1. A Mapping Approach to the Electron Cloud for LHC
T. Demma, LAL
S.Petracca, A. Stabile, Universtà del Sannio e INFN Salerno

2. Plan of Talk Electron Cloud Effect Cubic Map Formalism
Formulas for the Linear Coefficient
Conclusions and Outlook.

3. Electron Cloud Effect
The electron cloud develops quickly as photons striking the vacuum chamberwall knock out electrons that are then accelerated by the beam, gain energy, and strike the chamber again, producing more electrons.
The interaction between the electron cloud and a beam leads to the electron cloud effects such as single- and multi-bunch instability, tune shift, increase of pressure and so on.

4. Eclectron Cloud Buildup
max=1.7 (72 f, 28 e)
LHC dipole parameters
— ECLOUD (CERN) output
 bunch to bunch average

5. Maps for Electron Clouds
For a given beam pipe characteristics (SEY, chamber dimensions, etc.) the evolution of the electron density is only driven by the bunch passing by,and the existing electron density before the bunch passage:
m+1=F(m)
Simplify the e-cloud problem into a small number of mathematical parameters.
The bunch-to-bunch evolution of the e-cloud density is represented by a cubic map [U.Iriso, S.Peggs, Phys. Rev. ST-AB8, (2005)]:
m+1=a m+ b m2+ c m3
where m is the bunch to bunch average of the electron line density

6. Building the Cubic Map m+1=a m+ b m2+ c m3 max=1.7 (72 f, 28 e)
Lines corresponds to cubic fit of the form:
m+1=a m+ b m2+ c m3
Three sets of coefficients are needed to describe the e-cloud density evolution

7. Bunch Filling Patterns T. Demma et al. , Phys. Rev
Bunch Filling Patterns T.Demma et al., Phys. Rev. ST-AB 10, (2007)
(24f,12e, 36 f)
(12f,12e, 12f,12e,12f,12e)
(6f,6e)
Map results do not depends on the initial electron density
Relative error below 20%
Different bunch patterns can be described using the same Map coefficients

8. Cubic Map Coefficients
m+1=a m+ b m2+ c m3
The linear term a describes the exponential growth of the cloud density. For >>1:
1=a 0
2=a 1=a2 0 
n=a n-1=an 0
The quadratic term b is related to the saturation due to space charge
The cubic term is always very small and could be interpreted additional correction embodying small corrections.

9. e-cloud Buildup Dynamics
Reflected and secondary electrons at bunch m+1

10. Linear Term: Drift Regions (RHIC) U. Iriso and S. Pegg. Proc
Linear Term: Drift Regions (RHIC) U. Iriso and S. Pegg. Proc. of EPAC06, pp
In the absence of external electro- magnetic fields and assuming that the electron motion is limited to the transverse radial direction
For ultrarelativistic bunches:
simulations (bars)
Analytic calculation (lines)

11. Linear Term: LHC Dipole
In the limit of a large y-directed magnetic field (By=8.4T) we may consider only the (transverse) vertical motion of the electrons (the cyclotron radius of the particle elical trajectories is very small compared to the beampipe radius Rp).
Beampipe Geometry
δmax blue line and triangles
δmax black line and squares
ECLOUD simulations (bars)
Analytic calculation (lines)

12. Conclusions and Outlook
The electron-cloud buildup can be described by a cubic map.
Remarkably, if all other parameters (namely, the bunch charge N, the SEY, and the pipe parameters) are held fixed, the map coefficients basically do not depend on the filling pattern. The map can be thus used as a quick and (not so) dirty tool for finding filling patterns yielding electron-cloud densities compatible with safe operation.
An approximate formula has been derived for the linear coefficient in the map. The results are in acceptable agreement with numerical simulations obtained from ECLOUD.
The analytical result could be useful to determine safe regions in parameters space where to minimize the electron clouds.
Work is in order to compute the second order term taking into account the presence of the magnetic field.