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

The e�ect of RF on p olarization in a m uon storage ring Ra jendran Ra ja F ermi National A c c eler ator L ab or atory P�O� Box ��� Batavia� IL ����� Abstract W e in v estigate the preserv ation of p olarization in an idealized m uon storage rings with m uon energy �� GeV and �� GeV for b eams with momen tum ac� ceptance � p�p ���� The b eams dep olarize rap dily due to the di�eren tial rates of precession as a function of m uon momen tum� When an rf v oltage is applied� inducing sync hrotron oscillations� the p olarization is seen to b e preserv ed for sync hrotron tunes of the order of � � �� W e in v estigate the dep endence of dep o� larization as a function of sync hrotron tune and rf v oltage for �� GeV and �� GeV m uon storage rings� I� INTR ODUCTION The preserv ation of p olarization of m uons in a m uon storage ring is in v estigated� As the m uons circulate around the ring� the spin v ector of a m uon precesses faster than the momen tum v ector b y an amoun t � � � � g � �� � �� where g is the gyromagnetic ratio of the m uon and � is its Loren tz gamma factor� See ref ��� for a discussion of this in detail and references therein� Since there is a spread of momen ta in the ring� there will b e di�eren tial precession of the p olarization leading to a loss of the a v erage p olarization of the m uons as the m uons circulate the ring� If the momen tum of a m uon remains the same from turn to turn� the amoun t of spin precession will b e the same relativ e to the a v erage spin precession �

and the dep olarization will b e cum ulativ e� This e�ect has b een kno wn for a long time ��� and its remedy is to induce sync hrotron oscillations causing the individual m uon spin precession to v ary from turn to turn� W e will estimate the magnitude of the e�ect 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 oscillations prop erly � I I� SIMPLE SIMULA TION W e construct an idealized storage ring of �� GeV m uon energy with the parameters as sho wn in table I � W e inject ������ m uons in to the ring with � p�p of �� and allo w ed to circulate for ��� turns� The m uons are assumed to b e fully longitudinally p olarized and the p olarization is computed at the end of eac h turn for those m uons whose deca y electrons end up in a calorimeter of radial exten t �cm����cm placed around the b eam pip e� The p olarization is assumed to precess en tirely in the horizon tal plane and the magnitude of the a v erage p olarization 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 of p olarization as a function of turn n um b er for the �� GeV case in the absence of an y rf �Sync h���� The same m uons are used for the next turn with the same v alues of momen ta and the deca y is preformed again� The n um b er of m uons circulating turn b y turn is decreased b y the actual n um b er that w ould ha v e deca y ed� This is similar to the tec hnique used in ��� to determine the energy scale of the m uon collider� A� Generating sync hrotron oscillations A t the end of eac h turn� the m uons are sub jected to an v oltage and their momen ta rf will b e re�arranged dep ending on the phase of their arriv al at the rf � Eac h of the m uons will undergo a sync hrotron oscillation so that their energies E as a function of turn n um b er t will b e giv en b y the expression �

E � Acos � � � � � Q t � ����� � s where Q is the sync hrotron tune �fractional� and A is the amplitude of oscillation and s � is an arbitrary phase� It is imp ossible from kno wing the momen tum of eac h m uon � alone to generate sync hrotron oscillations� since one needs to compute b oth A and � � The � tric k is to generate the oscillations 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 or eac h m uon� generate t w o random n um b ers r and r suc h that b oth of � � these are Gaussian distributed with standard deviation � p � The t w o�dimensional distribution of these t w o Gaussian n um b ers is cylindrically symmetrical and is exp onen tially distributed � � in r � r � where r and r are pairs of Gaussian random n um b ers� The pro jection of this � � � � distribution along an y radial direction is Gaussian� Sync hrotron oscillations are obtained b y rotating this distribution b y the angle � � Q t and taking the x� comp onen t of the v ectors� s This will ha v e the e�ect of in tro ducing oscillations while at the same time preserving the mean and standard deviation of the distribution from turn to turn� B� � p�p �� � case W e in v estigate the case for a b eam energy spread of �� at the � � lev el� Figure � sho ws the p olarization as a function of turn n um b er for sync hrotron tunes ���� ������������ and ����� F or a sync hrotron tune of ����� the p olarization is stable for ��� turns� but con tains an oscillation term� These oscillations p ersist with increased frequency but m uc h reduced amplitude for sync hrotron tune v alues ��������� Figure � sho ws the corresp onding distributions for the �� GeV case� The loss of p olarization for the no case is slo w er in the rf �� GeV case as exp ected� since spins precess slo w er� �

� p�p ���� � C� case It has since b een p oin ted out that the curren t F ermilab design for a m uon storage ring en visages rms � p�p � ����� Figures � and � sho w the dep olarization as a function of turn n um b er for sync hrotron tunes of ���������� It can b e seen that in this case a sync hrotron tune of ���� is su�cien t to main tain p olarization up to ��� turns� I I I� MORE REALISTIC SIMULA TION The ab o v e sim ulation assumes that all the particles are in the buc k et and undergo rf sync hrotron oscillations with the same tune� In practice� this is not the case and w e will no w attempt to sim ulate realistic sync hrotron oscillations using the sync hrtorn oscillation di�erence equations ���� � E � � E � eV � sin� � sin� � ����� n �� n n s � � h� � E n �� � � � � ����� n �� n � � E s where � E is the di�erence b et w een the energy gained b y a particle at the end of turn n n �� as it tra v erses the with phase � � and a sync hronous particle as it tra v erses the with rf rf n � � phase � � The �slip factor� � is de�ned as � �� � � �� � where � is the Loren tz gamma s t t factor of particles at the transition energy and � is the Loren tz factor of the sync hronous particle� eV is the rf V oltage times the c harge of the particle� � is the v elo cit y of the particle expressed as a function of the v elo cit y of ligh t and h is the �harmonic n um b er�� whic h is the in tegral n um b er of rf oscillations during the time it tak es the sync hronous particle to tra v erse the ring� E is the energy of the sync hronous particle� One can sho w ��� that this s results in sync hrotron oscillations with the sync hrotron tune Q b eing giv en b y s s � h� eV cos� s Q � ����� s � � � E � s �