Overview of Beam Physics

Lecture 2

Electrodynamics

• Start with Maxwell equations (mks)

Where the fields obey the constitutive relations

• Continuity of sources implied

• Charged particles obey the Lorentz force equation,

With a generalization of the momentum p.

r

r B = 0

r

r D = e

r

r H =

r D

t+

r J e

r

r E =

r B

t

r D = (

r D )

r E

r B = µ

r H ( )

r H

dr p

dt= q

r E +

r v

r B ( )

r

r J e + e

t= 0

Relativistic dynamical quantities

• Velocity normalized to speed of light

• Momentum is defined relativistically as

with being the rest mass (nonrel. limit), and

• The energy U and momentum are related by

4-vector invariant length (like space-time)

r v

r c

r p = m0

r v

r m0c

m0

1

r v 2 / c2( )( )

1/ 2

r p 2c2 U 2 = m0c

2( )2

U = m0c2

Relativistic equations of motion

• Purely magnetic forces yield no change inenergy, as force is normal to instantaneousmotion,

• Transverse equation of motion has largerinertial mass

• Longitudinal acceleration with electricfield gives change in , different scaling of

inertial mass

dU =r v d

r p

m0

dr v

dt= q

r v

r B ( )

3m0

dv||dt

= qE0

EM Fields in special relativity

• Longitudinal fields are “Lorentz invariant”

• Transverse fields (i.e. self fields in beam)

• In beam rest frame and

“extra” factors of inertial mass…

r E =

r E

r v

r B ( )

r B =

r B + 1

c 2

r v

r E ( )

r B = 0

r F = q

r E +

r v

r B ( ) = q

r E 1 2( ) =

qr E 2

Hamiltonians in special relativity

• A Hamiltonian is a function of coordinates and“canonical momentum” which gives theequations of motion

• It is constructed from a Lagrangian

• A Hamiltonian can rigorously give equations ofmotion, and constants of the motion, i.e. H,

˙ x i =H

pi

, ˙ p i =H

xi

.

H

r x ,

r p ( ) =

r p

r ˙ x L pi

L

˙ x i

˙ H =H

t; H independent of time is constant.

Relativistic dynamics

• Need potentials for variational analysis

• The Lorentz force can be written as

• Relations between energy/momenta and

potentials can be seen

r E =

r e

r A

t

r B =

r

r A

e,

r A ( )

r F L =

dr p

dt= q

r E +

r v

r B ( )

= qr

e

r A

t

r v

r ( )r A

= q

r e

dr A

dt

pc, i = pi + qAi H =U + q e

The relativistic Hamiltonian

• From the 4-vector relation of energy and

momentum,

or

• Canonical variables simply related to

familiar mechanical variables (U,p)

H q e( )

2=

r p c q

r A ( )

2c2 + m0c

2( )2

H =

r p c q

r A ( )

2c2 + m0c

2( )2

+ q e

Transformations of Hamiltonian

• Transform to new canonical variables to

make the Hamiltonian constant?

• Example: wave with phase velocity

• Galilean transformation

• New Hamiltonian

• Example: traveling wave in accelerator…

v= z v t p = pz

˜ H , p( ) = H , p( ) v p .

Traveling wave Hamiltonian

• Field and vector potential

• 1D COM Hamiltonian:

• In terms of mechanical momentum forplotting (“algebraic” Hamiltonian, not forequations of motion)

H = pz,c +qE0

kzvcos kz z v t( )[ ]

2

c 2 + m0c2( )2

Ez z v t( ) =Az

t= E0 sin kz z v t( )[ ], Az z v t( ) =

E0

kzvcos kz z v t( )[ ],

˜ H ,p( ) = p 2c2 + m0c2( )

2v p +

qE0

kz

cos kz[ ]

Phase plane plots

• With algebraic form,for a given H, choose ,

can determine p.

• Plot motion alongconstant H contours

• Example: longitudinalmotion in small“potential” case, typicalof ion linacs andcircular accelerators

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.0 0.2 0.4 0.6 0.8 1.0

kz/2

p/p0

Phase space, general considerations

• Phase space density

distribution:

• Number of particles near a

phase space point is

• In beam physics, we often

will have 2D projections of

phase space, i.e. (x,px)

• Motion uncoupled in x,y,

and z.

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

p

x

f

r x ,

r p ,t( )

f

r x ,

r p ,t( )d3xd3p

Particle distribution in phase spaceprojection, approximated by f

Liouville theorem

• Vlasov equation

• For Hamiltonian systems:

• If particle number is conserved

• Conservation of phase space density:

df

dt=

f

t+

r ˙ x r

r x f +r ˙ p

r r p f

df

dt=

f

t+

dxidt

f

xi+dpidt

f

pi

i

=f

t+

H

pi

f

xi

H

xi

f

pi

i

=f

t+

H

pi

H

xi

df

dH

H

xi

H

pi

df

dH

i

=f

t

f

t= 0

df

dt= 0.

Design trajectory and paraxial rays

• Trajectories measuredwith respect to idealizeddesign

• Small angles assumed;paraxial rays

• Equations of motionparameterized using z asindependent variableinstead of t:

Design trajectory

paraxial rays

px, y << pz

r p

x =dx

dz=

1

vz

dx

dt=˙ x

vz

Example: equations of motion in

magnetic quadrupole

• Magnetic field (normal)

• Magnetic force

• Equation of motion

• Simple harmonicoscillator form:

v B 2 = B (yˆ x + xˆ y )

r F = qvz

r z

r B 2 = qvz B (yˆ y xˆ x )

x =Fx

m0v02

=q B

p0x y =

Fy

m0v02

=q B

p0y

x + 02x = 0

x = x0 cos 0 z z0( )[ ] + x 0

0

sin 0 z z0( )[ ]Solution:

02 q B / p0 = B /BR

Motion in uniform magnetic field

x

y

z

• Field

• Lorentz force

• Uniform motion in z

• Transverse circle atcyclotron frequency.

• Radius related to

Define:

• Helix in general, angle

r B = B0 ˆ z

dpz

dt= 0,

dr p

dt= q

r v

r B ( ) =

qB0

m0

r p ˆ z ( ).

d2vxdt 2 + c

2vx = 0, d2vydt 2 + c

2vy = 0. c

qB0m0

p (MeV/c) = 299.8 B0(T)R(m)p

pz / p

Solenoid (large pitch)

Circular accelerator (small pitch)

BR p (MeV/c) /299.8

Solenoid focusing

• Particles pick up infringe field

or

• Integrate, to find

• Result known as Busch’stheorem

• Assume particles are“born” in region with nomagnetic field…

Symmetry axis

Coils

Magnetic fieldlines

Charged particle test trajectory

Iron for magnetic flux return p

B2

Bzz

=0

1B =

Bzz

p q vzBt1

t2

dt = q Bz1

z2

dz = q 0

2

Bzzz1

z2

=0

dz = q 0

2

dBzdzz1

z2

=0

dz

= q 0

2Bz z2( ) Bz z1( )[ ] = q 0

2B0

The Larmor Frame

particletrajectory

initial position

xLc

later position

y

yL

xL

• Note radius of curvature R is1/2 of offset . Particle passesthrough origin!

• A frame defined by particleand origin rotates withLarmor frequency

• In Larmor frame, equations ofmotion are simple harmonic

L = c / 2

L

d L

dt= c

2=qB02 m0

˙ ̇ x L + L2 xL = 0,

˙ ̇ y L + L2 yL = 0,

x L + kL2xL = 0,

y L + kL2yL = 0,

kL = L

vz

qB02p

=B02BR

Distributions and equilibrium

• A distribution can be in equilibrium under

application of linear force

• Separability:

• Vlasov equilibrium

• Separated equations:

• Momentum solution

• Solution for linear restoring force in x:

• Self-forces (space-charge) are nonlinear in x,

Maxwell-Vlasov equilibria not Gaussian.

f

r x ,

r p ( ) = Nfx x,px( ) fy y,py( ) fz z ,pz( )

˙ x dX

dxP + FxX

dP

dpx

= 0fx x,px( ) = X x( )P px( )

P px( ) = Cp exppx2

2 0m0kBTx

Cp exp

px2

2 px

2

1

XFx x( )

dX

dx= 0m0

pP

dP

dpx= s

X x( ) = Cx exp0m0v0

202x 2

2kBTx

Cx exp

x 2

2 x2

RMS envelope equation

• Want information on extent of distributions

• Look at RMS envelope

• Take derivatives

x2 = x 2 = x 2 fx x, x ( )dxd x ,

d x

dz=

d

dzx 2 =

1

2 x

d

dzx 2

=1

2 x

d

dzx 2 fx x, x ( )dxd x

=1

x

x x fx x, x ( )dxd x = x x

x

.

d2x

dz 2=

d

dzx x

x

=1

x

d x x

dzx x 2

x3

=1

x

d

dzx x fx x, x ( )dxd x x x

2

x3

= x

2 + x x

x

x x 2

x3

RMS Emittance and the Envelope

• The RMS envelope equation can be put instandard form by noting that the rms emittance

is constant under of linear forces

• Also, with this assumption,

and

• The emittance enters the envelope equation as apressure-like term

• Example solution: equilibrium

x,rms2

= x 2 x 2 x x 2

x + x2x = 0

x x = x2 x 2 = x

2x2

x + x2

x =x, rms2

x3.

eq = x,rms

x

d2x

dz 2=

d

dzx x

x

=1

x

d x x

dzx x 2

x3

=1

x x

2x2

x2 0 0( )

0 0x x

x x 2

x3

• In electron sources we have strong acceleration

• Acceleration introduces “adiabatic” damping

• This enters into the envelope as

• In standard form

where we have introduced the normalizedemittance

Inclusion of Acceleration

d2x

dz2 =d

dz

px

pz

= x2x +

( ) x , ( ) =

Fz

p

d2 x

dz2+

( ) d x

dz+ x

2x = n,x

2

( )2

x3

n,x x,rms

Busch’s theorem:

magnetization emittance

• If particles are “born” in magnetic field,then they have canonical angularmomentum. Upon leaving field, they haverms transverse momentum

• This can be translated to normalizedemittance

p

qB02 x(y )

n.xp

m0cx

qB02m0c

x2 150B0 T( ) x

2

Reading references

• Review Chapter 1 sections (all) concerning

dynamics and phase space

• Review Chapter 2 sections (1,3-6)

concerning motion in magnetic fields and

linear focusing

• Review Chapter 5 sections (1-3, 5)

concerning distributions and envelopes