Unit 4 - Lectures 11 & 12 Acceleration by RF waves

Unit 4 - Lectures 11 & 12 Acceleration by RF waves

William A. Barletta

Director, United States Particle Accelerator School Dept. of Physics, MIT

Sources: USPAS Course notes by F. Sannibale High Energy Electron Linacs by P. Wilson

US Particle Accelerator School

Example of 3 coupled cavities

x j = i j 2Lo and = normal mode frequency

US Particle Accelerator School

Write the coupled circuit equations in matrix form

Lx q

=

1 q2

xq

where

L

=

1/

2 o

k

/

2

2 o

0

k

/

2 o

1

/

2 o

k

/

2 o

0

k

/

2

2 o

and

1/

2 o

xq

=

x1 x2 x3

Compute eigenvalues & eigenvectors to find the three normal modes

1

Mode q = 0 : zero mode

0 =

o 1+ k

x0 = 11

1

Mode q = 1: /2 mode

1 = o

x1 = 01

Mode q = 2 : mode

2 =

o 1 k

1 x2 = 11

US Particle Accelerator School

Exercise:

B fields can change the trajectory of a particle

F

=

q

E

+

1 c

v

B

Show that B fields cannot change its energy

We will cast our discussion in terms of the E field

US Particle Accelerator School

Electromagnetic waves

From Maxwell equations, we can derive

for electromagnetic waves in free space (no charge or current distributions present).

The plane wave is a particular solution of the EM wave equation

when

Phase of the wave =

= c k

US Particle Accelerator School

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