction, denoting y. It is straightforward to include the other direction. Also, we do not consider effects of wakefield in this report. There were various past studies on analytic and semianalytic estimations of orbit change due to random misalignments in high energy linacs (for example, [1–6]). Our method for beam orbit is basically the same as in these past works and gives similar formulas. However, our method and formulas for emittance growth are new. Reference [1] extensively studied analytic estimation of emittance growth in high energy linear accelerators. Though it gave a formula of emittance growth induced by cavity tilt, it did not consider beam energy spread, which is dominantly important in the ILC main linac. This reference also gave emittance growth with correlated misalignment of magnets and emittance after orbit corrections, but no formula for random misalignment of quadrupole magnets was given. Reference [3] gives expressions of chromatic dependence of the orbit with random misalignment of quadrupole 1098-4402=11=14(1)=014401(8)

II. EFFECT OF TILT OF ACCELERATING CAVITY A. Transverse kick by one cavity with tilt angle In a linac, if a cavity axis deviates from the designed beam direction, beam particles are kicked transversely by the accelerating field of the cavity. For ultrarelativistic beam, the transverse momentum change in a cavity with voltage Vc and tilt angle can be expressed as py;in ¼

eVc sin: c

(1)

We should also consider edge fields of the cavity. (The effect of the edge field can be derived applying Maxwell’s equations to field of an accelerating structure. See, for example, [7].) Using a hard edge model, transverse momentum change at the entrance and the exit of a cavity can be expressed as py;ent ¼ yent

eVc ; 2Lc

(2)

and py;ext ¼ yext

eVc ; 2Lc

(3)

respectively, where yent and yext are transverse offset of the particle with respect to the cavity center, and L is length of the cavity. Then, py;ent þ py;ext ¼ ðyent yext Þ

eVc : 2Lc

(4)

Assuming the beam orbit angle is negligibly small compared with the cavity tilt angle,

014401-1

yent yext ¼ L sin;

(5)

Ó 2011 The American Physical Society

KIYOSHI KUBO

Phys. Rev. ST Accel. Beams 14, 014401 (2011) TABLE I. List of assumptions and approximations for analytic formulas.

and py;ent þ py;ext ¼

eVc 1 sin ¼ py;in : 2 2c

(6)

It means the edge fields reduce the transverse momentum change by a half of that from the field inside the cavity. As total momentum change, we have py ¼

eVc sin: 2c

(7)

Then, transverse angle change (kick angle) is transverse momentum change divided by E=c for ultrarelativistic particles, ¼

eVc eV sin c ; 2E 2E

(8)

where E is the beam energy at the cavity.

Same mean square of tilt angle for all cavities Same mean square of offset for all quadrupole magnets Energy deviation, E, is constant for each particle Large number of components, for taking averages. Uniform FODO lattice Same energy gain in all cavities Take lowest order of energy spread, 2E , for emittance growth

Here, we derive a more simple expression with some approximations. For a long linac with many cavities with the same accelerating voltage, assuming beta function does not depend on the beam energy, we can take averages of i , Ei , and h2i i separately. We also assume the expected tilt angle square is the same for all cavities. Then, we can have i ! ;

B. Orbit due to random tilt of many cavities Summing up effects of all cavities in a linac, the vertical position and angle at the end of linac can be expressed as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ yf ¼ i f sini Ei =Ef i ; i (9) qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ i =f ðcosi f sini Þ Ei =Ef i ; y0f ¼

h2i i ! h2 i;

(12)

where is the average of the beta function. And Z Ef dE X1 1 ¼ ! logðEf =E0 Þ; eVc E0 eVc E i Ei

(13)

where E0 is the initial beam energy. Then,

i

hJf i

eVc logðEf =E0 Þh2 i: 8Ef

(14)

where the summation is taken for all cavities in the linac, f and f are the alpha and beta function at the end of linac, i is the beta function at the ith cavity, i is the betatron phase advance between the ith cavity to the linac end, Ei is the beam energy at the ith cavity, Ef is the beam energy at the end of linac, and i is the kick angle at the ith qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cavity. The factor Ei =Ef represents adiabatic damping due to acceleration. Assuming i is random and independent for each cavity, expected (average) action at the linac end is

Here, we have obtained a simple formula for expected orbit change due to random tilt angle of accelerating cavities. Assumptions and approximations used in this article are listed in Table I. Some of them will be used later.

hJf i hðf y2f þ 2f yf y0f þ f y2f Þ=2i X ¼ i Ei =Ef h2i i=2

C. Emittance growth due to random tilt of many cavities

i

X ¼ i Ei =Ef hðeVc i =2Ei Þ2 i=2 i

¼

1 X ðeVc Þ2 i h2i i ; 8Ef i Ei

(10)

where h i denote average over many sets of random errors and f ð1 þ 2f Þ=f . We used hi j i ¼ ij h2i i:

It is convenient to multiply this by (energy factor) for comparison with normalized emittance, hJf i

(15)

Here, we consider emittance growth from the dispersive effect. Effects of wakefield are ignored, which are usually not very important for superconducting cavities using relatively low rf frequencies, such as for ILC. Dispersive effect is estimated from orbit difference between particles with different energies. Let us assume that position and angle deviation are proportional to energy deviation, as follow: y ¼ E=E

(11)

This formula is suitable for numerical calculation to estimate expected orbit for a given design of a linac, using a computer.

eVc logðEf =E0 Þh2 i: 8mc2

y0 ¼ 0 E=E;

(16)

where E=E is relative energy deviation of a particle, and and 0 are dispersion and angle dispersion, respectively. The square of emittance with such deviation is expressed as

014401-2

ESTIMATION OF ORBIT CHANGE AND EMITTANCE . . . 2 ðy0 y0 Þ2 ½ðy yÞðy 0 y0 Þ2 2 ¼ ðy yÞ ¼ ðy0 þ y y0 yÞ2 ðy00 þ y0 y00 y0 Þ2 2 0 0 0 0 ðy0 þ y y0 yÞðy0 þ y y0 y Þ ¼ 20 þ 0 ½y ðyÞ2 þ 2y yy0 þ y ðy0 Þ2 ¼ 20 þ 0 ½y ðÞ2 þ 2y 0 þ y ð0 Þ2 ðE=EÞ2 ; (17) where overlines denote average over all particles, y0 , y00 , and 0 are position, angle, and emittance without deviation due to the dispersion, y , y , and y are Twiss parameters. So, for evaluating emittance growth, we calculate ½y ðyÞ2 þ 2y yy0 þ y ðy0 Þ2 or ½y ðÞ2 þ 2y 0 þ y ð0 Þ2 , at the end of the linac. First, angle change due to cavity tilt is proportional to the inverse of particle energy. Therefore, for orbit difference due to energy difference, we replace in Eq. (8) by ðE=EÞ, assuming E < E. Then, qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ yf;1 ¼ i f sini Ei =Ef ðE=Ei Þi ; i

y0f;1

X qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ i =f ðcosi f sini Þ ¼

(18)

i

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ei =Ef ðE=Ei Þi :

Phys. Rev. ST Accel. Beams 14, 014401 (2011)

magnet, and q is the phase advance between each quadrupole magnet to the linac end. From Eq. (9), qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ yq ¼ (22) q i siniq Ei =Eq i ; i