Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length...
Transcript of Bandwidth of FFAGs 10 October 2019...Number of cell 1000 Radius 1910 m FODO cell length 12 m Length...
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Bandwidth of FFAGs
10 October 2019
Shinji MachidaSTFC Rutherford Appleton Laboratory
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Contents
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• Background
• Why FFA?
• New Idea of FFA: Vertical excursion FFA
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Remark
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I am using acronym, FFAG, for Fixed Field Alternating Gradient Accelerator in the following presentation.
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FFAs for muon acceleration
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Historical background
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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
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Why FFA for muon acceleration?
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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.
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EMMA achievements (2012)
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Fast accelerationTune change accordingly
Fast accelerationOnly horizontal orbit change
Acceleration in serpentine channelPhase and momentum change as expected
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New development of FFA
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Vertical excursion FFAconcept
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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
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Vertical excursion FFAequation of motion
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For a fixed momentum p, linearised eq. of motion,
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z<latexit sha1_base64="HDzXchlsPlmuEyZZ/9zFJ+iVC6I=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cW7Ae0oWy2k3btZhN2N0IN/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3H1FpHst7M0nQj+hQ8pAzaqzUeOqXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqN3kcRTiBUzgHD66gBndQhyYwQHiGV3hzHpwX5935WLQWnHzmGP7A+fwB6UGM/g==</latexit><latexit sha1_base64="HDzXchlsPlmuEyZZ/9zFJ+iVC6I=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cW7Ae0oWy2k3btZhN2N0IN/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3H1FpHst7M0nQj+hQ8pAzaqzUeOqXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqN3kcRTiBUzgHD66gBndQhyYwQHiGV3hzHpwX5935WLQWnHzmGP7A+fwB6UGM/g==</latexit><latexit sha1_base64="HDzXchlsPlmuEyZZ/9zFJ+iVC6I=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cW7Ae0oWy2k3btZhN2N0IN/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3H1FpHst7M0nQj+hQ8pAzaqzUeOqXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqN3kcRTiBUzgHD66gBndQhyYwQHiGV3hzHpwX5935WLQWnHzmGP7A+fwB6UGM/g==</latexit><latexit sha1_base64="HDzXchlsPlmuEyZZ/9zFJ+iVC6I=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cW7Ae0oWy2k3btZhN2N0IN/QVePCji1Z/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3H1FpHst7M0nQj+hQ8pAzaqzUeOqXK27VnYOsEi8nFchR75e/eoOYpRFKwwTVuuu5ifEzqgxnAqelXqoxoWxMh9i1VNIItZ/ND52SM6sMSBgrW9KQufp7IqOR1pMosJ0RNSO97M3E/7xuasJrP+MySQ1KtlgUpoKYmMy+JgOukBkxsYQyxe2thI2ooszYbEo2BG/55VXSuqh6btVrXFZqN3kcRTiBUzgHD66gBndQhyYwQHiGV3hzHpwX5935WLQWnHzmGP7A+fwB6UGM/g==</latexit>
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
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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](https://reader034.fdocuments.in/reader034/viewer/2022050311/5f732eea31c87043982fc15e/html5/thumbnails/11.jpg)
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=">AAACQnicbVDPSxwxGM1Yrbpqu+qxl+AiKMgyUwoVQRB78WjBVcvOOmSy3+wGk8yQfCM7DvO39dK/wJt/gBcPLeLVg9ndOfijDwKP9973JXlxJoVF37/1Zj7Mzn2cX1hsLC2vfPrcXF07tWluOHR4KlNzHjMLUmjooEAJ55kBpmIJZ/Hlj7F/dgXGilSfYJFBT7GBFongDJ0UNX8dRmVRhRIS3LreGe0UNDRiMMTt/cPID2GUTS11XcuhzVVUin2/uihDoRMsqthtEPWKUR0rLkTUbPltfwL6ngQ1aZEax1HzJuynPFegkUtmbTfwM+yVzKDgEqpGmFvIGL9kA+g6qpkC2ysnFVR00yl9mqTGHY10or6cKJmytlCxSyqGQ/vWG4v/87o5Jru9UugsR9B8elGSS4opHfdJ+8IAR1k4wrgR7q2UD5lhHF3rDVdC8PbL78np13bgt4Of31oHe3UdC+QL2SBbJCDfyQE5IsekQzj5Te7IX/LP++Pdew/e4zQ649Uz6+QVvKdn44uyMw==</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](https://reader034.fdocuments.in/reader034/viewer/2022050311/5f732eea31c87043982fc15e/html5/thumbnails/12.jpg)
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](https://reader034.fdocuments.in/reader034/viewer/2022050311/5f732eea31c87043982fc15e/html5/thumbnails/13.jpg)
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](https://reader034.fdocuments.in/reader034/viewer/2022050311/5f732eea31c87043982fc15e/html5/thumbnails/14.jpg)
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](https://reader034.fdocuments.in/reader034/viewer/2022050311/5f732eea31c87043982fc15e/html5/thumbnails/15.jpg)
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](https://reader034.fdocuments.in/reader034/viewer/2022050311/5f732eea31c87043982fc15e/html5/thumbnails/16.jpg)
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](https://reader034.fdocuments.in/reader034/viewer/2022050311/5f732eea31c87043982fc15e/html5/thumbnails/17.jpg)
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](https://reader034.fdocuments.in/reader034/viewer/2022050311/5f732eea31c87043982fc15e/html5/thumbnails/18.jpg)
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](https://reader034.fdocuments.in/reader034/viewer/2022050311/5f732eea31c87043982fc15e/html5/thumbnails/19.jpg)
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](https://reader034.fdocuments.in/reader034/viewer/2022050311/5f732eea31c87043982fc15e/html5/thumbnails/20.jpg)
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
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To summerise
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![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](https://reader034.fdocuments.in/reader034/viewer/2022050311/5f732eea31c87043982fc15e/html5/thumbnails/22.jpg)
Summary
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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](https://reader034.fdocuments.in/reader034/viewer/2022050311/5f732eea31c87043982fc15e/html5/thumbnails/23.jpg)
Thank you for your attention.
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