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Particle Transport, Beams, and Emittance
Mike Syphers Northern Illinois University & Fermilab
Operator Lectures April 2020
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The Problem§ 1927: Lord Rutherford requested a “copious supply” of projectiles
“more energetic than natural alpha and beta particles” for use in scientific studies of nature
2M. Syphers Ops Lecture 1 SPRING 2020
requirements:position (X, Y, Z )angles (x’, y’ )time (t)kinetic energy (W)…
source
• For given type of particle, create an ideal system to provide particles to a final location with desired trajectory, desired kinetic energy per particle, at the desired time
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The Problem§ 1927: Lord Rutherford requested a “copious supply” of projectiles
“more energetic than natural alpha and beta particles” for use in scientific studies of nature
2M. Syphers Ops Lecture 1 SPRING 2020
requirements:position (X, Y, Z )angles (x’, y’ )time (t)kinetic energy (W)…
source
• For given type of particle, create an ideal system to provide particles to a final location with desired trajectory, desired kinetic energy per particle, at the desired timeand within tolerable spreads of these quantities
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The Problem§ 1927: Lord Rutherford requested a “copious supply” of projectiles
“more energetic than natural alpha and beta particles” for use in scientific studies of nature
2M. Syphers Ops Lecture 1 SPRING 2020
requirements:position (X, Y, Z )angles (x’, y’ )time (t)kinetic energy (W)…
source
• For given type of particle, create an ideal system to provide particles to a final location with desired trajectory, desired kinetic energy per particle, at the desired timeand within tolerable spreads of these quantities
within dX, dY, dt, dW, …
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A Few Words on Particle Sources…
• Electrons — relatively easy‣ filaments; photocathodes, laser driven plasmas,…
• Protons — not “too” hard ‣ ionized hydrogen gas, plasma sources,…
• Ions — similar techniques‣ ovens, plasma sources, ECRs — plus, separation
• Even more exotic particles: target, separate, collect‣ heavy ion isotopes‣ pions, muons, antiprotons, neutrinos,…
• Also polarized sources, …
3M. Syphers Ops Lecture 1 SPRING 2020
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Single-Pass vs. Repetitive Systems
§ Beam Transport (from point A to point B)
§ Acceleration along the way • single-pass with acceleration
• multi-pass acceleration
4M. Syphers Ops Lecture 1 SPRING 2020
may need motion in such a system to be stable for many (millions or more?) revolutions
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Stability of Motion Near the Ideal
• Not all particles (any??) begin “on” the design trajectory with exactly the ideal energy/momentum
• We wish to have a system that will keep particles near the ideal conditions as they are transported (and possibly accelerated) through the system
• Particles emerge from their “source” with a slight divergence and will need to be guided back toward the ideal trajectory
• Also, particles with different energies/momenta will travel at different speeds, and hence may not arrive at cavities, experiments, etc., at the ideal time
5M. Syphers Ops Lecture 1 SPRING 2020
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Reduction of the Problem
• Will treat transverse motion of particles through the accelerator as independent of the longitudinal motion, and study these two cases separately. Must show along the way that this is viable approach.
• Certainly not always be the case...‣ electric fields used for focusing at low energies can also
accelerate the particles as well;‣ fields in the gaps of cavities will have focusing effects; etc.
• However, much of the “cross talk” can be minimized, and for much of the particle’s journey, especially at higher energies, the major transverse focusing can be performed by magnetic fields -- particle’s energy not changed
• Look at “linear” fields, i.e. linear restoring forces
6M. Syphers Ops Lecture 1 SPRING 2020
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Outline§ Today: Particle Transport, Beams, and Emittance
• Investigate transverse motion of single particle in magnetic fields • Discuss properties of a “beam” of particles • Phase space and the beam “ellipse” • Transport of beam phase space properties
• Courant-Snyder Approach to Linear Transverse Dynamics
• Acceleration and Longitudinal Dynamics
• Linear and Nonlinear Imperfections and Adjustments
7M. Syphers Ops Lecture 1 SPRING 2020
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Equations of Motion§ Lorentz Force:
§ Magnetic Rigidity • particle of unit charge, q = e:
§ Reference Trajectory • Local Coordinate System
8M. Syphers Ops Lecture 1 SPRING 2020
xy
s
B⇢ ⌘ p
q=
p
e
~F = q( ~E + ~v ⇥ ~B)
(γm)v2
ρ= qvB ⟶ γmv/q = B ⋅ ρ
Bρ[T ⋅ m] = (10/2.9979) ⋅ p[GeV/c]
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Linear Magnetic Fields for Guiding & Focusing
Xd(z) = designX(z) = actual
z
X(z)
Xd(z)ρ
9M. Syphers Ops Lecture 1 SPRING 2020
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Linear Magnetic Fields for Guiding & Focusing
Xd(z) = designX(z) = actual
z
X(z)
Xd(z)
X(z)
ρ
9M. Syphers Ops Lecture 1 SPRING 2020
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Linear Magnetic Fields for Guiding & Focusing
Xd(z) = designX(z) = actual
z
X(z)
Xd(z)
X(z)s
x
ρ
9M. Syphers Ops Lecture 1 SPRING 2020
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Linear Magnetic Fields for Guiding & Focusing
Xd(z) = designX(z) = actual
z
X(z)
Xd(z)
X(z)s
x
x’ = dx/ds
ρ
9M. Syphers Ops Lecture 1 SPRING 2020
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Linear Transverse Fields (Bz = 0)
M. Syphers Ops Lecture 1 SPRING 2020
Drift Space Bx = 0 By = 0
Bending Region (dipole magnet) Bx = 0 By = B0
Focusing Region (quadrupole magnet) Bx = B’ y By = B’ x
(electrostatic quadrupole) Ey = -E’ y Ex = E’ x
Combined Function Magnet Bx = B’y By = B0 + B’ x
Uniform magnet + ES quad (a’ la Muon g-2 ring)
Bx = 0,Ey = -E’ y
By = B0,Ex = E’ x
Note: B’, E’ are “gradients” — change in field per distance: T/m, (V/m)/m
Think “simple harmonic oscillator”: F ∝ − x ⟶ ··x + k ⋅ x = 0 ⟶ x ∼ a sin(ωt) + b cos(ωt)
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• Quadrupole magnet: • field = 0 on longitudinal axis • varies linearly with transverse position
The Quadrupole Magnet• Dipole magnet: • uniform bend field
11M. Syphers Ops Lecture 1 SPRING 2020
Dipole,FNAL
B = ∇Φm ⟶ ∇2Φm = 0
Φm = Crn sin nϕ
Br = nCrn−1 sin nϕBϕ = nCrn−1 cos nϕ
2n-pole magnetn = 2 (quadrupole):
Bx = 2C y ≡ B′ yBy = 2C x ≡ B′ x
Φm = constant
Field: unit of Tesla (T)
B’ = Gradient of Field: unit of T/m
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Linear Restoring Forces — Short Version
12M. Syphers Ops Lecture 1 SPRING 2020
(γm)v2
ρ= evB0
ρ
B0ρ =(γm)v
e= p/e
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Linear Restoring Forces — Short Version
12M. Syphers Ops Lecture 1 SPRING 2020
(γm)v2
ρ= evB0
ρ
B0ρ =(γm)v
e= p/e
Δθ ρ
Δs
Δθ =Δsρ
=B ⋅ Δs
Bρ
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Linear Restoring Forces — Short Version
12M. Syphers Ops Lecture 1 SPRING 2020
(γm)v2
ρ= evB0
ρ
B0ρ =(γm)v
e= p/e
Δθ ρ
Δs
Δθ =Δsρ
=B ⋅ Δs
Bρ
B = B′ ⋅ xsuppose:
Look relative to the ideal trajectory:
s
Δx′ = −BΔsBρ
= −B′ ΔsBρ
⋅ x
Δs
(x, x′ )
x′ ≡ dx /ds
x(s)here, By = B′ ⋅ x
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Linear Restoring Forces — Short Version
12M. Syphers Ops Lecture 1 SPRING 2020
(γm)v2
ρ= evB0
ρ
B0ρ =(γm)v
e= p/e
Δθ ρ
Δs
Δθ =Δsρ
=B ⋅ Δs
Bρ
Δx′
Δs≡ x′ ′ = −
B′
Bρ⋅ x
x′ ′ +B′ (s)Bρ
x = 0
B = B′ ⋅ xsuppose:
Look relative to the ideal trajectory:
s
Δx′ = −BΔsBρ
= −B′ ΔsBρ
⋅ x
Δs
(x, x′ )
x′ ≡ dx /ds
x(s)here, By = B′ ⋅ x
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Linear Restoring Forces - Details• Assume general linear guide fields: --
• Look at radial motion:
By = B0 + B’xBx = B’y
xds
ρ
dθ
dt =⇢+ x
⇢
ds
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sha1_base64="lImfj94tsVaKF00SGkkH89qStMA=">AAACEXicbZDLSsNAFIYn9VbrLerSzWARCmJJ3FgQsehClxXsBZpQJpNJO3RyYWZSWkJewY2v4saFIm5d6c6Vr+I0caGtB4b5+P9zmDm/EzEqpGF8aoWFxaXlleJqaW19Y3NL395piTDmmDRxyELecZAgjAakKalkpBNxgnyHkbYzvJz67RHhgobBrZxExPZRP6AexUgqqadXXAnPoOVxhBOLD8LDcZrdqXWai65Ik1FPpD29bFSNrOA8mD9Qrp9/vY+PaleNnv5huSGOfRJIzJAQXdOIpJ0gLilmJC1ZsSARwkPUJ12FAfKJsJNsoxQeKMWFXsjVCSTM1N8TCfKFmPiO6vSRHIhZbyr+53Vj6dXshAZRLEmA84e8mEEZwmk80KWcYMkmChDmVP0V4gFSQUgVYkmFYM6uPA+t46qp+EalcQHyKoI9sA8qwAQnoA6uQQM0AQZ34AE8gWftXnvUXrTXvLWg/czsgj+lvX0D2JShUg==</latexit><latexit sha1_base64="lImfj94tsVaKF00SGkkH89qStMA=">AAACEXicbZDLSsNAFIYn9VbrLerSzWARCmJJ3FgQsehClxXsBZpQJpNJO3RyYWZSWkJewY2v4saFIm5d6c6Vr+I0caGtB4b5+P9zmDm/EzEqpGF8aoWFxaXlleJqaW19Y3NL395piTDmmDRxyELecZAgjAakKalkpBNxgnyHkbYzvJz67RHhgobBrZxExPZRP6AexUgqqadXXAnPoOVxhBOLD8LDcZrdqXWai65Ik1FPpD29bFSNrOA8mD9Qrp9/vY+PaleNnv5huSGOfRJIzJAQXdOIpJ0gLilmJC1ZsSARwkPUJ12FAfKJsJNsoxQeKMWFXsjVCSTM1N8TCfKFmPiO6vSRHIhZbyr+53Vj6dXshAZRLEmA84e8mEEZwmk80KWcYMkmChDmVP0V4gFSQUgVYkmFYM6uPA+t46qp+EalcQHyKoI9sA8qwAQnoA6uQQM0AQZ34AE8gWftXnvUXrTXvLWg/czsgj+lvX0D2JShUg==</latexit><latexit sha1_base64="kTwAwBkfNhcozY+Hn5OuPnfqEUk=">AAACEXicbZBNS8MwGMfT+TbnW9Wjl+AQBsJovSiIMPTicYJ7gbWUNE23sLQpSTocpV/Bi1/FiwdFvHrz5rcx63rQzQdCfvz/z0Py/P2EUaks69uorKyurW9UN2tb2zu7e+b+QVfyVGDSwZxx0feRJIzGpKOoYqSfCIIin5GeP76Z+b0JEZLy+F5NE+JGaBjTkGKktOSZjUDBK+iEAuHMESN++pAXd+5czsVA5tnEk7ln1q2mVRRcBruEOiir7ZlfTsBxGpFYYYakHNhWotwMCUUxI3nNSSVJEB6jIRlojFFEpJsVG+XwRCsBDLnQJ1awUH9PZCiSchr5ujNCaiQXvZn4nzdIVXjhZjROUkViPH8oTBlUHM7igQEVBCs21YCwoPqvEI+QDkLpEGs6BHtx5WXonjVtzXdWvXVdxlEFR+AYNIANzkEL3II26AAMHsEzeAVvxpPxYrwbH/PWilHOHII/ZXz+AF32nf8=</latexit>
ds
dt= vs
⇢
r<latexit sha1_base64="RZYPlfknIAg9No/0xLWjyqR/4Jg=">AAACCnicbZC7SgNBFIbPxluMt1VLm9EgWIVdG9MIQRvLCOYC2bDMzs4mQ2YvzMwGwrK1ja9iY6GIrQ8gdla+ipNsCk38YeDjP+dw5vxewplUlvVllFZW19Y3ypuVre2d3T1z/6At41QQ2iIxj0XXw5JyFtGWYorTbiIoDj1OO97oelrvjKmQLI7u1CSh/RAPIhYwgpW2XPPYCQQmmS/zzFc5ukRjVxaWI4ZxnoncNatWzZoJLYM9h2qj/v2BAKDpmp+OH5M0pJEiHEvZs61E9TMsFCOc5hUnlTTBZIQHtKcxwiGV/Wx2So5OteOjIBb6RQrN3N8TGQ6lnISe7gyxGsrF2tT8r9ZLVVDvZyxKUkUjUiwKUo5UjKa5IJ8JShSfaMBEMP1XRIZYB6F0ehUdgr148jK0z2u25ludxhUUKsMRnMAZ2HABDbiBJrSAwD08wjO8GA/Gk/FqvBWtJWM+cwh/ZLz/AG3ynZM=</latexit><latexit sha1_base64="AwngOTz7fF2TqXiQI7UPIRVSrZM=">AAACCnicbZA9S8NAGMcv9a3Wt6iDiMtpEZxK4mIXoejiWMG+QBvC5XJpj15y4e5SKCGzi1/FxUERVz+AuOni6sfw2nTQ1j8c/Pg/z8Nzz9+LGZXKsj6MwsLi0vJKcbW0tr6xuWVu7zQlTwQmDcwZF20PScJoRBqKKkbasSAo9BhpeYPLcb01JEJSHt2oUUycEPUiGlCMlLZc87AbCIRTX2aprzJ4DoeuzK2u6PMsFZlrlq2KNRGcB3sK5Vr1623v83u/7prvXZ/jJCSRwgxJ2bGtWDkpEopiRrJSN5EkRniAeqSjMUIhkU46OSWDx9rxYcCFfpGCE/f3RIpCKUehpztDpPpytjY2/6t1EhVUnZRGcaJIhPNFQcKg4nCcC/SpIFixkQaEBdV/hbiPdBBKp1fSIdizJ89D87Ria77WaVyAXEVwAI7ACbDBGaiBK1AHDYDBLbgHj+DJuDMejGfjJW8tGNOZXfBHxusP3UufZQ==</latexit><latexit sha1_base64="4W+8vIsz3Gf03coZSJnmxIlSTAM=">AAACCnicbZC7TsMwFIYdrqXcAowshgqJqUpYYEGqYGEsEr1IbRQ5jtNadZzIPqlURZlZeBUWBhBi5QnYeBvcNgO0/JKlT/85R8fnD1LBNTjOt7Wyura+sVnZqm7v7O7t2weHbZ1kirIWTUSiugHRTHDJWsBBsG6qGIkDwTrB6HZa74yZ0jyRDzBJmReTgeQRpwSM5dsn/UgRmoe6yEMo8DUe+3pu9dUwKXJV+HbNqTsz4WVwS6ihUk3f/uqHCc1iJoEKonXPdVLwcqKAU8GKaj/TLCV0RAasZ1CSmGkvn51S4DPjhDhKlHkS8Mz9PZGTWOtJHJjOmMBQL9am5n+1XgbRlZdzmWbAJJ0vijKBIcHTXHDIFaMgJgYIVdz8FdMhMUGASa9qQnAXT16G9kXdNXzv1Bo3ZRwVdIxO0Tly0SVqoDvURC1E0SN6Rq/ozXqyXqx362PeumKVM0foj6zPH1onm0w=</latexit>
13M. Syphers Ops Lecture 1 SPRING 2020
d2
dt2=
✓ds
dt
◆2 d2
ds2=
⇣vs
⇢
r
⌘2 d2
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Use cylindrical coordinates
r = ρ + x, θ, y
acceleration:
··y =ev2
s Bx
p
··r − r ·θ2 = −ev2
s By
p
![Page 22: Particle Transport, Beams, and Emittancenicadd.niu.edu/~syphers/OpsTalks/pdfs/OpsLecture1.pdf · 2020. 5. 28. · § Today: Particle Transport, Beams, and Emittance • Investigate](https://reader035.vdocuments.us/reader035/viewer/2022070221/6137724d0ad5d20676489ffd/html5/thumbnails/22.jpg)
Linear Restoring Forces - Details• Assume general linear guide fields: --
• Look at radial motion:
By = B0 + B’xBx = B’y
14M. Syphers Ops Lecture 1 SPRING 2020
r � r✓2 = �ev2sBy
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v2s
⇣⇢r
⌘2x00 � (⇢+ x)
⇣vsr
⌘2= �ev2sBy
p<latexit sha1_base64="HsRjQ9rECbcrPd6mORnmpEU+Ob8=">AAACT3icbVHLSiNBFL0dHR9x1DizdFMoYkSU7mx0I4qzcalgVEjHprpyOymsflB1WwxNw3ygG2c3q/mH2bhwGKxOAj4PFBzOObcep8JMSUOu+9upTU1/mZmdm68vfF1cWm6sfLswaa4FtkWqUn0VcoNKJtgmSQqvMo08DhVehjc/Kv/yFrWRaXJOwwy7Me8nMpKCk5WCRnQbmOuWrzCiph9pLgpfD9Ky0CXztewPaOu6xe42N9kOa1YO277bep2241X4JXvAdsYOjnY+DoZlkZVBY93ddUdgH4k3IetHh38efgLAadD45fdSkceYkFDcmI7nZtQtuCYpFJZ1PzeYcXHD+9ixNOExmm4x6qNkG1bpsSjVdiXERurriYLHxgzj0CZjTgPz3qvEz7xOTtF+t5BJlhMmYnxQlCtGKavKZT2pUZAaWsKFlvauTAy4bYPsF9RtCd77J38kF61dz/Iz28YxjDEHq7AGTfBgD47gBE6hDQLu4S88wT/nwXl0/tcm0ZozId/hDWrzz9PYthY=</latexit><latexit sha1_base64="JZzp+aWNJUgos4BaT8xkBcyRvR0=">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</latexit><latexit sha1_base64="JZzp+aWNJUgos4BaT8xkBcyRvR0=">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</latexit><latexit sha1_base64="9pnSPXf1U95+ppBzZRgm3XD6Ikg=">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</latexit>
x00 � ⇢+ x
⇢2= �eBy
p
✓r
⇢
◆2
<latexit sha1_base64="IJwpHrnp/ThR02O6e+c8lIFFHNw=">AAACNXicbZBNSyNBEIZrXNeP7KpxPXppFNFFlJlc9LLgx8WDBwWjQiaGnk5N0qTng+6axTDMn9KD/8OTHgQV8epfsJPx4FdBw8P7VlFdb5Aqach1b52RH6M/x8YnJiu/fk9Nz1Rn/xybJNMC6yJRiT4NuEElY6yTJIWnqUYeBQpPgt7uwD/5j9rIJD6iforNiHdiGUrByUqt6j47X15ma8wPNRe5r7sJWz0vhnBWK9g/tlY6uNPqF3la+ApDWik1XfYVvpadLv09q7Wqi+66Oyz2Fbw3WNzadi/vAeCgVb3224nIIoxJKG5Mw3NTauZckxQKi4qfGUy56PEONizGPELTzIdXF2zJKm0WJtq+mNhQfT+R88iYfhTYzohT13z2BuJ3XiOjcLOZyzjNCGNRLgozxShhgwhZW2oUpPoWuNDS/pWJLreJkA26YkPwPp/8FY5r657lQ5vGDpQ1AfOwACvgwQZswR4cQB0EXMANPMCjc+XcOU/Oc9k64rzNzMGHcl5eAYIUrZ0=</latexit><latexit sha1_base64="nAsn/dTSYW0b+kMdleOaUGz5uJE=">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</latexit><latexit sha1_base64="nAsn/dTSYW0b+kMdleOaUGz5uJE=">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</latexit><latexit sha1_base64="OO/pB3TjTnTCE8WAyrKfmXGiOaQ=">AAACNXicbZA9SwNBEIb3/IzxK2ppsxhERZS7NNoIoo2FRQSjgVwMe5u53OLeB7tzYjjuT9n4P6y0sFDE1r/gJpdCowMLD+87w+y8XiKFRtt+sSYmp6ZnZktz5fmFxaXlysrqlY5TxaHBYxmrpsc0SBFBAwVKaCYKWOhJuPZuTwf+9R0oLeLoEvsJtEPWi4QvOEMjdSrn9H5ri+5R11eMZ64KYrp7nw/hppbTI7pXOHDS6edZkrsSfNwuNFX05a4SvQB3bmqdStXet4dF/4IzgioZVb1TeXK7MU9DiJBLpnXLsRNsZ0yh4BLysptqSBi/ZT1oGYxYCLqdDa/O6aZRutSPlXkR0qH6cyJjodb90DOdIcNAj3sD8T+vlaJ/2M5ElKQIES8W+amkGNNBhLQrFHCUfQOMK2H+SnnATCJogi6bEJzxk//CVW3fMXxhV49PRnGUyDrZINvEIQfkmJyROmkQTh7IM3kj79aj9Wp9WJ9F64Q1mlkjv8r6+gZzsatZ</latexit>
linearize…
x00 � 1
⇢� x
⇢2= �B0 +B0x
B⇢
✓1 +
2x
⇢+
x2
⇢2
◆
<latexit sha1_base64="eAVXDyT13+RmjJg6Jz+etNxbdJ8=">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</latexit><latexit sha1_base64="Y/s/gc01fFwp3HKSzK/5SkqYbyk=">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</latexit><latexit sha1_base64="Y/s/gc01fFwp3HKSzK/5SkqYbyk=">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</latexit><latexit sha1_base64="kF+Gore5vEOhrfZYSkzZX1SYU9Y=">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</latexit>
x00 +
✓1
⇢2+
B0
B⇢
◆x = 0
<latexit sha1_base64="wtFmIrOODee4qshQ+mVCcaEMSVU=">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</latexit><latexit sha1_base64="7Xr+WdYlS/1ena2bUvED2hCe6WY=">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</latexit><latexit sha1_base64="7Xr+WdYlS/1ena2bUvED2hCe6WY=">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</latexit><latexit sha1_base64="NMJDOpi0Z5/DEQ78gNik5TVFq7s=">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</latexit>
r = ρ + x
curved coordinate system local focusing field
Note: both and are functions of location s
ρ B′
![Page 23: Particle Transport, Beams, and Emittancenicadd.niu.edu/~syphers/OpsTalks/pdfs/OpsLecture1.pdf · 2020. 5. 28. · § Today: Particle Transport, Beams, and Emittance • Investigate](https://reader035.vdocuments.us/reader035/viewer/2022070221/6137724d0ad5d20676489ffd/html5/thumbnails/23.jpg)
Hill’s Equation• Now, for vertical motion:
• So we have, • to lowest order,
GeneralFormHill’s Equation
15M. Syphers Ops Lecture 1 SPRING 2020
x′′+
(
B′
Bρ+
1
ρ2
)
x = 0
y′′−
(
B′
Bρ
)
y = 0
By = B0 + B’xBx = B’y
y00 � eBx
p
✓1 +
x
⇢
◆2
= 0<latexit sha1_base64="zjhVDim0/N22GrOgtUzyNtHqwlU=">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</latexit><latexit sha1_base64="Sw+rq0cXn+RCd78tJbKzwovGNIc=">AAACIHicbZA9SwNBEIb3/DZ+RS1tFkWMiHJno43gR2OpYBIhF8PeZi5Z3Ns7dufEcNxPsfEv2NvYWCiinYL/xU1iodEXFl6emWF23iCRwqDrvjtDwyOjY+MTk4Wp6ZnZueL8QsXEqeZQ5rGM9XnADEihoIwCJZwnGlgUSKgGl0fdevUKtBGxOsNOAvWItZQIBWdoUaO4Qztra3TTDzXjGRw2rvMsyX0JIZa8jT61yNftOPe1aLVx/WKb7lG3UVxxt9ye6F/jfZuV/QP37nP+Xp00im9+M+ZpBAq5ZMbUPDfBesY0Ci4hL/ipgYTxS9aCmrWKRWDqWe/AnK5a0qRhrO1TSHv050TGImM6UWA7I4ZtM1jrwv9qtRTD3XomVJIiKN5fFKaSYky7adGm0MBRdqxhXAv7V8rbzKaCNtOCDcEbPPmvqWxvedaf2jQOSV8TZIkskxLxyA7ZJ8fkhJQJJzfkgTyRZ+fWeXRenNd+65DzPbNIfsn5+ALGGqXF</latexit><latexit sha1_base64="Sw+rq0cXn+RCd78tJbKzwovGNIc=">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</latexit><latexit sha1_base64="sFE1nE/IdPA5XCfXaXh1nEeOCcU=">AAACIHicbZDLSgNBEEV7fBtfoy7dNAaJIoaZbOJGkLhxGcGokImhp1OTNOl50F0jhmE+xY2/4saFIrrTr7HzWGj0QsPlVBXVdf1ECo2O82nNzM7NLywuLRdWVtfWN+zNrSsdp4pDg8cyVjc+0yBFBA0UKOEmUcBCX8K13z8b1q/vQGkRR5c4SKAVsm4kAsEZGtS2q3RQKtEjL1CMZ1Br3+dZknsSAtx3D8fUIE/14txTotvDg9sKPaFO2y46ZWck+te4E1MkE9Xb9ofXiXkaQoRcMq2brpNgK2MKBZeQF7xUQ8J4n3WhaWzEQtCtbHRgTvcM6dAgVuZFSEf050TGQq0HoW86Q4Y9PV0bwv9qzRSD41YmoiRFiPh4UZBKijEdpkU7QgFHOTCGcSXMXynvMZMKmkwLJgR3+uS/5qpSdo2/cIqntUkcS2SH7JJ94pIqOSXnpE4ahJMH8kReyKv1aD1bb9b7uHXGmsxsk1+yvr4BFI+iRw==</latexit>
y00 =eBx
p
✓r
⇢
◆2
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linearize…y00 � eB0y
p= 0
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x′ ′ + Kx(s) x = 0
··y =ev2
s Bx
p
and y′ ′ + Ky(s) y = 0
![Page 24: Particle Transport, Beams, and Emittancenicadd.niu.edu/~syphers/OpsTalks/pdfs/OpsLecture1.pdf · 2020. 5. 28. · § Today: Particle Transport, Beams, and Emittance • Investigate](https://reader035.vdocuments.us/reader035/viewer/2022070221/6137724d0ad5d20676489ffd/html5/thumbnails/24.jpg)
Determining K — Examples§ Quadrupole Magnets
§ Sector Bend Dipole Magnets
§ Combined Function Magnet:
§ Sector Bends with Electrostatic Focusing
§ Other Considerations • Rectangular Bend • Bend with arbitrary Edge Angles
16M. Syphers Ops Lecture 1 SPRING 2020
Kx =1
⇢2
Ky = 0
Kx =1
⇢2� E0
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Ky =E0
v(B⇢)<latexit sha1_base64="lku1TQloRunp4ufMdFD4viIVEVg=">AAACAnicbZC7SgNBFIbPxluMt1UrsRkMYmzCro02QogIgk0Ec4FsCLOT2WTI7IWZ2cCyBBtfxcZCEVsLn8HON7DzFZxcCk38YeDjP+dw5vxuxJlUlvVpZBYWl5ZXsqu5tfWNzS1ze6cmw1gQWiUhD0XDxZJyFtCqYorTRiQo9l1O627/YlSvD6iQLAxuVRLRlo+7AfMYwUpbbXPvup2gc+R4ApP08miYDgplR/TC42HbzFtFayw0D/YU8qXS99c7AFTa5ofTCUns00ARjqVs2lakWikWihFOhzknljTCpI+7tKkxwD6VrXR8whAdaqeDvFDoFyg0dn9PpNiXMvFd3elj1ZOztZH5X60ZK++slbIgihUNyGSRF3OkQjTKA3WYoETxRAMmgum/ItLDOg2lU8vpEOzZk+ehdlK0Nd/oNMowURb24QAKYMMplOAKKlAFAnfwAE/wbNwbj8aL8TppzRjTmV34I+PtB8V1mVo=</latexit><latexit sha1_base64="0ubWj5tzsyOo0wAVzm7ulpe7zRg=">AAACAnicbZDLSsNAFIYnXmu9RQVB3AxWsW5K4kY3QqkIgpsK9gJNCJPppB06uTAzKYQQ3PgYbt24UMRtn8Kdb+DOV3B6WWjrDwMf/zmHM+d3I0aFNIxPbW5+YXFpObeSX11b39jUt7brIow5JjUcspA3XSQIowGpSSoZaUacIN9lpOH2Lof1Rp9wQcPgTiYRsX3UCahHMZLKcvS9GyeBF9DyOMLp1XGW9osVi3fDk8zRC0bJGAnOgjmBQrn8/TXYfTysOvqH1Q5x7JNAYoaEaJlGJO0UcUkxI1neigWJEO6hDmkpDJBPhJ2OTsjgkXLa0Au5eoGEI/f3RIp8IRLfVZ0+kl0xXRua/9VasfTO7ZQGUSxJgMeLvJhBGcJhHrBNOcGSJQoQ5lT9FeIuUmlIlVpehWBOnzwL9dOSqfhWpVEBY+XAPjgARWCCM1AG16AKagCDe/AEXsCr9qA9a2/a+7h1TpvM7IA/0gY/3fqaLA==</latexit><latexit sha1_base64="0ubWj5tzsyOo0wAVzm7ulpe7zRg=">AAACAnicbZDLSsNAFIYnXmu9RQVB3AxWsW5K4kY3QqkIgpsK9gJNCJPppB06uTAzKYQQ3PgYbt24UMRtn8Kdb+DOV3B6WWjrDwMf/zmHM+d3I0aFNIxPbW5+YXFpObeSX11b39jUt7brIow5JjUcspA3XSQIowGpSSoZaUacIN9lpOH2Lof1Rp9wQcPgTiYRsX3UCahHMZLKcvS9GyeBF9DyOMLp1XGW9osVi3fDk8zRC0bJGAnOgjmBQrn8/TXYfTysOvqH1Q5x7JNAYoaEaJlGJO0UcUkxI1neigWJEO6hDmkpDJBPhJ2OTsjgkXLa0Au5eoGEI/f3RIp8IRLfVZ0+kl0xXRua/9VasfTO7ZQGUSxJgMeLvJhBGcJhHrBNOcGSJQoQ5lT9FeIuUmlIlVpehWBOnzwL9dOSqfhWpVEBY+XAPjgARWCCM1AG16AKagCDe/AEXsCr9qA9a2/a+7h1TpvM7IA/0gY/3fqaLA==</latexit><latexit sha1_base64="UGlmyQfmBIrY7YBRNrlIuYgT1QU=">AAACAnicbZDLSsNAFIYnXmu9RV2Jm8Ei1k1J3OhGKBVBcFPBXqAJYTKdtEMnmTAzKYQQ3Pgqblwo4tancOfbOG2z0NYfBj7+cw5nzu/HjEplWd/G0vLK6tp6aaO8ubW9s2vu7bclTwQmLcwZF10fScJoRFqKKka6sSAo9Bnp+KPrSb0zJkJSHj2oNCZuiAYRDShGSlueeXjnpfAKOoFAOLs5zbNxteGIIT/LPbNi1ayp4CLYBVRAoaZnfjl9jpOQRAozJGXPtmLlZkgoihnJy04iSYzwCA1IT2OEQiLdbHpCDk+004cBF/pFCk7d3xMZCqVMQ193hkgN5XxtYv5X6yUquHQzGsWJIhGeLQoSBhWHkzxgnwqCFUs1ICyo/ivEQ6TTUDq1sg7Bnj95EdrnNVvzvVWpN4o4SuAIHIMqsMEFqINb0AQtgMEjeAav4M14Ml6Md+Nj1rpkFDMH4I+Mzx+Zy5ZD</latexit>
x00 +Kx x = 0
y00 +Ky y = 0<latexit sha1_base64="NDPi869kT6TdFAvpTAZjSnC70oY=">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</latexit><latexit sha1_base64="NDPi869kT6TdFAvpTAZjSnC70oY=">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</latexit><latexit sha1_base64="NDPi869kT6TdFAvpTAZjSnC70oY=">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</latexit><latexit sha1_base64="NDPi869kT6TdFAvpTAZjSnC70oY=">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</latexit>
(g-2 arrangement)
ex: in Main Injector, 400 m;
so, / m2 ρ ≈
Kx ≈ few × 10−6 ≈ 0
Kx =1ρ2
+B′
Bρ
Ky = −B′
Bρ
Kx =B′
Bρ
Ky = −B′
Bρ
ex: in Main Injector,
/ m2
Kx ≈ 0.04
![Page 25: Particle Transport, Beams, and Emittancenicadd.niu.edu/~syphers/OpsTalks/pdfs/OpsLecture1.pdf · 2020. 5. 28. · § Today: Particle Transport, Beams, and Emittance • Investigate](https://reader035.vdocuments.us/reader035/viewer/2022070221/6137724d0ad5d20676489ffd/html5/thumbnails/25.jpg)
Piecewise Method of Solution• Hill’s Equation:
• Though K(s) changes along the design trajectory, it is typically constant, in a piecewise fashion, through individual elements (drift, dipole mag, quad, ...)
• K = 0:
• K > 0:
• K < 0:
17M. Syphers Ops Lecture 1 SPRING 2020
x00 +K(s)x = 0(Here, x could be x or y, and K could be +/- )
![Page 26: Particle Transport, Beams, and Emittancenicadd.niu.edu/~syphers/OpsTalks/pdfs/OpsLecture1.pdf · 2020. 5. 28. · § Today: Particle Transport, Beams, and Emittance • Investigate](https://reader035.vdocuments.us/reader035/viewer/2022070221/6137724d0ad5d20676489ffd/html5/thumbnails/26.jpg)
Piecewise Method of Solution• Hill’s Equation:
• Though K(s) changes along the design trajectory, it is typically constant, in a piecewise fashion, through individual elements (drift, dipole mag, quad, ...)
• K = 0:
• K > 0:
• K < 0:
17M. Syphers Ops Lecture 1 SPRING 2020
x00 +K(s)x = 0(Here, x could be x or y, and K could be +/- )
![Page 27: Particle Transport, Beams, and Emittancenicadd.niu.edu/~syphers/OpsTalks/pdfs/OpsLecture1.pdf · 2020. 5. 28. · § Today: Particle Transport, Beams, and Emittance • Investigate](https://reader035.vdocuments.us/reader035/viewer/2022070221/6137724d0ad5d20676489ffd/html5/thumbnails/27.jpg)
Piecewise Method of Solution• Hill’s Equation:
• Though K(s) changes along the design trajectory, it is typically constant, in a piecewise fashion, through individual elements (drift, dipole mag, quad, ...)
• K = 0:
• K > 0:
• K < 0:
17M. Syphers Ops Lecture 1 SPRING 2020
x00 +K(s)x = 0(Here, x could be x or y, and K could be +/- )
![Page 28: Particle Transport, Beams, and Emittancenicadd.niu.edu/~syphers/OpsTalks/pdfs/OpsLecture1.pdf · 2020. 5. 28. · § Today: Particle Transport, Beams, and Emittance • Investigate](https://reader035.vdocuments.us/reader035/viewer/2022070221/6137724d0ad5d20676489ffd/html5/thumbnails/28.jpg)
Piecewise Method of Solution• Hill’s Equation:
• Though K(s) changes along the design trajectory, it is typically constant, in a piecewise fashion, through individual elements (drift, dipole mag, quad, ...)
• K = 0:
• K > 0:
• K < 0:
drift, or dipole
17M. Syphers Ops Lecture 1 SPRING 2020
x00 +K(s)x = 0(Here, x could be x or y, and K could be +/- )
![Page 29: Particle Transport, Beams, and Emittancenicadd.niu.edu/~syphers/OpsTalks/pdfs/OpsLecture1.pdf · 2020. 5. 28. · § Today: Particle Transport, Beams, and Emittance • Investigate](https://reader035.vdocuments.us/reader035/viewer/2022070221/6137724d0ad5d20676489ffd/html5/thumbnails/29.jpg)
Piecewise Method of Solution• Hill’s Equation:
• Though K(s) changes along the design trajectory, it is typically constant, in a piecewise fashion, through individual elements (drift, dipole mag, quad, ...)
• K = 0:
• K > 0:
• K < 0:
drift, or dipole
Quad,
Gradient
Magnet,
...
17M. Syphers Ops Lecture 1 SPRING 2020
x00 +K(s)x = 0(Here, x could be x or y, and K could be +/- )
![Page 30: Particle Transport, Beams, and Emittancenicadd.niu.edu/~syphers/OpsTalks/pdfs/OpsLecture1.pdf · 2020. 5. 28. · § Today: Particle Transport, Beams, and Emittance • Investigate](https://reader035.vdocuments.us/reader035/viewer/2022070221/6137724d0ad5d20676489ffd/html5/thumbnails/30.jpg)
Piecewise Method -- Matrix Formalism•Write solution to each piece in matrix form
• for each, assume K = const. from s=0 to s=L
• K = 0:
• K > 0:
• K < 0:
18M. Syphers Ops Lecture 1 SPRING 2020
![Page 31: Particle Transport, Beams, and Emittancenicadd.niu.edu/~syphers/OpsTalks/pdfs/OpsLecture1.pdf · 2020. 5. 28. · § Today: Particle Transport, Beams, and Emittance • Investigate](https://reader035.vdocuments.us/reader035/viewer/2022070221/6137724d0ad5d20676489ffd/html5/thumbnails/31.jpg)
“Thin Lens” Quadrupole• If quadrupole magnet is short enough, particle’s offset through the
quad does not change by much, but the slope of the trajectory does -- acts like a “thin lens” in geometrical optics
• Take limit as L --> 0, while KL remains finite
• (similarly, for defocusing quadrupole)
• Valid approx., if F >> L
x(s)
sF
19M. Syphers Ops Lecture 1 SPRING 2020
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Piecewise Method -- Matrix Formalism• Arbitrary trajectory, relative to the design trajectory, can be computed via
matrix multiplication
s0sN
x
20M. Syphers Ops Lecture 1 SPRING 2020
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Example: A Beam Line Calculation§ Will consider two particle trajectories, starting with
• (x,x’ ) = (0, 0.5 mrad), and (x,x’ ) = (5 mm, 0)
§ A distance 6 m later, the trajectories enter a thin lens quadrupole of focal length F = 3 m. This is followed by a second quadrupole of focal length -F, a distance 1 m later.
• Find the trajectories (x,x’ ) for each case at the exit of the second quad
21M. Syphers Ops Lecture 1 SPRING 2020
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Example: A Beam Line Calculation§ Will consider two particle trajectories, starting with
• (x,x’ ) = (0, 0.5 mrad), and (x,x’ ) = (5 mm, 0)
§ A distance 6 m later, the trajectories enter a thin lens quadrupole of focal length F = 3 m. This is followed by a second quadrupole of focal length -F, a distance 1 m later.
22M. Syphers Ops Lecture 1 SPRING 2020
✓xx0
◆=
✓1 1 m1
3 m43
◆✓1 6 m
� 13 m �1
◆✓x0
x00
◆
✓xx0
◆=
✓23 5 m
� 19 m
23
◆✓x0
x00
◆
✓xx0
◆=
✓1 01
3 m 1
◆✓1 1 m0 1
◆✓1 0
� 13 m 1
◆✓1 6 m0 1
◆✓x0
x00
◆
x0=0mm,x’0=0.5mr—>x=2.5mm,x’=0.33mr
x0=5mm,x’0=0.0mr—>x=3.3mm,x’=-0.56mr
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Can now make LARGE accelerators!§ Since the lens spacing can be made arbitrarily short, with
corresponding focusing field strengths, then in principal can make beam transport systems (and linacs and synchrotrons, for instance) of arbitrary size
§ Instrumental in paving the way for very large accelerators, both linacs and especially synchrotrons, where the bending and focusing functions can be separated into distinct magnet types
23M. Syphers Ops Lecture 1 SPRING 2020
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The Notion of an Amplitude Function…
• Can trace single particle trajectories through a periodic system
• Can represent either • multiple passages around a
circular accelerator, or • multiple particles through a
beam line
24M. Syphers Ops Lecture 1 SPRING 2020
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The Notion of an Amplitude Function…
• Can trace single particle trajectories through a periodic system
• Can represent either • multiple passages around a
circular accelerator, or • multiple particles through a
beam line
24M. Syphers Ops Lecture 1 SPRING 2020
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Particle Beams and Phase Space
25M. Syphers Ops Lecture 1 SPRING 2020
x’
source
2a
2𝜃
PhaseSpace:
a-a
𝜃
-𝜃x
x
sangle:x’=dx/ds
Shape,orientationofdistributionin“phasespace”willchangeasparticlesprogressdownstream,buteffective“area”ofdistributionwillremainconstant(Liouville);correlationswillnaturallydevelop
downstream
Transverse coordinates:
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Emittance in Terms of Moments
the “rms emittance”
§ Considering the general equation of an ellipse, the area enclosed by the ellipse is related to its coefficients by:
§ Can define quantities scaled by an area, 𝜖, of our elliptical distribution:
26M. Syphers Ops Lecture 1 SPRING 2020
x
x’ ax2 + bxx0 + cx02 = 1
A =2⇡p
4ac� b2
↵ ⌘ �hxx0i✏/⇡
� ⌘ hx2i✏/⇡
� ⌘ hx02i✏/⇡
�x2 + 2↵xx0 + �x02 = ✏/⇡
✏ = ⇡p
hx2ihx02i � hxx0i2
The ellipse (red curve above) that contains ~95% has area ~6𝜖
So, equation of the blue curve above:
𝛼, 𝛽, γcollectively are called the Courant-Snyder parameters, or Twiss parameters
(for Gaussian distribution)
area of ellipse:
see: http://nicadd.niu.edu/~syphers/tutorials/ellipseNotes.html
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Linear Optics§ Let x be the transverse (horizontal, say) displacement of a particle from
the ideal beam trajectory. Let the angle it makes to the ideal trajectory be x’ = dx/ds, where s is the distance along the ideal trajectory. Transport through a magnetic element is then described by a matrix M, such that
§ An arbitrary trajectory, relative to the design trajectory, can be computed via matrix multiplication for elements all along the beam line…
27M. Syphers Ops Lecture 1 SPRING 2020
~X = M ~X0 ~X =
✓xx0
◆
s0sN
x
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TRANSPORT of Beam Moments• Transport of particle state vector downstream from position 0
• Create a “covariance matrix” of the resulting vector…
• … then, by averaging over all the particles in the distribution,
28M. Syphers Ops Lecture 1 SPRING 2020
~X = M ~X0~X =
✓xx0
◆
~X ~XT =
✓x2 xx0
x0x x02
◆= M ~X0(M ~X0)
T = M ~X0~XT0 M
T
⌃ =
✓hx2i hxx0ihx0xi hx02i
◆⌃ = M⌃0M
Tweget:
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TRANSPORT of Beam Moments• So, since
• where
• then,
• If know matrices M, then can “transport” beam parameters from one point to any point downstream, which determines beam distribution along the way.
29M. Syphers Ops Lecture 1 SPRING 2020
K ⌘✓
� �↵�↵ �
◆
⌃ =
✓hx2i hxx0ihx0xi hx02i
◆⇠
✓✏� �✏↵�✏↵ ✏�
◆= ✏ ·K
K = M K0 MT
� ⌘ hx2i✏/⇡
xrms(s) =p
✏�(s)/⇡
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Conservation of Emittance§ Note that from
§ then,
§ and
§ Thus, the emittance is conserved upon transport through the system
30M. Syphers Ops Lecture 1 SPRING 2020
⌃ = M⌃0MT
K ⌘✓
� �↵�↵ �
◆⌃ = ✏ ·K
det⌃ = detM det⌃0 detMT = det⌃0
det⌃ = ✏2 detK = ✏2(�� � ↵2) = ✏2note:detM=1
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Computer Codes§ Complicated arrangements can be fed into now-standard computer codes for analysis
• TRANSPORT, MAD, DIMAD, TRACE, TRACE3D, COSY, SYNCH, CHEF, many more ...
31M. Syphers Ops Lecture 1 SPRING 2020
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Let’s Think About the Numbers & Units…
§ If <x2> ~ mm2, and <x’2> ~ mrad2, • then the emittance can have units of mm-mrad (also = 𝜇m)
§ Courant-Snyder parameters
32M. Syphers Ops Lecture 1 SPRING 2020
� =⇡hx2i✏
↵ = �⇡hxx0i✏
� =⇡hx02i
✏
mm2/(mm-mrad)~mm/mrad=m
(mm-mrad)/(mm-mrad)=dimensionless
mrad2/(mm-mrad)~mrad/mm=1/m
The“𝜋”comesfromourdefinitionofemittanceasanareainphasespace;emittanceisoftenexpressedinunitsof“𝜋mm-mrad”
✏ = ⇡p
hx2ihx02i � hxx0i2
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Summary§ Given an initial particle distribution in phase space at the input to a
beam transport system, can describe that distribution (sometimes not all that well, but we try…) using Courant-Snyder parameters:
§ The C-S parameters can then be computed downstream, using
33M. Syphers Ops Lecture 1 SPRING 2020
✏ = ⇡p
hx2ihx02i � hxx0i2
� =⇡hx2i✏
↵ = �⇡hxx0i✏
� =⇡hx02i
✏
⌃ = M⌃0MT