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Bandwidth of FFAGs 10 October 2019 Shinji Machida STFC Rutherford Appleton Laboratory

Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

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Page 1: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Bandwidth of FFAGs

10 October 2019

Shinji MachidaSTFC Rutherford Appleton Laboratory

Page 2: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Contents

2

• Background

• Why FFA?

• New Idea of FFA: Vertical excursion FFA

Page 3: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Remark

3

I am using acronym, FFAG, for Fixed Field Alternating Gradient Accelerator in the following presentation.

Page 4: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

FFAs for muon acceleration

4

Page 5: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Historical background

5

Year Publications Experiments

1997

Proposal of FFA for muon acceleration by F. Mills and C. Johnstone at 4th International conference physics potential and development of mu+mu- colliders, UCLA.

2000 PoP FFA at KEK

2001 Neutrino factory design based on FFAG at Nufact01

2006 FFAG for ADSR at Kyoto Univ.

2009 International Design Study (IDS) for the Neutrino Factory

2012 EMMA at Daresbury

Page 6: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Why FFA for muon acceleration?

6

No time to ramp the magnetic field. FFA has wide momentum aperture with fixed field (DC) magnets.

Fast acceleration of muons which have short life time.

No resonance build up within a few or several 10s of turns.

Ring lattice can be simplified with strong quadrupole magnets.This is called linear non-scaling FFA.

EMMA was constructed as a proof of principle.

Electron Model of Muon Acceleration (Many Applications)Fast acceleration of electrons with fixed RF frequency in a serpentine channel.Fast crossing of resonances.

Page 7: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

EMMA achievements (2012)

7

Fast accelerationTune change accordingly

Fast accelerationOnly horizontal orbit change

Acceleration in serpentine channelPhase and momentum change as expected

Page 8: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

New development of FFA

8

Page 9: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Vertical excursion FFAconcept

9

Orbit moves up when the beams are accelerated.

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

z

x and y

hFFAvFFA

Bird’s-eye view

Focusing action in the main body is skew quadrupole alike.

skew

normal

Page 10: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Vertical excursion FFAequation of motion

10

For a fixed momentum p, linearised eq. of motion,

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mid-planeoff on offwhere

beam

d2y

d✓2+ y � nz = 0

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d2z

d✓2+ ny = 0

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✓@⇢

@p

x

= 0<latexit sha1_base64="pcNxgeHg1EjHyfKtdYRgH8rb2fY=">AAACHHicbVDLSgMxFM3UV62vUZdugkWomzKjgt0IBTcuK9hW6JQhk2Y6oZnMkNwRy9APceOvuHGhiBsXgn9j+gC19UDg5Jx7b3JPkAquwXG+rMLS8srqWnG9tLG5tb1j7+61dJIpypo0EYm6DYhmgkvWBA6C3aaKkTgQrB0MLsd++44pzRN5A8OUdWPSlzzklICRfPvUEyyECvZCRWjupUQBJwJ7KkpGP9d0ZBTej+DYv79wfLvsVJ0J8CJxZ6SMZmj49ofXS2gWMwlUEK07rpNCNx8Pp4KNSl6mWUrogPRZx1BJYqa7+WS5ET4ySg+HiTJHAp6ovztyEms9jANTGROI9Lw3Fv/zOhmEtW7OZZoBk3T6UJgJDAkeJ4V7XDEKYmgIoYqbv2IaERMTmDxLJgR3fuVF0jqpuk7VvT4r12uzOIroAB2iCnLROaqjK9RATUTRA3pCL+jVerSerTfrfVpasGY9++gPrM9v0uehxA==</latexit><latexit sha1_base64="pcNxgeHg1EjHyfKtdYRgH8rb2fY=">AAACHHicbVDLSgMxFM3UV62vUZdugkWomzKjgt0IBTcuK9hW6JQhk2Y6oZnMkNwRy9APceOvuHGhiBsXgn9j+gC19UDg5Jx7b3JPkAquwXG+rMLS8srqWnG9tLG5tb1j7+61dJIpypo0EYm6DYhmgkvWBA6C3aaKkTgQrB0MLsd++44pzRN5A8OUdWPSlzzklICRfPvUEyyECvZCRWjupUQBJwJ7KkpGP9d0ZBTej+DYv79wfLvsVJ0J8CJxZ6SMZmj49ofXS2gWMwlUEK07rpNCNx8Pp4KNSl6mWUrogPRZx1BJYqa7+WS5ET4ySg+HiTJHAp6ovztyEms9jANTGROI9Lw3Fv/zOhmEtW7OZZoBk3T6UJgJDAkeJ4V7XDEKYmgIoYqbv2IaERMTmDxLJgR3fuVF0jqpuk7VvT4r12uzOIroAB2iCnLROaqjK9RATUTRA3pCL+jVerSerTfrfVpasGY9++gPrM9v0uehxA==</latexit><latexit sha1_base64="pcNxgeHg1EjHyfKtdYRgH8rb2fY=">AAACHHicbVDLSgMxFM3UV62vUZdugkWomzKjgt0IBTcuK9hW6JQhk2Y6oZnMkNwRy9APceOvuHGhiBsXgn9j+gC19UDg5Jx7b3JPkAquwXG+rMLS8srqWnG9tLG5tb1j7+61dJIpypo0EYm6DYhmgkvWBA6C3aaKkTgQrB0MLsd++44pzRN5A8OUdWPSlzzklICRfPvUEyyECvZCRWjupUQBJwJ7KkpGP9d0ZBTej+DYv79wfLvsVJ0J8CJxZ6SMZmj49ofXS2gWMwlUEK07rpNCNx8Pp4KNSl6mWUrogPRZx1BJYqa7+WS5ET4ySg+HiTJHAp6ovztyEms9jANTGROI9Lw3Fv/zOhmEtW7OZZoBk3T6UJgJDAkeJ4V7XDEKYmgIoYqbv2IaERMTmDxLJgR3fuVF0jqpuk7VvT4r12uzOIroAB2iCnLROaqjK9RATUTRA3pCL+jVerSerTfrfVpasGY9++gPrM9v0uehxA==</latexit><latexit sha1_base64="pcNxgeHg1EjHyfKtdYRgH8rb2fY=">AAACHHicbVDLSgMxFM3UV62vUZdugkWomzKjgt0IBTcuK9hW6JQhk2Y6oZnMkNwRy9APceOvuHGhiBsXgn9j+gC19UDg5Jx7b3JPkAquwXG+rMLS8srqWnG9tLG5tb1j7+61dJIpypo0EYm6DYhmgkvWBA6C3aaKkTgQrB0MLsd++44pzRN5A8OUdWPSlzzklICRfPvUEyyECvZCRWjupUQBJwJ7KkpGP9d0ZBTej+DYv79wfLvsVJ0J8CJxZ6SMZmj49ofXS2gWMwlUEK07rpNCNx8Pp4KNSl6mWUrogPRZx1BJYqa7+WS5ET4ySg+HiTJHAp6ovztyEms9jANTGROI9Lw3Fv/zOhmEtW7OZZoBk3T6UJgJDAkeJ4V7XDEKYmgIoYqbv2IaERMTmDxLJgR3fuVF0jqpuk7VvT4r12uzOIroAB2iCnLROaqjK9RATUTRA3pCL+jVerSerTfrfVpasGY9++gPrM9v0uehxA==</latexit>

✓@n

@p

x

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n = � ⇢

Bz

@Bz

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m =1

Bz

@Bz

@z<latexit sha1_base64="1rdyW6sQDh/30ASkZByw676w2as=">AAACF3icbVBNS8MwGE7n15xfVY9egkPwVFoR3EUYevE4wX3AWkqapVtYmpYkFbbSf+HFv+LFgyJe9ea/Md0K6uYDged9nvdN8j5BwqhUtv1lVFZW19Y3qpu1re2d3T1z/6Aj41Rg0sYxi0UvQJIwyklbUcVILxEERQEj3WB8XfjdeyIkjfmdmiTEi9CQ05BipLTkm1Z06YYC4czJsyt/ms8LN0FCUcRgIf1U09w367ZlzwCXiVOSOijR8s1PdxDjNCJcYYak7Dt2orysuBAzktfcVJIE4TEakr6mHEVEetlsrxyeaGUAw1jowxWcqb8nMhRJOYkC3RkhNZKLXiH+5/VTFTa8jPIkVYTj+UNhyqCKYRESHFBBsGITTRAWVP8V4hHSySgdZU2H4CyuvEw6Z5ZjW87teb3ZKOOogiNwDE6BAy5AE9yAFmgDDB7AE3gBr8aj8Wy8Ge/z1opRzhyCPzA+vgFREqCa</latexit><latexit sha1_base64="1rdyW6sQDh/30ASkZByw676w2as=">AAACF3icbVBNS8MwGE7n15xfVY9egkPwVFoR3EUYevE4wX3AWkqapVtYmpYkFbbSf+HFv+LFgyJe9ea/Md0K6uYDged9nvdN8j5BwqhUtv1lVFZW19Y3qpu1re2d3T1z/6Aj41Rg0sYxi0UvQJIwyklbUcVILxEERQEj3WB8XfjdeyIkjfmdmiTEi9CQ05BipLTkm1Z06YYC4czJsyt/ms8LN0FCUcRgIf1U09w367ZlzwCXiVOSOijR8s1PdxDjNCJcYYak7Dt2orysuBAzktfcVJIE4TEakr6mHEVEetlsrxyeaGUAw1jowxWcqb8nMhRJOYkC3RkhNZKLXiH+5/VTFTa8jPIkVYTj+UNhyqCKYRESHFBBsGITTRAWVP8V4hHSySgdZU2H4CyuvEw6Z5ZjW87teb3ZKOOogiNwDE6BAy5AE9yAFmgDDB7AE3gBr8aj8Wy8Ge/z1opRzhyCPzA+vgFREqCa</latexit><latexit sha1_base64="1rdyW6sQDh/30ASkZByw676w2as=">AAACF3icbVBNS8MwGE7n15xfVY9egkPwVFoR3EUYevE4wX3AWkqapVtYmpYkFbbSf+HFv+LFgyJe9ea/Md0K6uYDged9nvdN8j5BwqhUtv1lVFZW19Y3qpu1re2d3T1z/6Aj41Rg0sYxi0UvQJIwyklbUcVILxEERQEj3WB8XfjdeyIkjfmdmiTEi9CQ05BipLTkm1Z06YYC4czJsyt/ms8LN0FCUcRgIf1U09w367ZlzwCXiVOSOijR8s1PdxDjNCJcYYak7Dt2orysuBAzktfcVJIE4TEakr6mHEVEetlsrxyeaGUAw1jowxWcqb8nMhRJOYkC3RkhNZKLXiH+5/VTFTa8jPIkVYTj+UNhyqCKYRESHFBBsGITTRAWVP8V4hHSySgdZU2H4CyuvEw6Z5ZjW87teb3ZKOOogiNwDE6BAy5AE9yAFmgDDB7AE3gBr8aj8Wy8Ge/z1opRzhyCPzA+vgFREqCa</latexit><latexit sha1_base64="1rdyW6sQDh/30ASkZByw676w2as=">AAACF3icbVBNS8MwGE7n15xfVY9egkPwVFoR3EUYevE4wX3AWkqapVtYmpYkFbbSf+HFv+LFgyJe9ea/Md0K6uYDged9nvdN8j5BwqhUtv1lVFZW19Y3qpu1re2d3T1z/6Aj41Rg0sYxi0UvQJIwyklbUcVILxEERQEj3WB8XfjdeyIkjfmdmiTEi9CQ05BipLTkm1Z06YYC4czJsyt/ms8LN0FCUcRgIf1U09w367ZlzwCXiVOSOijR8s1PdxDjNCJcYYak7Dt2orysuBAzktfcVJIE4TEakr6mHEVEetlsrxyeaGUAw1jowxWcqb8nMhRJOYkC3RkhNZKLXiH+5/VTFTa8jPIkVYTj+UNhyqCKYRESHFBBsGITTRAWVP8V4hHSySgdZU2H4CyuvEw6Z5ZjW87teb3ZKOOogiNwDE6BAy5AE9yAFmgDDB7AE3gBr8aj8Wy8Ge/z1opRzhyCPzA+vgFREqCa</latexit>

Make the orbit and optics independent of p

: same shape of orbit

: invariance of optics

Magnetic field should satisfy

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Finally, vertical field on the mid-plane

Page 11: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Vertical excursion FFAmagnet

11

Magnet is upright rectangular (or sector)Off mid-plane field is expandedas polynomial with y.

Bz (z, x, y) = B0 exp (mz)1X

i=0

bzi (x) yi

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Bx (z, x, y) = B0 exp (mz)1X

i=0

bxi (x) yi

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By (z, x, y) = B0 exp (mz)1X

i=0

byi (x) yi

<latexit sha1_base64="wGkTm8AzpiAyHpJaR0M3i7TAcO4=">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</latexit><latexit sha1_base64="wGkTm8AzpiAyHpJaR0M3i7TAcO4=">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</latexit><latexit sha1_base64="wGkTm8AzpiAyHpJaR0M3i7TAcO4=">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</latexit><latexit sha1_base64="wGkTm8AzpiAyHpJaR0M3i7TAcO4=">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</latexit>

bz0 (x) = g (x)<latexit sha1_base64="v28xb/uMOgXtfoMUEm9+FCf69MY=">AAACEHicbZDLSsNAFIYn9VbrLerSzWAR66YkIiiCUHDjsoK9QBvCZDpJh04uzJyINfQR3Pgqblwo4talO9/GaRtEW38Y+PnOOZw5v5cIrsCyvozCwuLS8kpxtbS2vrG5ZW7vNFWcSsoaNBaxbHtEMcEj1gAOgrUTyUjoCdbyBpfjeuuWScXj6AaGCXNCEkTc55SARq556LnZvTXqCuZD5Q53JQ/6cHQRTAH+Ia5ZtqrWRHje2Lkpo1x11/zs9mKahiwCKohSHdtKwMmIBE4FG5W6qWIJoQMSsI62EQmZcrLJQSN8oEkP+7HULwI8ob8nMhIqNQw93RkS6KvZ2hj+V+uk4J85GY+SFFhEp4v8VGCI8Tgd3OOSURBDbQiVXP8V0z6RhILOsKRDsGdPnjfN46ptVe3rk3LtPI+jiPbQPqogG52iGrpCddRAFD2gJ/SCXo1H49l4M96nrQUjn9lFf2R8fANrVJwk</latexit><latexit sha1_base64="v28xb/uMOgXtfoMUEm9+FCf69MY=">AAACEHicbZDLSsNAFIYn9VbrLerSzWAR66YkIiiCUHDjsoK9QBvCZDpJh04uzJyINfQR3Pgqblwo4talO9/GaRtEW38Y+PnOOZw5v5cIrsCyvozCwuLS8kpxtbS2vrG5ZW7vNFWcSsoaNBaxbHtEMcEj1gAOgrUTyUjoCdbyBpfjeuuWScXj6AaGCXNCEkTc55SARq556LnZvTXqCuZD5Q53JQ/6cHQRTAH+Ia5ZtqrWRHje2Lkpo1x11/zs9mKahiwCKohSHdtKwMmIBE4FG5W6qWIJoQMSsI62EQmZcrLJQSN8oEkP+7HULwI8ob8nMhIqNQw93RkS6KvZ2hj+V+uk4J85GY+SFFhEp4v8VGCI8Tgd3OOSURBDbQiVXP8V0z6RhILOsKRDsGdPnjfN46ptVe3rk3LtPI+jiPbQPqogG52iGrpCddRAFD2gJ/SCXo1H49l4M96nrQUjn9lFf2R8fANrVJwk</latexit><latexit sha1_base64="v28xb/uMOgXtfoMUEm9+FCf69MY=">AAACEHicbZDLSsNAFIYn9VbrLerSzWAR66YkIiiCUHDjsoK9QBvCZDpJh04uzJyINfQR3Pgqblwo4talO9/GaRtEW38Y+PnOOZw5v5cIrsCyvozCwuLS8kpxtbS2vrG5ZW7vNFWcSsoaNBaxbHtEMcEj1gAOgrUTyUjoCdbyBpfjeuuWScXj6AaGCXNCEkTc55SARq556LnZvTXqCuZD5Q53JQ/6cHQRTAH+Ia5ZtqrWRHje2Lkpo1x11/zs9mKahiwCKohSHdtKwMmIBE4FG5W6qWIJoQMSsI62EQmZcrLJQSN8oEkP+7HULwI8ob8nMhIqNQw93RkS6KvZ2hj+V+uk4J85GY+SFFhEp4v8VGCI8Tgd3OOSURBDbQiVXP8V0z6RhILOsKRDsGdPnjfN46ptVe3rk3LtPI+jiPbQPqogG52iGrpCddRAFD2gJ/SCXo1H49l4M96nrQUjn9lFf2R8fANrVJwk</latexit><latexit sha1_base64="v28xb/uMOgXtfoMUEm9+FCf69MY=">AAACEHicbZDLSsNAFIYn9VbrLerSzWAR66YkIiiCUHDjsoK9QBvCZDpJh04uzJyINfQR3Pgqblwo4talO9/GaRtEW38Y+PnOOZw5v5cIrsCyvozCwuLS8kpxtbS2vrG5ZW7vNFWcSsoaNBaxbHtEMcEj1gAOgrUTyUjoCdbyBpfjeuuWScXj6AaGCXNCEkTc55SARq556LnZvTXqCuZD5Q53JQ/6cHQRTAH+Ia5ZtqrWRHje2Lkpo1x11/zs9mKahiwCKohSHdtKwMmIBE4FG5W6qWIJoQMSsI62EQmZcrLJQSN8oEkP+7HULwI8ob8nMhIqNQw93RkS6KvZ2hj+V+uk4J85GY+SFFhEp4v8VGCI8Tgd3OOSURBDbQiVXP8V0z6RhILOsKRDsGdPnjfN46ptVe3rk3LtPI+jiPbQPqogG52iGrpCddRAFD2gJ/SCXo1H49l4M96nrQUjn9lFf2R8fANrVJwk</latexit>

bx0 (x) =1

m

@g

@x<latexit sha1_base64="TvdWKHKM6bFzDqtQj3WRrg4F+fE=">AAACJ3icbZDLSgMxFIYz9VbrrerSTbAIdVNmRFAEpeDGZQV7gc4wZNJMG5q5kJyRlmHexo2v4kZQEV36JqbtgNr6Q+DjP+ckOb8XC67AND+NwtLyyupacb20sbm1vVPe3WupKJGUNWkkItnxiGKCh6wJHATrxJKRwBOs7Q2vJ/X2PZOKR+EdjGPmBKQfcp9TAtpyy1eem47MzBbMh+oI25L3B3B8afuS0NTK0iCboR0TCZwI3M9+eJS55YpZM6fCi2DlUEG5Gm75xe5FNAlYCFQQpbqWGYOTTi6kgmUlO1EsJnRI+qyrMSQBU0463TPDR9rpYT+S+oSAp+7viZQESo0DT3cGBAZqvjYx/6t1E/DPnZSHcQIspLOH/ERgiPAkNNzjklEQYw2ESq7/iumA6FxAR1vSIVjzKy9C66RmmTXr9rRSv8jjKKIDdIiqyEJnqI5uUAM1EUUP6Am9ojfj0Xg23o2PWWvByGf20R8ZX9/3Hac2</latexit><latexit sha1_base64="TvdWKHKM6bFzDqtQj3WRrg4F+fE=">AAACJ3icbZDLSgMxFIYz9VbrrerSTbAIdVNmRFAEpeDGZQV7gc4wZNJMG5q5kJyRlmHexo2v4kZQEV36JqbtgNr6Q+DjP+ckOb8XC67AND+NwtLyyupacb20sbm1vVPe3WupKJGUNWkkItnxiGKCh6wJHATrxJKRwBOs7Q2vJ/X2PZOKR+EdjGPmBKQfcp9TAtpyy1eem47MzBbMh+oI25L3B3B8afuS0NTK0iCboR0TCZwI3M9+eJS55YpZM6fCi2DlUEG5Gm75xe5FNAlYCFQQpbqWGYOTTi6kgmUlO1EsJnRI+qyrMSQBU0463TPDR9rpYT+S+oSAp+7viZQESo0DT3cGBAZqvjYx/6t1E/DPnZSHcQIspLOH/ERgiPAkNNzjklEQYw2ESq7/iumA6FxAR1vSIVjzKy9C66RmmTXr9rRSv8jjKKIDdIiqyEJnqI5uUAM1EUUP6Am9ojfj0Xg23o2PWWvByGf20R8ZX9/3Hac2</latexit><latexit sha1_base64="TvdWKHKM6bFzDqtQj3WRrg4F+fE=">AAACJ3icbZDLSgMxFIYz9VbrrerSTbAIdVNmRFAEpeDGZQV7gc4wZNJMG5q5kJyRlmHexo2v4kZQEV36JqbtgNr6Q+DjP+ckOb8XC67AND+NwtLyyupacb20sbm1vVPe3WupKJGUNWkkItnxiGKCh6wJHATrxJKRwBOs7Q2vJ/X2PZOKR+EdjGPmBKQfcp9TAtpyy1eem47MzBbMh+oI25L3B3B8afuS0NTK0iCboR0TCZwI3M9+eJS55YpZM6fCi2DlUEG5Gm75xe5FNAlYCFQQpbqWGYOTTi6kgmUlO1EsJnRI+qyrMSQBU0463TPDR9rpYT+S+oSAp+7viZQESo0DT3cGBAZqvjYx/6t1E/DPnZSHcQIspLOH/ERgiPAkNNzjklEQYw2ESq7/iumA6FxAR1vSIVjzKy9C66RmmTXr9rRSv8jjKKIDdIiqyEJnqI5uUAM1EUUP6Am9ojfj0Xg23o2PWWvByGf20R8ZX9/3Hac2</latexit><latexit sha1_base64="TvdWKHKM6bFzDqtQj3WRrg4F+fE=">AAACJ3icbZDLSgMxFIYz9VbrrerSTbAIdVNmRFAEpeDGZQV7gc4wZNJMG5q5kJyRlmHexo2v4kZQEV36JqbtgNr6Q+DjP+ckOb8XC67AND+NwtLyyupacb20sbm1vVPe3WupKJGUNWkkItnxiGKCh6wJHATrxJKRwBOs7Q2vJ/X2PZOKR+EdjGPmBKQfcp9TAtpyy1eem47MzBbMh+oI25L3B3B8afuS0NTK0iCboR0TCZwI3M9+eJS55YpZM6fCi2DlUEG5Gm75xe5FNAlYCFQQpbqWGYOTTi6kgmUlO1EsJnRI+qyrMSQBU0463TPDR9rpYT+S+oSAp+7viZQESo0DT3cGBAZqvjYx/6t1E/DPnZSHcQIspLOH/ERgiPAkNNzjklEQYw2ESq7/iumA6FxAR1vSIVjzKy9C66RmmTXr9rRSv8jjKKIDdIiqyEJnqI5uUAM1EUUP6Am9ojfj0Xg23o2PWWvByGf20R8ZX9/3Hac2</latexit>

by0 (x) = 0<latexit sha1_base64="4tWfJEOn9lWyOxNFwB51GsfpM24=">AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIdVMmIiiCUHDjsoJ9QBPCZDpph04ezNyIIXbhr7hxoYhbf8Odf+O0zUJbD1w4nHMv997jJ4IrwPjbKC0tr6yuldcrG5tb2zvm7l5bxamkrEVjEcuuTxQTPGIt4CBYN5GMhL5gHX90PfE790wqHkd3kCXMDckg4gGnBLTkmQe+l2d47AgWQO3BkXwwhJMr7JlVXMdTWIvELkgVFWh65pfTj2kasgioIEr1bJyAmxMJnAo2rjipYgmhIzJgPU0jEjLl5tP7x9axVvpWEEtdEVhT9fdETkKlstDXnSGBoZr3JuJ/Xi+F4MLNeZSkwCI6WxSkwoLYmoRh9blkFESmCaGS61stOiSSUNCRVXQI9vzLi6R9Wrdx3b49qzYuizjK6BAdoRqy0TlqoBvURC1E0SN6Rq/ozXgyXox342PWWjKKmX30B8bnD0+/lZg=</latexit><latexit sha1_base64="4tWfJEOn9lWyOxNFwB51GsfpM24=">AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIdVMmIiiCUHDjsoJ9QBPCZDpph04ezNyIIXbhr7hxoYhbf8Odf+O0zUJbD1w4nHMv997jJ4IrwPjbKC0tr6yuldcrG5tb2zvm7l5bxamkrEVjEcuuTxQTPGIt4CBYN5GMhL5gHX90PfE790wqHkd3kCXMDckg4gGnBLTkmQe+l2d47AgWQO3BkXwwhJMr7JlVXMdTWIvELkgVFWh65pfTj2kasgioIEr1bJyAmxMJnAo2rjipYgmhIzJgPU0jEjLl5tP7x9axVvpWEEtdEVhT9fdETkKlstDXnSGBoZr3JuJ/Xi+F4MLNeZSkwCI6WxSkwoLYmoRh9blkFESmCaGS61stOiSSUNCRVXQI9vzLi6R9Wrdx3b49qzYuizjK6BAdoRqy0TlqoBvURC1E0SN6Rq/ozXgyXox342PWWjKKmX30B8bnD0+/lZg=</latexit><latexit sha1_base64="4tWfJEOn9lWyOxNFwB51GsfpM24=">AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIdVMmIiiCUHDjsoJ9QBPCZDpph04ezNyIIXbhr7hxoYhbf8Odf+O0zUJbD1w4nHMv997jJ4IrwPjbKC0tr6yuldcrG5tb2zvm7l5bxamkrEVjEcuuTxQTPGIt4CBYN5GMhL5gHX90PfE790wqHkd3kCXMDckg4gGnBLTkmQe+l2d47AgWQO3BkXwwhJMr7JlVXMdTWIvELkgVFWh65pfTj2kasgioIEr1bJyAmxMJnAo2rjipYgmhIzJgPU0jEjLl5tP7x9axVvpWEEtdEVhT9fdETkKlstDXnSGBoZr3JuJ/Xi+F4MLNeZSkwCI6WxSkwoLYmoRh9blkFESmCaGS61stOiSSUNCRVXQI9vzLi6R9Wrdx3b49qzYuizjK6BAdoRqy0TlqoBvURC1E0SN6Rq/ozXgyXox342PWWjKKmX30B8bnD0+/lZg=</latexit><latexit sha1_base64="4tWfJEOn9lWyOxNFwB51GsfpM24=">AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIdVMmIiiCUHDjsoJ9QBPCZDpph04ezNyIIXbhr7hxoYhbf8Odf+O0zUJbD1w4nHMv997jJ4IrwPjbKC0tr6yuldcrG5tb2zvm7l5bxamkrEVjEcuuTxQTPGIt4CBYN5GMhL5gHX90PfE790wqHkd3kCXMDckg4gGnBLTkmQe+l2d47AgWQO3BkXwwhJMr7JlVXMdTWIvELkgVFWh65pfTj2kasgioIEr1bJyAmxMJnAo2rjipYgmhIzJgPU0jEjLl5tP7x9axVvpWEEtdEVhT9fdETkKlstDXnSGBoZr3JuJ/Xi+F4MLNeZSkwCI6WxSkwoLYmoRh9blkFESmCaGS61stOiSSUNCRVXQI9vzLi6R9Wrdx3b49qzYuizjK6BAdoRqy0TlqoBvURC1E0SN6Rq/ozXgyXox342PWWjKKmX30B8bnD0+/lZg=</latexit>

where

Non zero longitudinal field on mid-plane is something we did not have in a conventional accelerator.

-4

-3

-2

-1

0

1

2

3

4

-1 -0.5 0 0.5 1

Bx,y,z [T]

x [m]

BxByBz

Slightly off mid-plane

Page 12: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Why vertical excursion is good for muons?

12

Page 13: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Vertical excursion FFAInventors

13

Proposal of “electron cyclotron”

“If path length can be kept constant in the fixed field accelerator, ultra relativistic particles like electrons and muons can be accelerated continuously with fixed RF frequency.”

Stephen Brooks (RAL and now BNL) reinvented vFFA 50 years later.

Bull. APS 30, 20 (1955)

by Tihiro Ohkawa

Page 14: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Advantage of vFFAsummary

14

Fixed field No need to ramp the magnetic field unlike RCS. It has huge momentum acceptance.

Zero chromaticity

Wide momentum range of acceleration such as a factor of 30 with fixed field magnets within a reasonable aperture.

Fixed path length like linac

For ultra relativistic particles, revolution frequency is constant. High-Q fixed frequency cavity can be used.

Scaling condition is

separate from geometry

Shape of the ring is independent of scaling condition. Long straight insertion is a possibility.

Page 15: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

First design out of one day work10 to 300 GeV/c muon accelerator

15

Momentum range 10 - 300 GeV/cCircumference 12 km*Maximum field 12 T*Number of cell 1000

Radius 1910 mFODO cell length 12 m

Length of straight section 2 mLength of magnets 4 m

Field index m 8Orbit excursion 0.425 m

Tune per cell in decoupled space (0.3109, 0.2239)

* this is simply because the design is not optimised.

Page 16: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Orbitsper cell

16

Top view

Side view

(Expanded)

0.45

m

0.36 degree bend per cell

Bd Bf

Page 17: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Magnetic field along the orbitadjusting operation

17

Bx: Longitudinal

By: Horizontal

Bz: Vertical+15 T

-15 T

+15 T

-15 T

+1.5 T

-1.5 T

Significant cancellation of the normal bending by reverse bending in this design.

12 m

Page 18: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Superconducting magnetsseveral configurations

18

Several options to realise the fields.

Flat coil

Bedstead

Two solenoids

beambeam

beam

(I. Rodriguez, et al)

Page 19: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

As a proton drive

19

Page 20: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Preliminary design of FFA proton driver

20

Kinetic energy 0.4 to 1.2 GeVNumber of cell 25

Cell length 6 mStraight length 2.4 m

Field index 0.88 m^{-1} +/- ?%Magnet length 2 x 0.25 m with 0.1 m space

Bd/Bf ?Angle between 2 segments 0 for Bd, 0 for Bf (see figure)

Relative displacement between Bd and Bf 0.350 +/- ? mInner aperture ? mm^2Coupling angle Depends on tune

Nominal cell tune (0.2252, 0.1366)Orbit excursion 0.79 m when m=0.88

Fringe extent of Tanh 0.30 m

Page 21: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

To summerise

21

Page 22: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Summary

22

Fixed field magnet of FFA is advantage for fast acceleration of muons.Fixed RF frequency of vFFA for muon acceleration is another advantage.vFFA can have wide momentum acceptance with reasonable aperture. A factor of 30 is shown as an example.

Circumference factor (~R/rho) is relatively large due to reverse bend. Elimination or minimisation of reverse bends should be looked at.The present lattice does not have long straight section for injection, extraction, RF, collimation, etc. This should be possible, but need to be demonstrated.

Page 23: Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length of straight section 2 m Length of magnets 4 m Field index m 8 Orbit excursion 0.425

Thank you for your attention.

23