LEP Optics with (x ; y ) = (108 ; 90 )
J.M.. Jowett, CERN, Geneva, Switzerland
Abstract
An optics with phase advances of (x ; y ) = (108 ; 90 )
in the arc cells has a number of advantages over one with
(x ; y ) = (108; 60). In particular its dynamic aperture
is larger and the single-bunch current limit is higher. The
reasons for these differences are now well understood. Experiments on an optics with odd integer parts of the tunes
confirmed these expectations. A preliminary version of a
new optics with even integer parts of the tunes is presented
and shown to have a good horizontal dynamic aperture.
With further work it may meet most other criteria for an operational LEP2 optics.
1
MOTIVATION
The maximum operating energy of LEP will be determined
by the available dynamic aperture. It has been shown previously [1] that the (90 ; 60) optics used in physics for
the last two years has a barely adequate horizontal dynamic
aperture for energies above 90 GeV. The (108; 60) optics, despite the advantage of a lower horizontal emittance,
is similarly—but even more severely—limited. It has been
shown recently [2] that a (108 ; 90) optics is horizontally
limited at a larger value by a different physical effect.
Another significant advantage of any optics with a vertical phase advance y = 90 per arc cell is a higher threshold for transverse mode-coupling instability than an optics
with y = 60.
On the other hand, there are some well-known disadvantages of this choice: it is expected to be harder to correct
the vertical orbit and the polarization level is expected to
be lower. Indeed these were the principal reasons for our
switch to (90; 60 ) for LEP1 here in 1993. The reduced
polarization would reduce the maximum energy where we
can do energy calibration [9].
The sextupoles in arcs would need to be re-cabled at any
switch to a (108 ; 90) optics.
1.1 Understanding Dynamic Aperture Plots
In this talk, as well as in [3, 4], a number of dynamic aperture plots will be shown. I would like to pause here to recall [1] what they mean. Ideally, these plots should show the
6 dimensions of particle phase space but present graphical
displays are limited to 2, 3 or 4 dimensions. When limited
to three dimensions, it is most natural to take the three action variables, I = (I1 ; I2; I3) ' (Ix ; Iy ; It) of the modes
of linear oscillation around the closed orbit and suppress
the phases. Here the approximate equality applies in the familiar case where the first two modes can be approximately
identified with horizontal and vertical “betatron” motion and
the third with “synchrotron” oscillations. Most of the time,
we track with fixed initial phases = (0; 0; 3=2) and, most
of the time, this is justifiable. Sometimes, however, it is important also to average over the phase of the third mode in
order to evaluate the momentum acceptance.
By convention, the plots are made, not in terms of the
actions themselves, but the square roots of closely related
quantities (Ax ; Ay ; At) = (2I1 ; 2I2; 2I3 ). In terms of
these, we can say that:
p
103 Ax =m
is the maximum amplitude of
horizontal betatron oscillations, expressed in
mm, at a place where x = 1 m. (Similarly for
Ay )p
At =% is the maximum amplitude of energy
oscillation (expressed in %).
Thus, the physical horizontal displacement of a particle
from the closed orbit is given by the expression:
x
'
p A cos (2Q s=C + (s) + )
x px
x
x
x
+D A cos (2Q s=C + (s) + )
x
t
s
t
t (1)
(These simplifications are not strictly correct unless the ring
is uncoupled, flat, etc. The proper description in terms of
the projections of eigenmodes of linear oscillation around
closed orbit is always used in the calculations. Strictly
speaking, we give amplitude variables for modes 1, 2 and
3.)
Most of our dynamic aperture plots (e.g., Figure 1) show
two surfaces. The more irregular one is the dynamic aperture surface itself obtained by tracking particles with radiation damping until they are either lost or it is sure that they
started in the basin of attraction of the closed orbit (this is
our computational definition of the dynamic aperture). The
ellipsoidal surface represents the beam-stay-clear, taken as
an indication
of a comfortably adequate dynamic aperture.
p
In the Ax direction this ellipsoid corresponds to 10 of
the beam distribution corresponding to the horizontal emittance, x. In the Ay direction, it corresponds to 10 of
a beam distribution in which the vertical emittance is artificially set to y = x =2 (thus covering
p the extreme case of
maximal betatron coupling). In the At direction it corresponds to 7 of the energy distribution in the beam, enough
for adequate quantum lifetime. This projection of the dy-
p
namic aperture can often be increased by turning up the RF M05D46v2.lep2, 108/60 BT 1995, beta*x=2.5m, 91 GeV
voltage.
In all cases shown here, the tracking is done with radiation
damping (see [1]) switched on but without quantum fluctuations. The number of turns is chosen long enough that there
is no ambiguity about whether a given particle is stable or
1
not. At 91 GeV, 100 turns are more than sufficient.
Sqrt[At/%]
2
DYNAMIC APERTURE PROBLEM
FOR LEP2
2
1.5
1 10^3Sqrt[Ay/m]
0.5
0.5
0
2.1 Dynamic aperture of (108 ; 60 )
0
0.5
0
1 1.5
2
10^3Sqrt[Ax/m]
Second surface is {10, 10, 7}sigma ellipsoid
M1_25_05D46v2.lep2, e+ 108/60 BT 1996, betax*=1.25
VRF=2286.4 MV
Figure 2: Dynamic aperture of a (108 ; 60) optics similar
to that given in Figure 1 except that the experimental insertions are detuned to make x = 2:5 m.
91 GeV
1
Qs = 0.0913
Sqrt[At/%]
0.5
2
1.5
10^3Sqrt[Ay/m]
1
0.5
0
2
1.5
1
0.5
0
10^3Sqrt[Ax/m]
Second surface is {10, 10, 7}sigma ellipsoid
0
A y = A x / 2 in beam
distribution
Qy = 4 × 19
Sqrt[Ay/m]
lattice
90°/60°
90°/90°
108°/60°
108°/90°
2
1.5
1
0.5
q
The work described in [2] has provided a good analytical
understanding of the reasons for the instability. In particular we know that it is not sensitive to imperfections. However there is no cure so far for this optics. It was shown
that octupoles may improve the situation somewhat in the
case of the (90; 60) lattice and this may be worth trying
on (108; 60).
∂Qx/∂Ax
1,750
950
23,560
23,650
∂Qy/∂Ax
-27,500
-13,930
-81,180
-17,060
∂Qy/∂Ay
18,210
960
75,430
11,340
(depend on low β)
I6x
I6y
62.8
207.9
84.5
226.1
75.4
218.2
79.2
216.3
10^3Sqrt[Ax/m]
0.5
1
1.5
Figure 1: Dynamic aperture of a (108 ; 60) optics at
91 GeV. The inset plot shows the projection of the 3dimensional surfaces onto the horizontal plane of the two
“betatron” amplitudes.
Figure 1 shows the dynamic aperture of the latest
(108 ; 60) lattice proposed for use in 1996 [5]. It
does not allow 10x of clearance in the horizontal oscillation mode. As shown in [2] for a somewhat different
(108 ; 60) lattice, the limitation arises from the change
of the vertical tune with horizontal betatron amplitude,
@Qy . Table 1 shows how the value of this and other tune
@Ax
derivatives vary according to the phase advances of the arc
cells.
Some measurements of the dynamic aperture of a
(108 ; 60) optics were carried out last year [6]. The
interpretation of these measurements is a little contentious
but I interpret them as being in rather good quantitative
agreement with dynamic aperture calculations done in the
conditions of the measurements; this will be discussed
further tomorrow [4].
Table 1: Quantities determining detuning and RBSC instability at very high energy, reproduced from Table 4.1 of [2]
3
(108 ; 90 )
WITH ODD TUNES
@Q
Perturbation calculations [2] of @Axy shown in Figure 3 pro-
vided a useful guide to a potentially better choice of arc
phases. Last year this led us to make a first experimental
study of a (108; 90) optics. It was designed with odd tunes
simply because it was not practically possible to re-cable
the sextupoles in the arcs of LEP; at the time they were cabled into 2 SF and 3 SD families for the (90; 60) operational optics. The correction of chromatic effects had to be
done with all sextupoles grouped into just one SF and one
SD family (Normally 2 SD families would be used for an
optics with y = 90 .) and this was expected to work better with odd than with even tunes. Most recent LEP optics
have had even tunes since this is believed to provide better beam-beam performance; if the sextupoles are re-cabled
(which requires an intervention of about two days in the LEP
tunnel), there is no reason why an even-tune version cannot
also be developed (see Section 6).
Figure 2 shows a new 3-dimensional scan of the dynamic aperture of this optics, confirming the results of [2].
The horizontal dynamic aperture is considerably larger than
shown in Figure 1 in accordance with the reduced value of
@Qy indicated in Table 1. The horizontal dynamic aperture
@Ax
is in fact limited by another physical effect, the Radiative
Beta-Synchrotron Coupling (RBSC) instability, described
in [1].
A comparison of the quantities I6x and I6y (see Table 1)
that characterise this instability shows that the (108; 90)
optics is not significantly better or worse than the others.
It is true, however, that reducing the value of x will enhance this instability because of the additional radiation in
the quadrupoles of the interaction region. Figure 1 was computed for an optics with x = 1:25 m while Figure 4 was
done for x = 2:5 m. It has been pointed out that the comparison between the two made in my talk was somewhat unfair. To remedy this, I now include the dynamic aperture of
the (108 ; 60) optics with x = 2:5 m as Figure 2.1 While
the situation is a little better for x = 2:5 m, it can still only
be described as marginal.
I would like to direct your attention to Figure 8 of [2]
which shows how the addition of imperfections affects the
dynamic aperture of various lattices. It should be noted that
the dynamic aperture of the (108; 90) case is diminished
somewhat by imperfections while that of (108; 60) is not.
This can be understood on the basis of the physical effects
which determine the stability limit. Detuning with amplitude is not significantly changed while a perfect lattice limited by RBSC is more vulnerable to the imperfection-driven
resonances which appear at large amplitudes.
3.1 MD on (108 ; 90 ) in 1995
In 1995 we had two MD sessions with the odd-tune optics:
15/9/95, fill 1989.00 The beam circulated immediately
and we measured single-bunch instability limit (see Section 4). We ramped 4 e+ bunches of 0.1 mA to 45.6 GeV
with no losses.
∂Q
∂Qy y
∂A
∂Ax x
(units of 1000)
µ y = 60 o
∂Qy
Perturbation
calculations using
Hori-Deprit
algorithm
implented in
Mathematica
(Y. Alexahin,
CERN-SL-95-110
(AP).
JM J
tt (108° 90°)
µx /o
(units of 1000)
∂Ax
µy /o
1
1: µ x = 90 o
2
2: µ x = 108o
ti
LEP P f
W kh
Ch
i 16/1/96 P
6
Figure 3: How the choice of arc-cell phase advance can affect the critical derivative of the vertical tune with horizontal amplitude. The colour-coding of the dots (if you can see
it) serves to guide the eye and is defined in Table 1. This
figure is also reproduced from Figures 6 and 7 of [2].
108/90 optics 1995, Y05E46, 91GeV, odd tunes, stan
VRF=2464 MV
1.5
91 GeV
1
Sqrt[At/%]
0.5
00
2
1
10^3Sqrt[Ax/m]
2
1
10^3Sqrt[Ay/m]
3
4 0
7/10/95, fill 3062.00 Continuingfrom the previous MD,
we squeezed to y = 9 cm and measured dynamic aperture
using the injection kicker, with and without the emittance
wigglers. Afterwards, we squeezed to y = 5 cm and measured dynamic aperture again.
It proved to be possible to correct the closed orbit to
yRMS = 0:35 mm. Following this a transverse polarization of 10 % was measured without further optimisation of
beam conditions. At the end we lost the beam by switching on damping wigglers (this was expected in view of the
limited momentum acceptance).
1 This is by no means justice for all optics since the (108 ; 90 ) optics
in Figure 4 was tracked with a very asymmetric RF voltage distribution
while the (108 ; 60 ) optics in Figures 1 and 2 were privileged with perfect symmetry.
Second surface is {10, 10, 7}sigma ellipsoid
RBSC
Sqrt[Ay/m]
2.5
2
1.5
1
0.5
10^3Sqrt[Ax/m]
0.5
(108° 90°)
i
1 1.5
2 2.5
f
3
kh
Ch
i 16/1/96
8
Figure 4: Dynamic aperture of the 1995 (108; 90) optics
with odd tunes at 91 GeV. Otherwise this figure is analogous
to Figure 1.
The RMS vertical dispersion was measured to be
DyRMS = 5 cm.
The dynamic aperture measurements were found to be in
good agreement with tracking for the conditions of the experiment. By including the effect of imperfections it was
possible to model the experiment in detail (see [2], to be reviewed tomorrow in [4].
4
Measurement of 10% polarization in MD on
(108; 90).
Unexploited potential to improve the polarization by
spin harmonic compensation, etc.
So far there have been no measurements of polarization on
(108; 60) to make a comparison.
SINGLE-BUNCH INTENSITY
0.2 Optimum Qs for
A higher vertical phase advance reduces the average value
of y in the arcs. As discussed in other talks [7, 8] there is
a theoretical expectation that this reduces the effect of the
transverse impedance of bellows joining vacuum chamber
sections. Since these are give a major contribution to the total transverse impedance, we can expect to raise the threshold for the transverse mode-coupling instability (TMCI) by
changing from a lattice with y = 60 to one with y =
90 . Figures given in Table 4.2 of [2] indicate that the improvement expected should be of the order of 12 %.
In the first MD, we saw that the single-bunch current was
limited at Ib = 505 A in conditions (damping wiggler field
+ = 1:02 T, polarization wigglers off and Qs ' 0:085 )
BDW
where it had been limited at Ib = 430 A in the (108; 60)
optics. The instability limiting the current had the classic
hallmarks of TMCI.
The tune-shifts with current were also measured in the experiment [2].
The increased TMCI threshold will increase luminosity at
LEP2 provided that the two-beam current is not otherwise
limited (e.g., by parasitic beam-beam effects at injection). It
is worth mentioning that, when some of the copper RF cavities are removed, the factor by which we will gain by moving to a higher vertical phase advances will increase as the
bellows account for a larger fraction of the total transverse
impedance.
5 POLARIZATION
Polarization at energies up to about 65 GeV is desirable for
energy calibration by extrapolation to 90 GeV [9].
So far there have been no measurements of polarization
above 46 GeV or so on any lattice. Still, theoretical understanding is now good enough for quantitative prediction. It
consists essentially of linear depolarization theory for lattice
modelling supplemented by sideband theory for additional
depolarization due to synchrotron sideband resonances and
energy spread. The polarization levels attainable with y =
90 are expected to be lower than with y = 60. This is
partly due to the difficulty of correcting vertical orbit to sufficient level when pickups are spaced at intervals of 90 .
The reduced level of polarization is the biggest disadvantage of a (108 ; 90) optics. Despite all this, there are some
grounds for hope, namely:
Success with the vertical orbit in MD (yRMS
0:35 mm).
=
maximum P
P/%
20
Polarization at Z= 10%
0.1
15
0
10
E/GeV
50
60
70
80
90
5
E/GeV
0
50
60
70
80
90
P/%
20
15
Optimistic prediction
Polarization at Z= 20%
?
10
5
E/GeV
0
50
60
70
80
90
Figure 5: Extrapolations of measured polarization value
measured at 45 GeV to higher energy using the sideband
theory to account for the effect of the increasing energy
spread in the beam. At the same time, the synchrotron tune
is adjusted as a function of energy in order to minimise depolarization effects, as shown in the small plot at top right.
The upper left plot shows an extrapolation based on our
measured value P = 10% at 45.6 GeV while the lower plot
presents an extrapolation from an optimistic P = 20%.
6
NEW (108 ; 90 ) OPTICS WITH EVEN
TUNES
As mentioned in Section 3 and shown in [10], even integer
parts of the tunes are favoured for beam-beam reasons. A
(108; 90) optics with [Qx] = 102 and [Qx] = 96 is under
development. This part of my talk describes some preliminary results, on this new optics.
To allow a better chromatic correction the sextupoles
have been re-cabled (within MAD) into two SF and two
SD families (as they were back in 1992, otherwise the layout is for 1996). The matching started from the (108; 60)
optics proposed for 1996 [5] and many of the insertions
are identical. The present version needs some minor improvements (particularly to the dispersion suppressors) and
checks need to be made that it meets all constraints now imposed on an operational optics, is not too sensitive to imperfections and so on. However, because of its similarities to
the (108; 60) optics, one can say that it is basically bunchtrain compatible2 .
At the time of speaking, little effort has been made to improve the chromatic behaviour of this optics so the momentum acceptance is poor as indicated in Figure 6.
q Not yet well corrected ...
Preliminary
102.38
96.19
102.36
96.18
102.34
96.17
96.16
102.32
96.15
102.3
108/90 preliminary Qx=102,Qy=96, e+ betax*=2.5m, 9
2
1.5
1 Sqrt[At/%]
4
3
10^3Sqrt[Ay/m] 2
1
0
0.5
0
3
2
1
0
10^3Sqrt[Ax/m]
Second surface is {10, 10, 7}sigma ellipsoid
Figure 7: Dynamic aperture of a preliminary (108 ; 90)
optics with even tunes and x = 2:5 m at 91 GeV. See important comments in the text concerning the momentum acceptance. For this tracking, the RF voltage distribution was
moderately asymmetric.
96.14
102.28
102.26
Qx
Qy
96.13
96.12
102.24
7 CONCLUSIONS
96.11
102.22
96.1
At present it seems that a (108; 90) optics with even or
odd tunes is the strongest candidate for use at the highest
LEP2 energies. It is probably the best candidate to overcome the limitation of LEP energy by dynamic aperture, i.e.,
this may be the only way to get to LEP’s top energy. Or there
may be no way.
Other advantages accruing to this choice of optics include
higher single-bunch current which will help luminosity. The
main drawback is the reduced expectation for polarization.
First indications are that an even-tune version of a
(108; 90) optics is feasible. The preliminary version of
a physics optics has good horizontal dynamic aperture although some reduction should be expected from imperfections. Moreover the horizontal dynamic aperture will be
sensitive to the RF voltage available (because of RBSC instability). Its optical performance and detailed suitability
(check-list) remain to be checked.
Although it should be possible to match to x = 1:25 m,
this may reduce dynamic aperture (extra radiation in low-
quads may enhance the RBSC).
So far, in my opinion, experiments have failed to show
that tracking results are pessimistic. I.e., we must take dynamic predictions seriously (see also [4])!
102.2
-0.008 -0.006 -0.004 -0.002
0
96.09
0.002 0.004 0.006 0.008
Figure 6: Tune variation as a function of a fixed momentum deviation for non-radiating particles in a machine with
no RF system. Although it is only roughly related to the
physics of the problem, this is enough to show that the momentum acceptance of this optics needs to be improved.
The dynamic aperture of this preliminary optics is shown
in Figure 7. Although the momentum acceptance looks
good here, this is due to the fact that all particles start off
with the same synchrotron phase in the tracking. An averaging over synchrotron phase—for which there was not
enough time before this workshop—will reduce the aperture
for synchrotron oscillations. However note that exactly the
same tracking methodology was applied in all the cases reported in this talk so all other cases would suffer some reduction.
The main point of showing such a preliminary result is to
show that it is possible to maintain the improvement in horizontal dynamic aperture in a (108; 90) optics with even
tunes. Whether this optics can be developed further into a
workable solution for the highest operating energies of LEP
remains to be demonstrated.
2 When a final version is developed we will need to check non-local
side-effects of bunch train bumps and parasitic encounters, such as how
the vertical dispersion adds up at the IPs, etc.
Acknowledgements This talk has drawn heavily on the
work of Y. Alexahin [2]. Recent dynamic aperture calculations have been carried out very efficiently with the help of
new software developed with S. Tredwell.
8
REFERENCES
[1] J.M. Jowett, “Dynamic Aperture for LEP: Physics and Calculations”, in J. Poole (Ed.), Proceedings of the Fourth Work-
shop on LEP Performance, Chamonix, January 1994, CERN
SL/94-06 (DI) (1994).
[2] Y. Alexahin, “Improving the dynamic aperture of LEP2”,
CERN–SL–95–110 (AP) 1995.
[3] J.M. Jowett, “Problems expected from RF asymmetries”, this
workshop.
[4] F. Ruggiero, “Dynamic Aperture”, this workshop.
[5] M. Meddahi, private communication.
[6] C. Arimatea, D. Brandt, A. Hofmann, G. von Holtey, R. Jung,
M. Lamont, M. Meddahi, G. Morpurgo, F. Ruggiero, SL-MD
Note 199 (1995).
[7] A. Hofmann, “Bunch Intensity Limitations”, this workshop.
[8] K. Cornelis, “TMCI and what to do about it”, this workshop.
[9] M. Placidi, “Energy measurement possibilities for LEP2”,
this workshop.
[10] E. Keil, “Beam-beam effects as a function of the tunes”, this
workshop.