Transcript Focusing Properties of a Solenoid Magnet

Focusing Properties of a Solenoid
Magnet
Simon Jolly
UKNFIC Meeting, 12/05/05
Cylindrical Polar Coordinates
Dimensions given in (r,,z)
rather than (x,y,z). Therefore
vector in Cartesian coordinates
given by:
r cos


r  r sin  (1)

 z 

Simon Jolly
UKNFIC Meeting, 12/05/05
2
C.P. Unit Vectors

r
In general, unit vectors given by:

xn
xˆ n 
r
x n
So:
dr cos
sin
0






dr
ˆ
ˆr 
ˆ  0 (2)
 sin   cos
z


 
dr


0 




0
1






dr
Simon Jolly
UKNFIC Meeting, 12/05/05
3
C.P. Velocity
d

dt r cos 
Ýsin  
Ý
r
cos


r


 

dr  d
Ýcos
rÝ

r sin   rÝsin   r
 
dt dt

Ý
z
 zÝ  





ˆ  zÝzˆ
Ý
 rÝrˆ  r
Simon Jolly
UKNFIC Meeting, 12/05/05
(3)
4
C.P. Acceleration
Ýsin 
rÝcos  r

drÝ d 
Ýcos
Ý
rÝ
 rÝsin   r
dt dt 

Ý
z






(4)


2
 Ý
Ý
Ý r
Ý
Ýsin
rÝ r cos  2rÝ


Ý r
Ý
Ýcos  Ý
Ý2 sin
 2rÝ
rÝ r



Ý

zÝ


2
ˆÝ
Ý
Ý r
Ý
Ý
Ý
Ý
 r  r rˆ  2rÝ
zÝ
zˆ
Simon Jolly
UKNFIC Meeting, 12/05/05



 

5
Particle Motion in B-field
Particle acceleration, a, in Bfield, B, given by:
(charge q, mass m, velocity v)
In cylindrical
polar coordinates:
ma  qv  B
ˆ
rˆ 
zˆ
Ý zÝ (5)
mÝ
rÝ qrÝ B  q rÝ r
Br
 
  
ˆ  rÝB
ÝB  zÝB rˆ  zÝB  rÝB 
qr

2
ˆÝ
Ý
Ý r
Ý
Ý
Ý
Ý
 m r  r rˆ  2 rÝ
zÝzˆ 
z
Simon Jolly
UKNFIC Meeting, 12/05/05

r
z

B
Bz
(6)

ÝB zˆ
 r
r
6

Solenoid B-field
Solenoid field is
axially symmetric (no
-dependence), so:
B  0
Br r,,z  Br r,0,z
Define on-axis field:
B0 z  Bz 0,0,z
(7)
Components of Solenoid field:
Bz  B0
Simon Jolly
UKNFIC Meeting, 12/05/05

B  0
Br   12 B0r
(8)
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
Equations of Motion in Solenoid
Combine eqns. 6 & 8 and split particle motion
into r,  and z components:
 
Ý r
Ý
Ý qzÝB  rÝB 
m2rÝ
Ý2  qr
ÝB
(9) m Ý
rÝ r
z
(10)
ÝB
zÝ qr
(11) mÝ
r
r
z
(focusing)
(rotation)
(acceleration)

Simon Jolly
UKNFIC Meeting, 12/05/05
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Equations of Motion (2)
Since:
dB z dB z dz
Ý
Bz 

 BzzÝ
dt
dz dt
(12)
combining eqns. 8, 10 & 12 gives:
m
Ý r 2
Ý
Ý  qzÝB  rÝB 
Ý
2r
r
r
z
r
zÝr 
 q  

 q B0  rÝBz   rBz zÝ 2rÝBz 

 2
 2 
q Ý 2
(13)
 Bz r  2rrÝBz 
2r
Simon Jolly

UKNFIC Meeting, 12/05/05



9
Equations of Motion (3)
 
d 2 Ý 2Ý
Ý 2rrÝ
Ý
Now,
r   r 
dt
d 2
2Ý
and
r
B

r
Bz  2rrÝBz

z
dt
 So eqn. 13 becomes:
m d 2 Ý q d 2
r  
r Bz 

r dt
2r dt

d 2 Ý q d 2

r  
r Bz 

dt
2m dt
 
 
Jolly
Simon
UKNFIC Meeting, 12/05/05
(14)
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
Equations of Motion (4)
Integrating eqn. 14 with respect to time:
2 Ý q 2
r 
r Bz  c where c is a constant
of integration
2m
q
c
Ý
 
Bz  2
(15)
2m
r
For an on-axis beam, c=0, so eqn. 15 becomes:
q
Ý

Bz
2m

Simon Jolly
UKNFIC Meeting, 12/05/05
(16)
11
Equations of Motion (5)
Integrating eqn. 16 with respect to time:
q

Bz dt
(17)

2m

dz Bz
Since Bz dt  Bz dz  dz ,
dt
zÝ

q Bz

dz
(18)

2m zÝ
Simon Jolly
UKNFIC Meeting, 12/05/05
12
On-Axis Beam Rotation
The longitudinal kinetic energy T  12 mzÝ2 (19)
Therefore eqn. 18 becomes:
2
q
 
Bz dz


8Tm
(20)
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.
Simon Jolly
UKNFIC Meeting, 12/05/05
13
Transverse Beam Motion
Now insert eqn. 16 into 9:
q
2
2
Ý
Ý
Ý
 r  r 
rBz
2
2m
q 2
2
Ý2
Ý
rÝ
rB

r

z
2
2m
q 2 2
q2
2

rBz 
rBz
2
2
2m
4m


2
2
Ý
ÝB
Ý
Ý
(9) m r  r  qr
z



q 2
2
Ý
rÝ
rBz
2
4m
Simon Jolly
UKNFIC Meeting, 12/05/05
(21)
14
Transverse Beam Motion (2)


What we actually want is r (focusing
per unit
length):
dr dr dz
rÝ

 r zÝ
dt dz dt

d
d
Ý
rÝ rÝ rzÝ rzÝ2  rÝ
zÝ (22)
dt
dt

2
q
2
2
Ý
Ý
Ý




(23)
r z rz
rBz
2
4m
Simon Jolly
UKNFIC Meeting, 12/05/05
15
Solenoid Focusing Strength
2
Ý
Substituting eqn. 19 for z , and therefore setting
Ý
zÝ 0 , modifies eqn. 23 accordingly, giving the
radial ray equation:
q 2 2
(24)
r
Bz r

8Tm
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 Jolly
UKNFIC Meeting, 12/05/05
16
Solenoid Focal Length
The focal length, f, of the solenoid (using the thin
lens approximation) is given by:
2

1 r 1
q
 


rdz

f
r
r
8Tm
B
z
2
dz
(25)
Since the focal length is proportional to 1/q2, the
solenoid lens is only useful at low particle
momenta.
Simon Jolly
UKNFIC Meeting, 12/05/05
17