Longitudinal Motion 2 Eric Prebys, FNAL

RF Manipulations As you’ll show in homework, the synchrotron tune (longitudinal

oscillations/turn) is generally <<1. That means that if there are multiple RF cavities around the ring,

the orbiting particle will see the vector sum of the cavities.

We will clearly get the maximum energy gain if all phases are the same, so (assuming all voltages are the same)

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 9 - Longitudinal Motion 2 2

RF1 RF2

RF3R

F3 effeff

N

iii

V

Vdn

E

sin

)sin(1

fi is the phase angle at the arrival of the particle at cavity i

sNVdn

E sin0

Do we always want the maximum acceleration? As we’ll see, there are times when we want to change the

amplitude of the RF quickly. Because cavities represent stored energy, changing their amplitude

quickly can be difficult. Much quicker to change phase Standard technique is to divide RF cavities into two groups and

adjust the relative phase. In the simplest case, we put half the RF cavities into group “A” and half into group “B”. We can adjust the phases of these cavities relative to our nominal synchronous phase as

So

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 9 - Longitudinal Motion 2 3

seffeff

s

ssss

sseffeff

NVV

NV

N

VN

VN

V

;cos

sincos

sincoscossinsincoscossin2

sin2

sin2

sin

0

0

00

02

0 0

eff

eff

V

NVV

Like “turning RF off”

RF capture We can capture beam by increasing the RF voltage with no

accelerating phase

As we accelerate beam, t decreases. Recall

So as beam accelerates, bunches get narrower

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 9 - Longitudinal Motion 2 4

constant1 22 LLL

Et

4

1

20

2

4

12

02

cos

cos

sSrfLLLRMS

sSrfL

L

LRMS

EeVt

EeVE

Bucket to Bucket Transfer In general, the accelerating gradient of an RF structure is

So when bunches get short enough, it’s advantageous to transfer to a higher frequency. For example, in the Fermilab Linac

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 9 - Longitudinal Motion 2 5

breakdownRFRF

breakdown VV

L

V

0 1 2 3 4 5 6

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6

-1.5

-1

-0.5

0

0.5

1

1.5

200 MHz Linac 800 MHz Linac

Transfer beam to every 4th bucket

Bunch Rotation If we slowly change the RF voltage (or effective voltage by

phasing), we can adiabatically change the bunch shape

If we suddenly change the voltage, then the bunch will be mismatched and will rotate in longitudinal phase space

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 9 - Longitudinal Motion 2 6

4

1

0

4

1

20

2

4

1

0

4

12

02

cos

cos

VEeV

t

VEeV

E

sSrfLLLRMS

sSrfL

L

LRMS

0 1 2 3 4 5 6

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6

-1.5

-1

-0.5

0

0.5

1

1.5