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					           MEASUREMENT OF THE ELECTRON CLOUD IMPEDANCE

                                           K. Cornelis, CERN Switzerland


                                                                 the momentum deviation). This technique was tested [2]
ABSTRACT
                                                                 in order to measure chromaticity in an easy way.
The difference between the head and tail betatron phase
advance depends strongly on the impedance seen by the
bunch. Using a wide band pick up in the SPS this phase                                                   DP/P < 0
difference between the head and tail of an LHC bunch
can be measured. Comparing the data of bunches in the
beginning and at the end of the batch gives an empirical
measurement of the contribution of the electron cloud to
the total impedance.

                                                                  Head                                                    Tail

                1 INTRODUCTION
 Experimental results seem to indicate [1] that the nature
of the horizontal instability, created by the electron
cloud, is a low order coupled bunch instability. This
implies a long wake ranging over the distance of one or                             DP/P > 0
more bunch spacings.
                                                                 Fig. 1 : Due to the chromaticity the betatron oscillation
In the vertical plane the instability looks more like a
                                                                 for the particles moving forward will be slower than for
single bunch head-tail instability, indicating a short
                                                                 the particles moving backwards.
range wake field, created by the electron cloud. Using
the head-tail monitor one can get an idea of how this
                                                                 The instrument is consist of a coupler with a bandwidth
vertical wake field looks like.
                                                                 of 2 GHz, connected to a scope that can sample the
                                                                 transverse signal every turn with resolution of 0.5
   2 MEASUREMENT OF IMPEDANCE                                    nsec/point for about 300 turns.
      WITH HEAD –TAIL MOTION
                                                                           0
2.1 Head-tail Motion and Chromaticity                                           0     50       100        150       200      250
                                                                      -0.05


In order to understand better what follows, it is necessary              -0.1
                                                                 1
to explain the principle of the head-tail monitor as it was           -0.15
developed by R. Jones, in order to provide a quick
                                                                         -0.2
chromaticity measurement [2]. In fig. 1 the betatron
oscillation of the particles are represented by little clocks.        -0.25
By giving a transverse kick all the clocks are reset to the                                          t
same time. The big circle represents the position of the
particles in longitudinal phase space. With a positive           Fig. 2 : Evolution of the phase difference between head
chromaticity and above transition, the clocks moving             and tail due to chromaticity
backwards (upper half) will run slower and the clocks on
the lower part will run faster. So during the first half of
the synchrotron period the clocks arriving at the head           2.2 Head-Tail Motion and Impedance
will run behind, and those arriving at the tail will be
early. In the second half of the synchrotron period the          Once the intensity of the bunch increases, the transverse
slow clocks will catch up again and the fast clocks will         wake field created by the head of the bunch will also
be slowed down. It is easy to show that the phase                influence the phase advance of the tail and this in a
difference (time difference) between the clock at the head       different way than the chromaticity. The transverse wake
and the tail evolves like : C.(1 - cos(Zst) (fig 2) where        which depends on the position of the head part of the
the constant C depends on the product : [.DP/P, ([ being         bunch gives a driving force for the tail oscillator. The
the chromaticity, Zs is the synchrotron frequency, DP/P          phase difference between driving force and induced


Chamonix XI                                                                                                                      163
oscillation is 90°. So, if initially head and tail start of                                  3 DEPENDENCE OF THE WAKEFIELD
with the same phase and amplitude , the tail will start to                                     ON THE POSITION IN THE BATCH.
have an oscillator component that is 90° shifted from the
head and proportional to the wake created by the head                                       This technique can now be applied to different bunches
(fig. 3).                                                                                   in an LHC bunch train. Looking at the phase evolution
                                                                                            of a bunch in the beginning of the batch the result is
                                                                                            comparable to the single bunch measurement with the
                                                                                            same bunch intensity (fig 4). For a bunch at the end of
                                                                                            the batch the result is clearly different (fig 6).
                         a                                                           Tail
 Head
                                                                          -iWa
                                                                                                 1.40E+00

                                                                                                 9.00E-01

                                                                                                 4.00E-01

                                                                                                -1.00E-01
Fig. 3 : Influence of the head oscillator on the tail                                                             1     26 51 76 101 126 151 176 201 226
                                                                                                -6.00E-01
oscillator due to wake field.
                                                                                               -1.10E+00
This gives quit a different result than the chromaticity
                                                                                               -1.60E+00
measurement. In fig 4 the measured phase advance is
                                                  10
shown for a single bunch with an intensity of 6 10 .
                                                                                            Fig. 6 : Phase advance evolution for a bunch at the end
                                                                                                                                             10
                                                                                            of the batch for a bunch intensity of 6 10 and
  1 .40 E + 0 0
                                                                                            Qs=200turns.
   9 .00 E -0 1
                                                                                            The phase evolution in fig. 6 shows a higher frequency
   4 .00 E -0 1
                                                                                            component than in fig. 4. The only way to obtain this
  -1.0 0 E -0 1                                                                             with calculation is by adding a wake with a shorter
                   1     26 51          7 6 1 01 1 26 1 51 1 76 2 01 2 26                   interaction length (0.3 to 0.5) times the bunch length
  -6.0 0 E -0 1
                                                                                            (fig. 7).
 -1.1 0 E + 0 0

 -1.6 0 E + 0 0                                                                                                                            d iff

Fig. 4: Phase advance measurement for a single bunch                                          1.60 E + 00
       10
of 6 10 . Qs is 200 turns.                                                                          4 TRANSVERSE INSTABILITIES
                                                                                              1.20 E + 00
                                                                                               8.00 E -0 1
This can be compared to a calculation, using a simple                                          4.00 E -0 1
step-like wake with an interaction length larger or equal                                     0.00 E + 00
to the bunch length (fig 5).                                                                  -4. 00E -01
                                                                                                             1    17   33   49   65   81   97 113 129 145 161 177 193 209 225 241

                                                                                                             C
                                                                                              -8. 00E -01
                                                                                                                   0.00E+ 00
                                            d iff                                                            W
                                                                                              -1. 20E + 00
                                                                                                                   1.10E-01
                                                                                                             IL
1.60E + 00                                                                                    -1. 60E + 00         4.00E-01
1.20E + 00
 8.00E -01
 4.00E -01
                                                                                            Fig.7: Calculated phase advance with an interaction
0.00E + 00
                                                                                            length of 0.4 times the bunch length.
              1    17   33   49   65   81   97 113 129 145 161 177 193 209 225 241
-4.00E -01
              C
                       0.00E + 00
-8.00E -01
              W
-1.20E + 00
                       2.20E -02                                                                      4 THE WAKEFIELD PICTURE
              IL
-1.60E + 00            1.00E + 01
                                                                                            Summing up the measurements one can conclude the
Fig. 5: Calculated phase difference for a simple step                                       following:
wake function.                                                                              N The leading bunches in the batch see the same
                                                                                                 vertical wakefield as a single bunch. The interaction
                                                                                                 length is equal or larger than the bunch length.


164                                                                                                                                                                Chamonix XI
N   The trailing bunches in the batch see a vertical          is of the order of the time it takes for the electrons to
    wakefield with a shorter interaction length (0.3 to       travel through the bunch.
    0.5) times the bunch length.
N   The vertical wakefield for the trailing bunches
    gives a negative coherent tune shift. The measured                   20 to 40 nsec           1 to 2 nsec
    [1] positive tune shift must be incoherent in origin.
N   In the horizontal plane, the electron cloud acts like a
    long range wake covering one or more bunch
    spacings.                                                 Fig. 8: In the SPS dipoles the electron cloud looks like a
                                                              vertical ribbon with the electrons moving up and down.
The different behaviour between horizontal and vertical
plane can be understood from the geometry of the SPS                              REFERENCES
vacuum chamber which is flat in all the dipoles. In this      [1] G. Arduini, Observations of Transverse Instabilities,
case the electron cloud looks like vertical ribbon at the     Session 3, these proceedings.
same horizontal position as the bunch (Fig 8). When the       [2] R.Jones, Progress with Beam-Instrumentation,
bunch is displaced in the horizontal plane it will drag the   Session 3, these proceedings
ribbon with it, but the time it takes will be typical the
time it takes for the electrons to travel from surface to
surface which is, because of the multi-pacting of the
same order as the time between bunches. Changing the
vertical position will only change the electron cloud
density at level of the bunch size and this time constant




Chamonix XI                                                                                                         165

				
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posted:11/22/2011
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