transverse motion 2
DESCRIPTION
Transverse Motion 2. Eric Prebys, FNAL. Some Formalism. Let’s look at the Hill’ equation again… We can write the general solution as a linear combination of a “sine-like” and “cosine-like” term where - PowerPoint PPT PresentationTRANSCRIPT
Transverse Motion 2Eric Prebys, FNAL
Some Formalism Let’s look at the Hill’ equation again… We can write the general solution as a linear combination of a
“sine-like” and “cosine-like” term where
When we plug this into the original equation, we see that
Since a and b are arbitrary, each function must independently satisfy the equation. We further see that when we look at our initial conditions
So our transfer matrix becomes
USPAS, Knoxville, TN, Jan. 20-31, 2013Lecture 4 - Transverse Motion 1 2
0)( xsKx
)()()( sbSsaCsx
1)0(;0)0(0)0(;1)0(
SCSC
0)()()()()()( sSsKsSbsCsKsCa
0
0
)0()0()0()0()0()0(
xbbSbCaxxaabSaCx
0
0
0
0
)()()()(
)()(
)()(
xx
sSsCsSsC
sxsx
xx
sxsx
M
Calculating the Lattice functions If we know the transfer matrix or one period, we can explicitly
calculate the lattice functions at the ends
If we know the lattice functions at one point, we can use the transfer matrix to transfer them to another point by considering the following two equivalent things Going around the ring, starting and ending at point a, then proceeding to point b Going from point a to point b, then going all the way around the ring
USPAS, Knoxville, TN, Jan. 20-31, 2013Lecture 4 - Transverse Motion 1 3
2cos1sin;21cos
sincossinsinsincos
M
M
Tr
),(),(),(),(
),(),(),(),(1
abaaabbb
aaababbb
sssCssssCs
sCssssssCs
MMMM
MMMM
),()(),()(
),(2sin)(2cos),(2sin)(2cos
2sin)(2cos),(
1
1
abaabb
abaabb
ssssss
ssssss
ssCs
MJMJ
MJIMJIJIM
Recall:
)()(
)()()(
ssss
s
J
Calculating the Lattice functions (cont’d) Using
We can now evolve the J matrix at any point as
Multiplying this mess out and gathering terms, we get
USPAS, Knoxville, TN, Jan. 20-31, 2013Lecture 4 - Transverse Motion 1 4
1121
1222
2221
1211
)()()()(
)()()()(
)(mmmm
ssss
mmmm
ssss
saa
aa
bb
bbb
J
1121
12221
2221
1211 ),(),(mmmm
ssmmmm
ss abab MM
)()()(
22
)()()(
222
2212221
212
2111211
2212211121122211
a
a
a
b
b
b
sss
mmmmmmmmmmmmmmmm
sss
Examples Drift of length L:
Thin focusing (defocusing) lens:
USPAS, Knoxville, TN, Jan. 20-31, 2013Lecture 4 - Transverse Motion 1 5
0200
0
00
0
0
0
212
1
112010
011
1101
ff
f
ff
f
f
M
Physical Implications The general expressions for motion are
We form the combination
If you don’t get out much, you recognize this as the general equation for an ellipse
USPAS, Knoxville, TN, Jan. 20-31, 2013Lecture 4 - Transverse Motion 1 6
sincos
)(;cos
Ax
sAx
x
'xA
A
Area = πA2
Particle will trace out the ellipse on subsequent revolutions
Interpretation (cont’d) As particles go through the lattice, the Twiss parameters will vary
periodically:
s
x
x
x
x
x
x
x
x
x
x
β = maxα = 0maximum
β = decreasingα >0focusing
β = minα = 0minimum
β = increasingα < 0defocusing
USPAS, Knoxville, TN, Jan. 20-31, 2013 7Lecture 4 - Transverse Motion 1
Motion at each point bounded by
)()( sAsx
Conceptual understanding of β It’s important to remember that the betatron function represents a
bounding envelope to the beam motion, not the beam motion itself
USPAS, Knoxville, TN, Jan. 20-31, 2013Lecture 4 - Transverse Motion 1 8
Normalized particle trajectory Trajectories over multiple turns
)(sin)()( 2/1 ssAsx
s
sdss
0 )()(
s is also effectively the local
wave number which determines the rate of phase advance
Closely spaced strong quads small β small aperture, lots of wigglesSparsely spaced weak quads large β large aperture, few wiggles
Betatron tune As particles go around a ring,
they will undergo a number of betatrons oscillations ν (sometimes Q) given by
This is referred to as the “tune”
We can generally think of the tune in two parts:
Ideal orbit
Particle trajectory
)(2
1sds
6.7Integer : magnet/aperture
optimization
Fraction: Beam Stability
USPAS, Knoxville, TN, Jan. 20-31, 2013 9Lecture 4 - Transverse Motion 1
Tune, stability, and the tune plane If the tune is an integer, or low order rational number, then the effect
of any imperfection or perturbation will tend be reinforced on subsequent orbits.
When we add the effects of coupling between the planes, we find this is also true for combinations of the tunes from both planes, so in general, we want to avoid
Many instabilities occur when something perturbs the tune of the beam, or part of the beam, until it falls onto a resonance, thus you will often hear effects characterized by the “tune shift” they produce.
y)instabilit(resonant integer yyxx kk
“small” integers
fract. part of X tune
frac
t. pa
rt of
Y tu
ne
Avoid lines in the “tune plane”
USPAS, Knoxville, TN, Jan. 20-31, 2013 10Lecture 4 - Transverse Motion 1
(We’ll talk about this in much more detail soon, but in general…)
Emittance
x
'xIf each particle is described by an ellipse with a particular amplitude, then an ensemble of particles will always remain within a bounding ellipse of a particular area:
Area =
Either leave the out, or include it explicitly as a “unit”. Thus• microns (CERN) and• -mm-mr (FNAL)
Are actually the same units (just remember you’ll never have to explicity use in the calculation)
USPAS, Knoxville, TN, Jan. 20-31, 2013 11Lecture 4 - Transverse Motion 1
or 22 xxxx
Definitions of Emittance and Admittance Because distributions normally have long tails, we have to adopt a
convention for defining the emittance. The two most common are Gaussian (electron machines, CERN):
95% Emmittance (FNAL):
It is also useful to talk about the “Admittance”: the area of the largest amplitude ellipese which can propagate through a beam line
USPAS, Knoxville, TN, Jan. 20-31, 2013Lecture 4 - Transverse Motion 1 12
beam theof 39% contains ;2
x
2dA
Limiting half-aperture
Adiabatic Damping In our discussions up to now, we assume that all fields scale with
momentum, so our lattice remains the same, but what happens to the ensemble of particles? Consider what happens to the slope of a particle as the forward momentum.
If we evaluate the emittance at a point where =0, we have
USPAS, Knoxville, TN, Jan. 20-31, 2013Lecture 4 - Transverse Motion 1 13
0p
xp
pp 0
xp0ppx x
0
00
1
ppxx
ppx
pppxx x
Normalized emittance
Printed copy has lots of typos!
Mismatch and Emittance Dilution In our previous discussion, we implicitly assumed that the
distribution of particles in phase space followed the ellipse defined by the lattice function
Once injected, these particles will follow the path defined by the lattice ellipse, effectively increasing the emittance
USPAS, Knoxville, TN, Jan. 20-31, 2013Lecture 4 - Transverse Motion 1 14
x
'x
Area =
…but there’s no guarantee What happens if this it’s not?
'x
x
Lattice ellipse
Injected particle distribution'x
xEffective (increased) emittance
Beam Lines In our definition and derivation of the lattice function, a closed path
through a periodic system. This definition doesn’t exist for a beam line, but once we know the lattice functions at one point, we know how to propagate the lattice function down the beam line.
USPAS, Knoxville, TN, Jan. 20-31, 2013Lecture 4 - Transverse Motion 1 15
in
in
in
out
out
out
mmmmmmmmmmmmmmmm
222
2212221
212
2111211
2212211121122211
22
in
in
in
out
out
out
inout,M
Establishing Initial Conditions When extracting beam from a ring, the initial optics of the beam
line are set by the optics at the point of extraction.
For particles from a source, the initial lattice functions can be defined by the distribution of the particles out of the source
USPAS, Knoxville, TN, Jan. 20-31, 2013Lecture 4 - Transverse Motion 1 16
in
in
in
in
in
in