Particle Transport, Beams, and Emittancenicadd.niu.edu/~syphers/OpsTalks/pdfs/OpsLecture1.pdf · 2020. 5. 28. · § Today: Particle Transport, Beams, and Emittance • Investigate

Particle Transport, Beams, and Emittance

Mike Syphers Northern Illinois University & Fermilab

Operator Lectures April 2020

[email protected]

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





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





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, …

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, …


Single-Pass vs. Repetitive Systems

§ Beam Transport (from point A to point B)

§ Acceleration along the way • single-pass with acceleration

• multi-pass acceleration


may need motion in such a system to be stable for many (millions or more?) revolutions

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


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


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


Equations of Motion§ Lorentz Force:

§ Magnetic Rigidity • particle of unit charge, q = e:

§ Reference Trajectory • Local Coordinate System


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]

Linear Magnetic Fields for Guiding & Focusing

Xd(z) = designX(z) = actual

z

X(z)

Xd(z)ρ




z

X(z)

Xd(z)

X(z)

ρ




z

X(z)

Xd(z)

X(z)s

x

ρ




z

X(z)

Xd(z)

X(z)s

x

x’ = dx/ds

ρ


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)

• Quadrupole magnet: • field = 0 on longitudinal axis • varies linearly with transverse position

The Quadrupole Magnet• Dipole magnet: • uniform bend field


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

Linear Restoring Forces — Short Version


(γm)v2

ρ= evB0

ρ

B0ρ =(γm)v

e= p/e



(γm)v2

ρ= evB0

ρ

B0ρ =(γm)v

e= p/e

Δθ ρ

Δs

Δθ =Δsρ

=B ⋅ Δs

Bρ



(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



(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

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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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>


d2

dt2=

✓ds

dt

◆2 d2

ds2=

⇣vs

⇢

r

⌘2 d2

ds2<latexit sha1_base64="uFXkTRWkDm4aMd1FzFtUUa3TNUE=">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</latexit><latexit sha1_base64="gfgkI1Io6nyIUwIqMXb3zaxdF9g=">AAACW3icdVFNT8IwGO6GH4ioqPHkpZGY4IVsXPRi4sfFIyaCJgxI13XQ0HVL+86ELPtV3v0RnvTAXzGWjwOCvkmbJ89H2j71E8E1OM6XZRc2Nre2izul3fLe/kHl8Kit41RR1qKxiNWLTzQTXLIWcBDsJVGMRL5gz/7ofqo/vzKleSyfYJywbkQGkoecEjBUv6K8UBGaBb1GngVgdnyNPcFCqC0EPeVzT/HBEC56jWW7Xra/9vVc89QwzjP1f6RfqTp1ZzZ4HbgLUL25dd4mh++y2a98eEFM04hJoIJo3XGdBLoZUcCpYHnJSzVLCB2RAesYKEnEdDebdZPjc8MEOIyVWRLwjF1OZCTSehz5xhkRGOpVbUr+pXVSCK+6GZdJCkzS+UFhKjDEeFo0DrhiFMTYAEIVN3fFdEhMFWC+o2RKcFefvA7ajbpr8KNp4w7Np4hO0RmqIRddohv0gJqohSj6RN/WtlW0JnbBLtnludW2Fplj9Gvskx9upLoF</latexit><latexit sha1_base64="gfgkI1Io6nyIUwIqMXb3zaxdF9g=">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</latexit><latexit sha1_base64="30+JmjP6e8bTxDAjxtJ4dafm5/Y=">AAACW3icdVFNT8IwGO6GCiIqajx5aSQmeCEbF72YEL14xEQ+Egak6zpo6D7SviMhy/6kJz34V4wd7ICib9LmyfORtk/dWHAFlvVhmKW9/YNy5bB6VDs+Oa2fnfdVlEjKejQSkRy6RDHBQ9YDDoINY8lI4Ao2cBdPuT5YMql4FL7CKmbjgMxC7nNKQFPTunR8SWjqTdpZ6oHe8QN2BPOhWQgq5zNH8tkcbiftbbvati+naqM5ch5lqfw/Mq03rJa1HrwL7AI0UDHdaf3N8SKaBCwEKohSI9uKYZwSCZwKllWdRLGY0AWZsZGGIQmYGqfrbjJ8oxkP+5HUKwS8ZrcTKQmUWgWudgYE5uq3lpN/aaME/PtxysM4ARbSzUF+IjBEOC8ae1wyCmKlAaGS67tiOie6CtDfUdUl2L+fvAv67Zat8YvV6DwWdVTQFbpGTWSjO9RBz6iLeoiid/RllI2K8WmWzKpZ21hNo8hcoB9jXn4DvQq2hw==</latexit>

Use cylindrical coordinates

r = ρ + x, θ, y

acceleration:

··y =ev2

s Bx

p

··r − r ·θ2 = −ev2

s By

p

Linear Restoring Forces - Details• Assume general linear guide fields: --

• Look at radial motion:



r � r✓2 = �ev2sBy

p<latexit sha1_base64="gn/isgKUMnyidEbUhBH9MW2smqo=">AAACGXicbZDLSsNAFIYnXmu9VV26GSyCGyUpgm4EqRuXFWwrNGmYTE/s4OTCzIkQQl7Dja/ixoUiLnXl2zi9LLT6w8DHf86ZmfMHqRQabfvLmptfWFxarqxUV9fWNzZrW9sdnWSKQ5snMlE3AdMgRQxtFCjhJlXAokBCN7i7GNW796C0SOJrzFPwInYbi1Bwhsbya7Y7GCRYqPJQUXdELg4BWdlv0DN66IaK8QLufd1vNP28LNLSr9XtI3ss+hecKdTJVC2/9mHu5VkEMXLJtO45dopewRQKLqGsupmGlPE7dgs9gzGLQHvFeLOS7htnQMNEmRMjHbs/JwoWaZ1HgemMGA71bG1k/lfrZRieeoWI0wwh5pOHwkxSTOgoJjoQCjjK3ADjSpi/Uj5kJg00YVZNCM7syn+h0zhyDF8d18+b0zgqZJfskQPikBNyTi5Ji7QJJw/kibyQV+vRerberPdJ65w1ndkhv2R9fgOnmqCx</latexit><latexit sha1_base64="gn/isgKUMnyidEbUhBH9MW2smqo=">AAACGXicbZDLSsNAFIYnXmu9VV26GSyCGyUpgm4EqRuXFWwrNGmYTE/s4OTCzIkQQl7Dja/ixoUiLnXl2zi9LLT6w8DHf86ZmfMHqRQabfvLmptfWFxarqxUV9fWNzZrW9sdnWSKQ5snMlE3AdMgRQxtFCjhJlXAokBCN7i7GNW796C0SOJrzFPwInYbi1Bwhsbya7Y7GCRYqPJQUXdELg4BWdlv0DN66IaK8QLufd1vNP28LNLSr9XtI3ss+hecKdTJVC2/9mHu5VkEMXLJtO45dopewRQKLqGsupmGlPE7dgs9gzGLQHvFeLOS7htnQMNEmRMjHbs/JwoWaZ1HgemMGA71bG1k/lfrZRieeoWI0wwh5pOHwkxSTOgoJjoQCjjK3ADjSpi/Uj5kJg00YVZNCM7syn+h0zhyDF8d18+b0zgqZJfskQPikBNyTi5Ji7QJJw/kibyQV+vRerberPdJ65w1ndkhv2R9fgOnmqCx</latexit><latexit sha1_base64="gn/isgKUMnyidEbUhBH9MW2smqo=">AAACGXicbZDLSsNAFIYnXmu9VV26GSyCGyUpgm4EqRuXFWwrNGmYTE/s4OTCzIkQQl7Dja/ixoUiLnXl2zi9LLT6w8DHf86ZmfMHqRQabfvLmptfWFxarqxUV9fWNzZrW9sdnWSKQ5snMlE3AdMgRQxtFCjhJlXAokBCN7i7GNW796C0SOJrzFPwInYbi1Bwhsbya7Y7GCRYqPJQUXdELg4BWdlv0DN66IaK8QLufd1vNP28LNLSr9XtI3ss+hecKdTJVC2/9mHu5VkEMXLJtO45dopewRQKLqGsupmGlPE7dgs9gzGLQHvFeLOS7htnQMNEmRMjHbs/JwoWaZ1HgemMGA71bG1k/lfrZRieeoWI0wwh5pOHwkxSTOgoJjoQCjjK3ADjSpi/Uj5kJg00YVZNCM7syn+h0zhyDF8d18+b0zgqZJfskQPikBNyTi5Ji7QJJw/kibyQV+vRerberPdJ65w1ndkhv2R9fgOnmqCx</latexit><latexit sha1_base64="gn/isgKUMnyidEbUhBH9MW2smqo=">AAACGXicbZDLSsNAFIYnXmu9VV26GSyCGyUpgm4EqRuXFWwrNGmYTE/s4OTCzIkQQl7Dja/ixoUiLnXl2zi9LLT6w8DHf86ZmfMHqRQabfvLmptfWFxarqxUV9fWNzZrW9sdnWSKQ5snMlE3AdMgRQxtFCjhJlXAokBCN7i7GNW796C0SOJrzFPwInYbi1Bwhsbya7Y7GCRYqPJQUXdELg4BWdlv0DN66IaK8QLufd1vNP28LNLSr9XtI3ss+hecKdTJVC2/9mHu5VkEMXLJtO45dopewRQKLqGsupmGlPE7dgs9gzGLQHvFeLOS7htnQMNEmRMjHbs/JwoWaZ1HgemMGA71bG1k/lfrZRieeoWI0wwh5pOHwkxSTOgoJjoQCjjK3ADjSpi/Uj5kJg00YVZNCM7syn+h0zhyDF8d18+b0zgqZJfskQPikBNyTi5Ji7QJJw/kibyQV+vRerberPdJ65w1ndkhv2R9fgOnmqCx</latexit>

v2s

⇣⇢r

⌘2x00 � (⇢+ x)

⇣vsr

⌘2= �ev2sBy

p<latexit sha1_base64="HsRjQ9rECbcrPd6mORnmpEU+Ob8=">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</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=">AAACNXicbZC7SgNBFIZnvRtvUUubQREVUXbTaCN4aSwsFIwK2RhmJ2eTIbOzy8xZMSz7UlrY+RBWWlgoYusrOMlaeDsw8PH/53Dm/EEihUHXfXIGBoeGR0bHxksTk1PTM+XZuTMTp5pDlccy1hcBMyCFgioKlHCRaGBRIOE86Bz0/PMr0EbE6hS7CdQj1lIiFJyhlRrlI3q9skI3qB9qxjNft2O6fp334bKS0x26UTiw3+jmWZL7EkJcLTRd9OW+Fq02rl1WGuUld9PtF/0L3hcs7e65t8+z9+q4UX7wmzFPI1DIJTOm5rkJ1jOmUXAJeclPDSSMd1gLahYVi8DUs/7VOV22SpOGsbZPIe2r3ycyFhnTjQLbGTFsm99eT/zPq6UYbtczoZIUQfFiUZhKijHtRUibQgNH2bXAuBb2r5S3mU0EbdAlG4L3++S/cFbZ9Cyf2DT2SVFjZIEsklXikS2ySw7JMakSTm7II3khr86d8+y8Oe9F64DzNTNPfpTz8QklS67X</latexit><latexit sha1_base64="nAsn/dTSYW0b+kMdleOaUGz5uJE=">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</latexit><latexit sha1_base64="OO/pB3TjTnTCE8WAyrKfmXGiOaQ=">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</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′

Hill’s Equation• Now, for vertical motion:

• So we have, • to lowest order,

GeneralFormHill’s Equation


x′′+

(

B′

Bρ+

1

ρ2

)

x = 0

y′′−

(

B′

Bρ

)

y = 0


y00 � eBx

p

✓1 +

x

⇢

◆2

= 0<latexit sha1_base64="zjhVDim0/N22GrOgtUzyNtHqwlU=">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</latexit><latexit sha1_base64="Sw+rq0cXn+RCd78tJbKzwovGNIc=">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</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

<latexit sha1_base64="Nc4B53nzwNtODD3Dy7GuFlFVfqE=">AAACHHicbZC7SgNBFIZnvRtvUUubwSCJjexGQRshaGOpYFTIxmV2cjY7OHth5qwYln0QG1/FxkIRGwvBt3FyKTTxh4GP/5zDmfP7qRQabfvbmpqemZ2bX1gsLS2vrK6V1zeudJIpDk2eyETd+EyDFDE0UaCEm1QBi3wJ1/7dab9+fQ9KiyS+xF4K7Yh1YxEIztBYXnmf9qpVekzdQDGew4n3UORp4UoIsDb0VJG7KkwKV4luiLu3deqVK/aePRCdBGcEFTLSuVf+dDsJzyKIkUumdcuxU2znTKHgEoqSm2lIGb9jXWgZjFkEup0PjivojnE6NEiUeTHSgft7ImeR1r3IN50Rw1CP1/rmf7VWhsFROxdxmiHEfLgoyCTFhPaToh2hgKPsGWBcCfNXykNmIkGTZ8mE4IyfPAlX9T3H8MVBpXEyimOBbJFtUiMOOSQNckbOSZNw8kieySt5s56sF+vd+hi2TlmjmU3yR9bXDxMHoWQ=</latexit><latexit sha1_base64="Nc4B53nzwNtODD3Dy7GuFlFVfqE=">AAACHHicbZC7SgNBFIZnvRtvUUubwSCJjexGQRshaGOpYFTIxmV2cjY7OHth5qwYln0QG1/FxkIRGwvBt3FyKTTxh4GP/5zDmfP7qRQabfvbmpqemZ2bX1gsLS2vrK6V1zeudJIpDk2eyETd+EyDFDE0UaCEm1QBi3wJ1/7dab9+fQ9KiyS+xF4K7Yh1YxEIztBYXnmf9qpVekzdQDGew4n3UORp4UoIsDb0VJG7KkwKV4luiLu3deqVK/aePRCdBGcEFTLSuVf+dDsJzyKIkUumdcuxU2znTKHgEoqSm2lIGb9jXWgZjFkEup0PjivojnE6NEiUeTHSgft7ImeR1r3IN50Rw1CP1/rmf7VWhsFROxdxmiHEfLgoyCTFhPaToh2hgKPsGWBcCfNXykNmIkGTZ8mE4IyfPAlX9T3H8MVBpXEyimOBbJFtUiMOOSQNckbOSZNw8kieySt5s56sF+vd+hi2TlmjmU3yR9bXDxMHoWQ=</latexit><latexit sha1_base64="Nc4B53nzwNtODD3Dy7GuFlFVfqE=">AAACHHicbZC7SgNBFIZnvRtvUUubwSCJjexGQRshaGOpYFTIxmV2cjY7OHth5qwYln0QG1/FxkIRGwvBt3FyKTTxh4GP/5zDmfP7qRQabfvbmpqemZ2bX1gsLS2vrK6V1zeudJIpDk2eyETd+EyDFDE0UaCEm1QBi3wJ1/7dab9+fQ9KiyS+xF4K7Yh1YxEIztBYXnmf9qpVekzdQDGew4n3UORp4UoIsDb0VJG7KkwKV4luiLu3deqVK/aePRCdBGcEFTLSuVf+dDsJzyKIkUumdcuxU2znTKHgEoqSm2lIGb9jXWgZjFkEup0PjivojnE6NEiUeTHSgft7ImeR1r3IN50Rw1CP1/rmf7VWhsFROxdxmiHEfLgoyCTFhPaToh2hgKPsGWBcCfNXykNmIkGTZ8mE4IyfPAlX9T3H8MVBpXEyimOBbJFtUiMOOSQNckbOSZNw8kieySt5s56sF+vd+hi2TlmjmU3yR9bXDxMHoWQ=</latexit><latexit sha1_base64="Nc4B53nzwNtODD3Dy7GuFlFVfqE=">AAACHHicbZC7SgNBFIZnvRtvUUubwSCJjexGQRshaGOpYFTIxmV2cjY7OHth5qwYln0QG1/FxkIRGwvBt3FyKTTxh4GP/5zDmfP7qRQabfvbmpqemZ2bX1gsLS2vrK6V1zeudJIpDk2eyETd+EyDFDE0UaCEm1QBi3wJ1/7dab9+fQ9KiyS+xF4K7Yh1YxEIztBYXnmf9qpVekzdQDGew4n3UORp4UoIsDb0VJG7KkwKV4luiLu3deqVK/aePRCdBGcEFTLSuVf+dDsJzyKIkUumdcuxU2znTKHgEoqSm2lIGb9jXWgZjFkEup0PjivojnE6NEiUeTHSgft7ImeR1r3IN50Rw1CP1/rmf7VWhsFROxdxmiHEfLgoyCTFhPaToh2hgKPsGWBcCfNXykNmIkGTZ8mE4IyfPAlX9T3H8MVBpXEyimOBbJFtUiMOOSQNckbOSZNw8kieySt5s56sF+vd+hi2TlmjmU3yR9bXDxMHoWQ=</latexit>

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

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


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=">AAACNnicbZDLSsNAFIYnXmu8VV26GSxeUJCkGwURim4EESrYWmhKmUxP69DJJM5MpCH0qdz4HO66caGIWx/BaZtFrf4w8POdczhzfj/iTGnHGVgzs3PzC4u5JXt5ZXVtPb+xWVVhLClUaMhDWfOJAs4EVDTTHGqRBBL4HO797uWwfv8EUrFQ3OkkgkZAOoK1GSXaoGb+xvOhw0QKj4JISZLDvt3b38dH+LrZw94Z7uG98z3sYM+zk4wnQ56MuQeiNTHbzBecY2ck/Ne4mSmgTOVm/tVrhTQOQGjKiVJ114l0IyVSM8qhb3uxgojQLulA3VhBAlCNdHR2H+8a0sLtUJonNB7RyYmUBEolgW86A6If1HRtCP+r1WPdPm2kTESxBkHHi9oxxzrEwwxxi0mgmifGECqZ+SumD0QSqk3StgnBnT75r6kWj13jb4uF0kUWRw5tox10gFx0gkroCpVRBVH0jAboHX1YL9ab9Wl9jVtnrGxmC/2S9f0DRWWmzg==</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

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:


x00 +K(s)x = 0(Here, x could be x or y, and K could be +/- )



• K = 0:

• K > 0:

• K < 0:





• K = 0:

• K > 0:

• K < 0:





• K = 0:

• K > 0:

• K < 0:

drift, or dipole





• K = 0:

• K > 0:

• K < 0:

drift, or dipole

Quad,

Gradient

Magnet,

...



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:


“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


Piecewise Method -- Matrix Formalism• Arbitrary trajectory, relative to the design trajectory, can be computed via

matrix multiplication

s0sN

x


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


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.


✓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

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


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


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


Particle Beams and Phase Space


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:

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:


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

http://nicadd.niu.edu/~syphers/tutorials/ellipseNotes.html

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…


~X = M ~X0 ~X =

✓xx0

◆

s0sN

x

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,


~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:

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.


K ⌘✓

� �↵�↵ �

◆

⌃ =

✓hx2i hxx0ihx0xi hx02i

◆⇠

✓✏� �✏↵�✏↵ ✏�

◆= ✏ ·K

K = M K0 MT

� ⌘ hx2i✏/⇡

xrms(s) =p

✏�(s)/⇡

Conservation of Emittance§ Note that from

§ then,

§ and

§ Thus, the emittance is conserved upon transport through the system


⌃ = M⌃0MT

K ⌘✓

� �↵�↵ �

◆⌃ = ✏ ·K

det⌃ = detM det⌃0 detMT = det⌃0

det⌃ = ✏2 detK = ✏2(�� ↵2) = ✏2note:detM=1

Computer Codes§ Complicated arrangements can be fed into now-standard computer codes for analysis

• TRANSPORT, MAD, DIMAD, TRACE, TRACE3D, COSY, SYNCH, CHEF, many more ...


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


� =⇡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


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


✏ = ⇡p


� =⇡hx2i✏

↵ = �⇡hxx0i✏

� =⇡hx02i

✏

⌃ = M⌃0MT

Documents

Particle Transport, Beams, and Emittancenicadd.niu.edu/~syphers/OpsTalks/pdfs/OpsLecture1.pdf · 2020. 5. 28. · § Today: Particle Transport, Beams, and Emittance • Investigate