Focusing Properties of a Solenoid Magnet

Focusing Properties of a Solenoid Magnet

Simon Jolly

UKNFIC Meeting, 12/05/05

Simon Jolly

UKNFIC Meeting, 12/05/05

Simon JollyUKNFIC Meeting, 12/05/05

2

Cylindrical Polar CoordinatesCylindrical Polar Coordinates

Dimensions given in (r,,z) rather than (x,y,z). Therefore vector in Cartesian coordinates given by:

€

r =

rcosθ

rsinθ

z

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

(1)

Simon JollyUKNFIC Meeting, 12/05/05

3

C.P. Unit VectorsC.P. Unit Vectors

In general, unit vectors given by:

€

ˆ x n ≡

∂r

∂xn

∂r

∂xnSo:

€

ˆ r =

dr

drdr

dr

=

cosθ

sinθ

0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

€

ˆ θ =

−sinθ

cosθ

0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

€

ˆ z =

0

0

1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

(2)

Simon JollyUKNFIC Meeting, 12/05/05

4

C.P. VelocityC.P. Velocity

€

˙ r ≡dr

dt=

d

dtrcosθ

d

dtrsinθ

˙ z

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

˙ r cosθ − r ˙ θ sinθ

˙ r sinθ + r ˙ θ cosθ

˙ z

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

€

=˙ r ̂ r + r ˙ θ ˆ θ + ˙ z ̂ z (3)

Simon JollyUKNFIC Meeting, 12/05/05

5

C.P. AccelerationC.P. Acceleration

€

˙ ̇ r ≡d˙ r

dt=

d

dt

˙ r cosθ − r ˙ θ sinθ

˙ r sinθ + r ˙ θ cosθ

˙ z

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

€

= ˙ ̇ r − r ˙ θ 2( )ˆ r + 2˙ r ˙ θ + r ˙ ̇ θ ( ) ˆ θ + ˙ ̇ z ̂ z

€

=

˙ ̇ r − r ˙ θ 2( )cosθ − 2˙ r ̇ θ + r ˙ ̇ θ ( )sinθ

2˙ r ˙ θ + r ˙ ̇ θ ( )cosθ + ˙ ̇ r − r ˙ θ 2( )sinθ

˙ ̇ z

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

(4)

Simon JollyUKNFIC Meeting, 12/05/05

6

Particle Motion in B-fieldParticle Motion in B-fieldParticle acceleration, a, in B-field, B, given by:(charge q, mass m, velocity v)

€

m˙ ̇ r = q˙ r × B = q

ˆ r ˆ θ ˆ z

˙ r r ˙ θ ˙ z

Br Bθ Bz

€

ma = qv × B

€

⇒ m ˙ ̇ r − r ˙ θ 2( )ˆ r + 2˙ r ˙ θ + r ˙ ̇ θ ( ) ˆ θ + ˙ ̇ z ̂ z [ ] =

€

q r ˙ θ Bz − ˙ z Bθ( )ˆ r + ˙ z Br − ˙ r Bz( ) ˆ θ + ˙ r Bθ − r ˙ θ Br( )ˆ z [ ]

In cylindrical polar coordinates:

(6)

(5)

Simon JollyUKNFIC Meeting, 12/05/05

7

Solenoid B-fieldSolenoid B-field

Solenoid field is axially symmetric (no -dependence), so:

€

Bθ = 0

€

Br r,θ,z( ) = Br r,0,z( )

Define on-axis field:

€

B0 z( ) = Bz 0,0,z( )

Components of Solenoid field:

€

Bz = B0

€

Bθ = 0

€

Br = − 12 B0

′r (8)

(7)

Simon JollyUKNFIC Meeting, 12/05/05

8

Equations of Motion in SolenoidEquations of Motion in Solenoid

Combine eqns. 6 & 8 and split particle motion into r, and z components:

€

m˙ ̇ z = −qr ˙ θ Br€

m ˙ ̇ r − r ˙ θ 2( ) = qr ˙ θ Bz

€

m 2˙ r ˙ θ + r ˙ ̇ θ ( ) = q ˙ z Br − ˙ r Bz( )

(9)

(10)

(11)

(focusing)

(rotation)

(acceleration)

Simon JollyUKNFIC Meeting, 12/05/05

9

Equations of Motion (2)Equations of Motion (2)

Since:

€

m

r2r˙ r ̇ θ + r2˙ ̇ θ ( ) = q ˙ z Br − ˙ r Bz( )

(13)

€

˙ B z =dBz

dt=

dBz

dz

dz

dt= Bz

′ ˙ z (12)

combining eqns. 8, 10 & 12 gives:

€

=−q˙ z r

2B0

′ + ˙ r Bz

⎛

⎝ ⎜

⎞

⎠ ⎟

€

=−q

2rBz

′ ˙ z + 2˙ r Bz

⎛ ⎝ ⎜ ⎞

⎠ ⎟

€

=−q

2r˙ B zr

2 + 2r˙ r Bz( )

Simon JollyUKNFIC Meeting, 12/05/05

10

Equations of Motion (3)Equations of Motion (3)

Now,

€

m

r

d

dtr2 ˙ θ ( ) =

−q

2r

d

dtr2Bz( )

So eqn. 13 becomes:€

d

dtr2 ˙ θ ( ) = r2˙ ̇ θ + 2r˙ r ˙ θ

and

€

d

dtr2Bz( ) = r2 ˙ B z + 2r˙ r Bz

€

⇒d

dtr2 ˙ θ ( ) =

−q

2m

d

dtr2Bz( ) (14)

Simon JollyUKNFIC Meeting, 12/05/05

11

Equations of Motion (4)Equations of Motion (4)

Integrating eqn. 14 with respect to time:

where c is a constant of integration

€

r2 ˙ θ =−q

2mr2Bz + c

€

⇒ ˙ θ =−q

2mBz +

c

r2 (15)

For an on-axis beam, c=0, so eqn. 15 becomes:

€

˙ θ =−q

2mBz (16)

Simon JollyUKNFIC Meeting, 12/05/05

12

Equations of Motion (5)Equations of Motion (5)

Integrating eqn. 16 with respect to time:

€

=−q

2mBzdt∫

Since ,

€

Bzdt = Bzdzdz

dt=

Bz

˙ z dz

(17)

€

=−q

2m

Bz

˙ z dz∫ (18)

Simon JollyUKNFIC Meeting, 12/05/05

13

On-Axis Beam RotationOn-Axis Beam Rotation

The longitudinal kinetic energy

€

T = 12 m˙ z 2

€

=−q2

8TmBzdz∫ (20)

Therefore eqn. 18 becomes:

This means that the outgoing beam is rotated with respect to the incoming beam, and this rotation is proportional to the integrated field, Bzdz, and the particle kinetic energy T.

(19)

Simon JollyUKNFIC Meeting, 12/05/05

14

Transverse Beam MotionTransverse Beam Motion

Now insert eqn. 16 into 9:

€

m ˙ ̇ r − r ˙ θ 2( ) = qr ˙ θ Bz(9)

€

⇒ ˙ ̇ r − r ˙ θ 2 =−q2

2m2rBz

2

€

⇒ ˙ ̇ r =−q2

2m2rBz

2 + r ˙ θ 2

€

=−q2

2m2rBz

2 +q2

4m2rBz

2

€

⇒ ˙ ̇ r =−q2

4m2rBz

2 (21)

Simon JollyUKNFIC Meeting, 12/05/05

15

Transverse Beam Motion (2)Transverse Beam Motion (2)What we actually want is (focusing per unit length):

€

˙ r =dr

dt=

dr

dz

dz

dt= ′ r ˙ z

€

⇒ ′ ′ r ˙ z 2 + ′ r ̇ ̇ z =−q2

4m2rBz

2 (23)

€

′ ′ r

€

˙ ̇ r =d

dt˙ r =

d

dt′ r ˙ z = ′ ′ r ˙ z 2 + ′ r ̇ ̇ z (22)

Simon JollyUKNFIC Meeting, 12/05/05

16

Solenoid Focusing StrengthSolenoid Focusing StrengthSubstituting eqn. 19 for , and therefore setting , modifies eqn. 23 accordingly, giving the radial ray equation:

€

′ ′ r =−q2

8TmBz

2r (24)€

˙ z 2

€

˙ ̇ z = 0

As such, aside from the instrinsic particle properties of charge, kinetic energy and mass (which the solenoid does not modify), the focusing strength of the solenoid lens is purely a function of the longitudinal B-field, Bz, and the radius r.

Simon JollyUKNFIC Meeting, 12/05/05

17

Solenoid Focal LengthSolenoid Focal LengthThe focal length, f, of the solenoid (using the thin lens approximation) is given by:

€

1

f=

− ′ r

r=

−1

r′ ′ r dz∫ =

q2

8TmBz

2dz∫ (25)

Since the focal length is proportional to 1/q2, the solenoid lens is only useful at low particle momenta.