An Introduction to Ion-Optics

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5

Inserting Unity Matrix I = RR-1 in equ. (6) it follows X0 T (RTRT-1) s0-1 (R-1 R) X0 = 1

from which we derive (RX0)T (Rs0 RT)-1 (RX0) = 1

(7)

(9)

(8)

The equation of the new ellipsoid after transformation becomes X1 T s1-1 X1 = 1

where s1 = Rs0 RT

(10)

(6)

Conclusion: Knowing the TRANSPORT matrix R that transports one ray through an ion-optical system using (7) we can now also transport the phase space ellipse describing the initial beam using (10)

Slide 21

The transport of rays and phase ellipses in a Drift and focusing Quadrupole, Lens

2.

3.

Matching of emittance and acceptance

Lens 2

Lens 3

Focus

Focus

Slide 22

Increase of Emittance e due to degrader

Focus

A degrader / target increases the

emittance e due to multiple scattering.

The emittance growth is minimal when

the degrader in positioned in a focus

As can be seen from the schematic

drawing of the horizontal x-Theta

Phase space.

for back-of-the-envelop discussions!

Slide 23

Emittance e measurement by tuning a quadrupole

Lee, p. 55

The emittance e is an important

parameter of a beam. It can be measured as shown below.

s11 (1 + s12 L/ s11 - L g) + (eL)2/s22

xmax =

Exercise 3:

In the accelerator

reference book s22 is printed as s11

Verify which is correct

¶Bz/¶x * l

Br

g =

(Quadr. field strength

l = eff. field length)

L = Distance between quadrupole and

beam profile monitor

Take minimum 3 measurements of

xmax(g) and determine Emittance e

(11)

(12)

Slide 24

Emittance e measurement by moving viewer method

The emittance e can also be measured in a drift space as shown below.

s11 + 2 L1 s12 + L1 2 s22

(xmax(V2))2 =

L = Distances between viewers

( beam profile monitors)

¾½¾¾¾½¾¾¾¾½¾®

L1 L2

Viewer V1 V2 V3

Beam

(xmax(V3))2 =

s11 + 2 (L1 + L2)s12 + (L1 + L2 )2 s22

where s11 = (xmax(V1))2

e = Ö

```````````

s11s22 - (s12) 2

Emittance:

Discuss practical aspects

No ellipse no e? Phase space!

(13)

(14)

Slide 25

Taylor expansion

Linear (1st order)TRANSPORT Matrix Rnm

,l

Note: Several notations are in use for 6 dim. ray vector & matrix elements.

Remarks:

Midplane symmetry of magnets

reason for many matrix element = 0

Linear approx. for “well” designed