Lecture Notes in Physics: Nonlinear Dynamics Aspects of Particle Accelerators

590 Pages·1986·12.22 MB·English

by  L. A. Radicati

GNINEPO SSERDDA
SNOITAREDISNOC NO EHT STATISTICAL NOITPIRCSED FO MAEB-DEGRAHC SAMSALP
L.A. Radicati
Scuola Normale Superiore di Pisa, Pisa, Italy
E. Picasso dna F. Ruggiero
,NREC Geneva, Switzerland
Presented by .E Picasso
i. NOITCUDORTNI
Accelerators are non-linear dynamical systems whose evolution is governed by
electromagnetic forces arising from external fields dna from the interaction of the
maeb particles gnoma sevlesmeht dna with their environment. These forces, qnoma
which ew must include those desuac yb external noise dna yb mutnauq effects, evah a
modnar tnenopmoc os that the time evolution of the maeb particles should eb properly
debircsed yb a metsys of emos 2101 coupled non-linear stochastic equations.
Solving such a system is, of course, na impossible task. What one usually does
is to substitute into the system of coupled equations a single stochastic equation,
describing the motion of one particle under the combined effect of the external
fields dna of the forces created by all the other particles, namely na equation of
the form:
: K(x,t;~) + Z R (x,t;6) ~ (t) . (1.1)
Here x = (~,2) are the single particle coordinates dna momenta, while @(x,t) is the
phase space particle density. The "forces" no the r.h.s, of Eq. (1.1) have been
split into a deterministic Hamiltonian component K dna a random component, depending
no the stochastic variables ~(t) which describe both external dna internal sour-
ces of noise. Since the correlation time of this noise is generally very short,
compared to the relaxation time of the distribution ~, the random variables {~(t)
can eb assumed to eb Gaussian. Then the stochastic equation (I.I) becomes equi-
valent to a kcnalP-rekkoF (F-P) equation for ,~ having the form of a continuity
equation in esahp spacel,2).
nA alternative procedure to investigate the time evolution of r ylnommoc adop-
ted in most simulation sedoc 3"5), is to track a given sample of particles (typi-
cally in the order of 301 ) yb snaem of the stochastic equation of motion (1.1). ehT
dohtem consists of na approximate integration of .qE (1.1) over subsequent small
time intervals At, thus relating the wen coordinates dna atnemom of the particles
xi(t+At ) to their previous values xi(t ) dna to r At each time step, a
proper smoothing of the sample distribution in phase space allows the computation of
the density r
ehT esu of the P-F equation6, )7 - or of its deterministic version, the Vlasov
g) equationS, in accelerator physics is not :wen it has, in particular, neeb
-
advocated yb J.D. noswaL )01 dna R.C. 11,12) Davidson ohw have dezisahpme the ana-
logy neewteb particle smaeb dna charged .samsalp In this paper, which sah didactic
purposes, ew want to show in a wef examples woh the P-F equation nac eb desu to
describe in a unified yaw emos of the statistical anemonehp that era especially
important in nredom high luminosity machines. ruO hcaorppa is similar to that in
Ref. ,6 but instead of the 2-particle correlation function, it is the single par-
ticle distribution ~ which plays a central r61e in view of a fluid description of
accelerator .smaeb
ehT paper is organized sa follows: in Section 2, ew recall the relation neewteb
the N-particle distribution function dna the density r dna discuss the main charac-
teristics of the maeb plasma; Section 3 contains a short description of the forces
acting no the ;maeb in Section 4 ew write nwod the P-F equation dna briefly discuss
the effects giving rise to stochastic forces. eW then apply the theory sketched in
the previous sections to a wef specific examples: radiation damping dna mutnauq
effects (see Section 5); intra-beam scattering (Section 6); maeb-maeb interaction
(Section 7); stochastic cooling (Section 8); dna Section 9 contains a wef tentative
conclusions.
2. CITSILIBABORP .NOITPIRCSED EHT MAEB AMSALP
ehT presence of stochastic dna collective effects suggests that the natural
description of the motion of the maeb particles should eb a statistical eno in terms
of na N-particle distribution function 7)
f : f(~z .....~ ,EI ..... )t,N__p , (2.1)
where the ~i dna ~i era the coordinates dna mechanical atnemom of the par-
ticles. ehT function f allows the calculation of the time-dependent average values
of yna observable physical quantities dna provides a complete probabilistic
description of the system. Unfortunately, this description is far too complicated
to eb useful. A more convenient yaw of describing a system of N indistinguishable
particles is to define a one-particle distribution function r obtained yb
integrating f over the variables of (N-l) particles. morF ~ eno then defines the
charge density p(q,t) dna the current density j(q,t) yb the relations
p(q,t) = e f p3d ~(q,p,t) ,
(2.2)
~(~,t) = e f p3d c(p 2 + )2c2m -I/2 p_ ~(q,p,t)
.
ehT functions (2.2) nac eb viewed sa the charge dna current density of a plasma con-
fined in the region occupied yb the .smaeb It is a plasma which differs in ynam
respects from a fusion plasma. Indeed:
i) It is charged, which snaem that there era long range interactions between par-
ticles without screening at the eybeD length sa in ordinary .samsalp
ii) ehT particle density n = p/e is hcum smaller than in a fusion plasma. Typical-
ly n = i0 B - 01 9 mc -3 sa detupmoc in the local rest frame of the .maeb roF
,nosirapmoc n ~ 4101 mc -3 in a fusion plasma.
iii) Although the fluid velocity ~ = en/__j is close to the speed of light, in the
local rest frame the plasma is non-relativistic. This snaem that the mutnemom
spreads era hcum smaller than .cm
iv) ehT fluid description of our plasma is bound to eb complicated, because of the
anisotropy introduced yb the external fields. This implies that the pressure
dna the temperature will eb represented yb anisotropic tensors.
.3 LANRETNI DNA LANRETXE SECROF
ehT forces acting no the plasma arise from the external electromagnetic fields
dna from those created yb the particles in the maeb itself. ehT external forces,
independent of ~, give rise to the so-called single particle effects, the only seno
usually taken into account to determine the equilibrium orbit. ehT stability of
this orbit, with respect to small deviations from the equilibrium values qo(t) dna
Po(t), nac eb studied yb linearizing the external forces around qo(t) dna
~o(t)13). However, a fully non-linear analysis is dedeen to discuss other impor-
tant properties of the machine, such sa the dynamical aperture or the size of the
RF-bucket14).