The eect of RF on p olarization in a m uon storage ring - - PDF document

SLIDE 1 The eect

• f

RF

• n

p

• larization

in a m uon storage ring Ra jendran Ra ja F ermi National A c c eler ator L ab

• r

atory PO Box

• Batavia

IL

• Abstract

W e in v estigate the preserv ation

• f

p

• larization

in an idealized m uon storage rings with m uon energy

• GeV

and

• GeV

for b eams with momen tum ac ceptance

• pp

The b eams dep

• larize

rap dily due to the dieren tial rates

• f

precession as a function

• f

m uon momen tum When an rf v

• ltage

is applied inducing sync hrotron

• scillations

the p

• larization

is seen to b e preserv ed for sync hrotron tunes

• f

the

• rder
• f
• W

e in v estigate the dep endence

• f

dep

• larization

as a function

• f

sync hrotron tune and rf v

• ltage

for

• GeV

and

• GeV

m uon storage rings I INTR ODUCTION The preserv ation

• f

p

• larization
• f

m uons in a m uon storage ring is in v estigated As the m uons circulate around the ring the spin v ector

• f

a m uon precesses faster than the momen tum v ector b y an amoun t

• g
• where

g is the gyromagnetic ratio

• f

the m uon and

• is

its Loren tz gamma factor See ref

• for

a discussion

• f

this in detail and references therein Since there is a spread

• f

momen ta in the ring there will b e dieren tial precession

• f

the p

• larization

leading to a loss

• f

the a v erage p

• larization
• f

the m uons as the m uons circulate the ring If the momen tum

• f

a m uon remains the same from turn to turn the amoun t

• f

spin precession will b e the same relativ e to the a v erage spin precession

SLIDE 2 and the dep

• larization

will b e cum ulativ e This eect has b een kno wn for a long time

• and

its remedy is to induce sync hrotron

• scillations

causing the individual m uon spin precession to v ary from turn to turn W e will estimate the magnitude

• f

the eect and its remedy using simple sim ulations and then in v estigate further using more realistic sim ulations that tak e in to accoun t sync hrotron

• scillations

prop erly

• I

I SIMPLE SIMULA TION W e construct an idealized storage ring

• f
• GeV

m uon energy with the parameters as sho wn in table I

• W

e inject

• m

uons in to the ring with

• pp
• f
• and

allo w ed to circulate for

• turns

The m uons are assumed to b e fully longitudinally p

• larized

and the p

• larization

is computed at the end

• f

eac h turn for those m uons whose deca y electrons end up in a calorimeter

• f

radial exten t cmcm placed around the b eam pip e The p

• larization

is assumed to precess en tirely in the horizon tal plane and the magnitude

• f

the a v erage p

• larization

v ector is plotted from turn to turn The m uons are deca y ed with a v ertex that is uniformly distributed in the deca y b eam pip e Figure

• sho

ws the loss

• f

p

• larization

as a function

• f

turn n um b er for the

• GeV

case in the absence

• f

an y rf Sync h The same m uons are used for the next turn with the same v alues

• f

momen ta and the deca y is preformed again The n um b er

• f

m uons circulating turn b y turn is decreased b y the actual n um b er that w

• uld

ha v e deca y ed This is similar to the tec hnique used in

• to

determine the energy scale

• f

the m uon collider A Generating sync hrotron

• scillations

A t the end

• f

eac h turn the m uons are sub jected to an rf v

• ltage

and their momen ta will b e rearranged dep ending

• n

the phase

• f

their arriv al at the rf Eac h

• f

the m uons will undergo a sync hrotron

• scillation

so that their energies E as a function

• f

turn n um b er t will b e giv en b y the expression

SLIDE 3 E

• Acos
• Q

s t

• where

Q s is the sync hrotron tune fractional and A is the amplitude

• f
• scillation

and

• is

an arbitrary phase It is imp

• ssible

from kno wing the momen tum

• f

eac h m uon alone to generate sync hrotron

• scillations

since

• ne

needs to compute b

• th

A and

• The

tric k is to generate the

• scillations

in suc h a w a y that the mean energy and its standard deviation are preserv ed from turn to turn This is accomplished b y the follo wing neat construction F

• r

eac h m uon generate t w

• random

n um b ers r

• and

r

• suc

h that b

• th
• f

these are Gaussian distributed with standard deviation

• p

The t w

• dimensional

distribution

• f

these t w

• Gaussian

n um b ers is cylindrically symmetrical and is exp

• nen

tially distributed in r

• r
• where

r

• and

r

• are

pairs

• f

Gaussian random n um b ers The pro jection

• f

this distribution along an y radial direction is Gaussian Sync hrotron

• scillations

are

• btained

b y rotating this distribution b y the angle

• Q

s t and taking the x comp

• nen

t

• f

the v ectors This will ha v e the eect

• f

in tro ducing

• scillations

while at the same time preserving the mean and standard deviation

• f

the distribution from turn to turn B

• pp

case W e in v estigate the case for a b eam energy spread

• f
• at

the

• lev

el Figure

• sho

ws the p

• larization

as a function

• f

turn n um b er for sync hrotron tunes

• and
• F
• r

a sync hrotron tune

• f
• the

p

• larization

is stable for

• turns

but con tains an

• scillation

term These

• scillations

p ersist with increased frequency but m uc h reduced amplitude for sync hrotron tune v alues

• Figure
• sho

ws the corresp

• nding

distributions for the

• GeV

case The loss

• f

p

• larization

for the no rf case is slo w er in the

• GeV

case as exp ected since spins precess slo w er

SLIDE 4 C

• pp

case It has since b een p

• in

ted

• ut

that the curren t F ermilab design for a m uon storage ring en visages rms

• pp
• Figures
• and
• sho

w the dep

• larization

as a function

• f

turn n um b er for sync hrotron tunes

• f
• It

can b e seen that in this case a sync hrotron tune

• f
• is

sucien t to main tain p

• larization

up to

• turns

I I I MORE REALISTIC SIMULA TION The ab

• v

e sim ulation assumes that all the particles are in the rf buc k et and undergo sync hrotron

• scillations

with the same tune In practice this is not the case and w e will no w attempt to sim ulate realistic sync hrotron

• scillations

using the sync hrtorn

• scillation

dierence equations

• E

n

• E

n

• eV

sin n

• sin

s

• n
• n
• h

E n

• E

s

• where

E n is the dierence b et w een the energy gained b y a particle at the end

• f

turn n as it tra v erses the rf with phase

• n
• and

a sync hronous particle as it tra v erses the rf with phase

• s
• The

slip factor

• is

dened as

• t
• where
• t

is the Loren tz gamma factor

• f

particles at the transition energy and

• is

the Loren tz factor

• f

the sync hronous particle eV is the rf V

• ltage

times the c harge

• f

the particle

• is

the v elo cit y

• f

the particle expressed as a function

• f

the v elo cit y

• f

ligh t and h is the harmonic n um b er whic h is the in tegral n um b er

• f

rf

• scillations

during the time it tak es the sync hronous particle to tra v erse the ring E s is the energy

• f

the sync hronous particle One can sho w

• that

this results in sync hrotron

• scillations

with the sync hrotron tune Q s b eing giv en b y Q s

• s

h eV cos s

• E

s

SLIDE 5 where Q s is dened as the fractional n um b er

• f

sync hrotron

• scillations

during

• ne

turn around the ring The rf v

• ltage

needed can b e written as eV

• E

s h cos s Q

• S
• W

e w an t to

• p

erate the rf suc h that there is no net acceleration This implies a sync hronous phase

• f
• radians

since w e are ab

• v

e transition and cos s

• This

is kno wn in the jargon as a stationary buc k et The fractional momen tum spread

• p
• p

at the

• lev

el supp

• rted

b y the stationary buc k et can b e written as

• p
• p
• Q

s hj j

• W

e w

• uld

lik e

• f

a Gaussian energy spread to b e con tained in the buc k et hence the notation Giv en the sync hrotron tune and

• p
• p
• ne

can write eV

• Q

s

• E

s

• p
• p
• for

the stationary buc k et W e tak e the transition energy in the ring to b e

• GeV

for m uons

• and

a total circumference

• f
• kilometers

for the ring Note that w e are no w lea ving the idealized parameters

• f

the previous sim ulation and inc hing closer to a realistic design This leads to the separatrix curv es sho wn in gure

• whic

h depict the turn b y turn motion

• f

particles with v arious inital conditions The plot sho ws t w

• rf

buc k ets with sync hronous phases

• and
• Inside

the separatrix whic h is a curv e that starts with zero initial phase the particles are con tained Outside the separatrix the particles are un b

• unded

and propagate around the ring with energies that do not

• scillate

around the sync hronous energy

• It

is the

• scillations

that tak e the particles

• n

either side

• f

the sync hronous energy that will enable the p

• larization

to b e main tained since these

• scillations

cause a particle to exp erience dieren t spin precessions from turn to turn whic h do not cum ulativ ely add up to v alues dieren t from the mean spin precession

• f

the sync hronous particle

• This

leads to a v alue

• f
• for

the

• function

for the

• GeV

ring and

• for

the

• GeV

ring

SLIDE 6 Figure

• sho

ws the

• scillations
• f

the particles in v arious con tours sho wn in gure

• starting

with the con tour closest to the sync hronous particle and mo ving further a w a y

• as

a function

• f

turn n um b er The particle

• n

the con tour closest to the sync hronous phase sho ws

• scillations

consisten t with the sync hronous tune whereas the

• scillations

slo w do wn as

• ne

gets further a w a y from the cen ter

• f

the buc k et On the separatrix the

• scillations

tak e an innite amoun t

• f

time Figure

• sho

ws the initial distribution

• f

the b eam as it is injected The b eam has a momen tum spread

• pp
• f
• at

the

• lev

el The b eam energy c hosen is

• GeV

Figure

• sho

ws the ev

• lution
• f

the b eam at the end

• f
• turns

Notice that the b eam shap e in the buc k et is main tained whereas the particles

• utside

the buc k et drift in phase A

• GeV

Results W e no w compute the p

• larization

for

• particles

turn b y turn assuming a longitudinal p

• larization
• f
• at

injection and precessing eac h particle b y the amoun t appropriate to it giv en its energy for that turn Figure

• sho

ws the v ariation

• f

p

• larization

as a function

• f

turn n um b er for Q s

• for

rf v

• ltages

ranging from

• MV

It can b e seen that

• MV

is to

• small

and do es not c hange the p

• larization

appreciably from the no rf case A t

• MV

the

• scillations

are more

• r

less

• ptimal

to main tain the p

• larization

for

• turns

A t

• MV
• ne

gets to

• m

uc h rf and to

• m

uc h energy gain and a greater amplitude in the p

• larization
• scillationFigures
• sho

w the corresp

• nding

curv es for Q s

• and
• resp

ectiv ely

• It

can b e seen that the

• ptim

um p

• larization

b eha vior is for Q s

• and

rf v

• ltage

in the vicinit y

• f

MVMVFigure

• sho

ws the curv e for Q s

• This

v alue

• f

sync hrotron tune is clearly to

• lo

w since increasing the rf v

• ltage

causes dep

• larization

due to the particles energy spread increasing as a result

• f

not tra v ersing their tra jectories fast enough in

• turns

T able I I summarizes the results for all the spin tunes for

• GeV

m uon storage ring energy

• F
• r

a

• GeV

storage ring

• f

the m uons surviv e after

• turns

P is the a v erage p

• larization
• v

er

• turns

and P W is the in tensit y

SLIDE 7 w eigh ted p

• larization
• v

er

• turns

When no rf is applied

• ne

gets P

• and

P W

• It

can again b e seen that for Q s

• and

rfMV

• ne

gets a buc k et that is capable

• f

supp

• rting
• at

the

• lev

el and

• ne
• btains

P

• and

P W

• B
• GeV

Results W e rep eat the ab

• v

e set

• f

calculations for the m uon storage ring energy

• f
• GeV

and

• pp
• at

the

• lev

el The v alues

• f

P and P W for the no rf case are

• and
• resp

ectiv ely

• This

is due to faster spin precession The spin tune for

• GeV

is

• whereas

for the

• GeV

ring it is

• Figures
• sho

w the v ariation

• f

p

• larization

as a function

• f

turn n um b er for rf v

• ltages

ranging from MV MV Figure

• again

sho ws that a Q s

• is

to

• small

and the presence

• f

rf actually mak es matters w

• rse

F

• r

a

• GeV

storage ring

• f

the m uons surviv e after

• turns

T able I I I summarizes the results for the

• GeV

case It app ears as though an rf v

• ltage
• f
• MV

and Q s

• yields

the b est results

• f

P

• and

P W

• IV

SCALING RELA TIONS Using equations

• and
• and

the results presen ted here

• ne

can arriv e at the follo wing expression for the rf needed and its frequency

• eV
• Q

s

• pp
• M

V

• r

f f r eq uency

• Q

S

• pp
• M

H z

• V

HIGHER OREDER EFFECTS W e ha v e so far considered an ideal storage ring whic h supp

• rts

a large momen tum ac ceptanceIn practice there will b e sextup

• le

elemen ts presen t whic h could in tro duce spin

SLIDE 8 P arameter V alue P arameter V alue Muon Energy

• GeV
• spin

precession in

• ne

turn

• radians

Magnetic eld

• T

esla radius

• f

ring

• meters

b eam circulation time E sec dilated m uon life time E sec turn b y turn deca y constan t E T ABLE I P arameters

• f

an idealized m uon storage ring tune dep endence

• n

the particle p

• sition

as w ell the momen tum This will in tro duce faster dep

• larizations

than sim ulated here F urther w

• rk

needs to b e done to in v estigate the eect and size

• f

rf needed for suc h realistic rings VI A CKNO WLEDGEMENTS The author wishes to ac kno wledge useful con v ersations with Ch uc k Ank en brandt Nor b ert Holtk amp Carol Johnstone Da vid Neuer and Alvin T

• llestrup

VI I CONCLUSIONS W e ha v e sho wn that loss

• f

p

• larization

due to dieren tial spin precession can b e stemmed b y the in tro duction

• f

an rf v

• ltage

and explored the dep endence

• n

spin tune Once it can b e sho wn that preserv ation

• f

p

• larization

in the ring extends signican tly the ph ysics reac h

• f

the mac hine w

• rk

in progress the case for the relativ ely mo dest amoun ts

• f

rf required to preserv e the p

• larization

will seem more comp elling

SLIDE 9 FIG

• Figure

sho ws the rate

• f

c hange

• f

p

• larization

with turn n um b er for a m uon storage ring

• f
• GeV

momen tum for v arious v alues

• f

the sync hroton

• scillation

tune

SLIDE 10 FIG

• Figure

sho ws the rate

• f

c hange

• f

p

• larization

with turn n um b er for a m uon storage ring

• f
• GeV

momen tum for v arious v alues

• f

the sync hroton

• scillation

tune

SLIDE 11 FIG

• Figure

sho ws the rate

• f

c hange

• f

p

• larization

with turn n um b er for a m uon storage ring

• f
• GeV

momen tum for v arious v alues

• f

the sync hroton

• scillation

tune

SLIDE 12 FIG

• Figure

sho ws the rate

• f

c hange

• f

p

• larization

with turn n um b er for a m uon storage ring

• f
• GeV

momen tum for v arious v alues

• f

the sync hroton

• scillation

tune

SLIDE 13 FIG

• Figure

sho ws the separatrix for an rf v

• ltage
• MV

and a sync hrotron tune

• f
• The

t w

• stationary

buc k ets sho wn ha v e sync hronous phases

• f
• resp

ectiv ely

• The

con tours sho w the path

• f

particles from turn to turn The particles circulate in a clo c kwise fashion along the closed con tours from righ t to left in the un b

• unded

con tours b elo w the separatrix and from left to righ t in the un b

• unded

con tours ab

• v

e the separatrix The

• scillation
• f

energy along the n um b ered con tours are sho wn in gure

SLIDE 14 FIG

• Figure

sho ws the

• scillations

in energy

• f

particles

• n

con tours n um b ered

• f

gure

• The

further

• ne

is from the sync hronous particle the slo w er the

• scillations

SLIDE 15 FIG

• The

b eam prole at injection sup erimp

• sed
• n

the separatrix

• f

gure

SLIDE 16 FIG

• The

shap e

• f

the b eam at the end

• f
• turns

The b eam inside the separatrix remains as is whereas the particles

• utside

drift in phase with the

• nes

with p

• sitiv

e E arriving at later times larger phases

SLIDE 17 FIG

• V

ariation

• f

p

• larization

as a function

• f

turn for v arious rf v

• ltages

ranging from MV to

• MV

Beam Energy

• GeV

sync hrotron tune

SLIDE 18 FIG

• V

ariation

• f

p

• larization

as a function

• f

turn for v arious rf v

• ltages

ranging from MV to

• MV

Beam Energy

• GeV

sync hrotron tune

SLIDE 19 FIG

• V

ariation

• f

p

• larization

as a function

• f

turn for v arious rf v

• ltages

ranging from MV to

• MV

Beam Energy

• GeV

sync hrotron tune

SLIDE 20 FIG

• V

ariation

• f

p

• larization

as a function

• f

turn for v arious rf v

• ltages

ranging from MV to

• MV

Beam Energy

• GeV

sync hrotron tune

SLIDE 21 FIG

• V

ariation

• f

p

• larization

as a function

• f

turn for v arious rf v

• ltages

ranging from MV to

• MV

Beam Energy

• GeV

sync hrotron tune

SLIDE 22 REFERENCES

• Calibrating

the energy

• f

a

• GeV

m uon collider using spin precession Ra jendran Ra ja and Alvin T

• llestrup

Ph ysRevD

• RRossmanith

T alk giv en at the

• F

ermilab W

• rkshop
• n

Ph ysics at the First Muon Collider and F ron t End

• f

a m uon Collider A Blondel Pro ceedings

• f

the NUF A CT W

• rkshop

Ly

• n

F rance

• An

in tro duction to the ph ysics

• f

high energy accelerators DJEdw ards MJSyphersJohn Wiley and Sons P age

SLIDE 23 FIG

• V

ariation

• f

p

• larization

as a function

• f

turn for v arious rf v

• ltages

ranging from MV to

• MV

Beam Energy

• GeV

sync hrotron tune

SLIDE 24 FIG

• V

ariation

• f

p

• larization

as a function

• f

turn for v arious rf v

• ltages

ranging from MV to

• MV

Beam Energy

• GeV

sync hrotron tune

SLIDE 25 FIG

• V

ariation

• f

p

• larization

as a function

• f

turn for v arious rf v

• ltages

ranging from MV to

• MV

Beam Energy

• GeV

sync hrotron tune

SLIDE 26 FIG

• V

ariation

• f

p

• larization

as a function

• f

turn for v arious rf v

• ltages

ranging from MV to

• MV

Beam Energy

• GeV

sync hrotron tune

SLIDE 27 FIG

• V

ariation

• f

p

• larization

as a function

• f

turn for v arious rf v

• ltages

ranging from MV to

• MV

Beam Energy

• GeV

sync hrotron tune

SLIDE 28 Q S r f MV r f freq MHz

• p
• p

P P W

• T

ABLE I I Summary

• f

results for

• GeV

m uon storage ring P is the a v erage p

• alrization

for

• turns

and P W is the in tensit y w eigh ted p

• larization

for

• turns

SLIDE 29 Q S r f MV r f freq MHz

• p
• p

P P W

• T

ABLE I I I Summary

• f

results for

• GeV

m uon storage ring P is the a v erage p

• alrization

for

• turns

and P W is the in tensit y w eigh ted p

• larization

for

• turns