Simulations of electron-cloud heat load for the cold arcs of the CERN Large Hadron Collider and its high-luminosity upgrade scenarios

CERN, BE Department, 1211 Geneva 23, Switzerland(Received 30 November 2010; revised manuscript received 5 December 2011; published 4 May 2012)The heat load generated by an electron cloud in the cold arcs of the Large Hadron Collider (LHC) is aconcern for operation near and beyond nominal beam current. We report the results of simulation studies,with updated secondary-emission models, which examine the severity of the electron heat load over arange of possible operation parameters, both for the nominal LHC and for various luminosity-upgradescenarios, such as the so-called ‘‘full crab crossing’’ and ‘‘early separation’’schemes, the ‘‘large Piwinskiangle’’ scheme, and a variant of the latter providing ‘‘compatibility’’ with the (upgraded) LHCbexperiment. The variable parameters considered are the maximum secondary-emission yield, the numberof particles per bunch, and the spacing between bunches. In addition, the dependence of the heat load onthe longitudinal bunch proﬁle is investigated.

Simulations of electron-cloud heat load for the cold arcs of the CERN Large Hadron Collider and its high-luminosity upgrade scenarios

I. INTRODUCTION
The performance of the Large Hadron Collider (LHC) and of its possible future upgrades may ultimately be limited by the electron cloud which is built up both through photoemission from synchrotron radiation and by a beaminduced multipacting process [1][2][3][4].
At 7 TeV beam energy, the critical energy of synchrotron radiation is about 44 eV, which is close to the energy where the photoemission yield is maximum for many materials.Some of the synchrotron radiation photons, hitting the metallic beam screen inside the beam pipe, produce photoelectrons.These photoelectrons are accelerated in the field of the bunch that has emitted the photons, passing by quasisimultaneously, to up to 200 eV and they reach the opposite beam pipe wall after a few ns, before the following bunch arrives.When these accelerated electrons hit the wall they produce secondary electrons, at a typical low energy of a few eV [1].The next bunch generates new photoelectrons, and its electric field accelerates both these photoelectrons and the secondary electrons left behind by the previous bunch.As a consequence of the resulting cascade, a significant electron cloud can be produced from the combination of synchrotron radiation, photoemission, and secondary emission.
Figure 1 shows a schematic picture of this avalanche process that leads to the electron cloud in the LHC.
The electron cloud, once created, can cause both single and coupled-bunch instabilities of the proton beam, give rise to incoherent beam losses [5], or lead to a vacuumpressure increase by several orders of magnitude due to electron-stimulated desorption [6].
A primary concern for the LHC is the additional heat load due to the electron cloud that is deposited on the beam screen, a perforated tube inserted into the cold bore of the superconducting magnets in order to protect the cold bore from synchrotron radiation and ion bombardment [7].The LHC beam screen has a round cross section with flat parts at the top and bottom, and pumping slots as shown in Fig. 2(a).For the simulations reported in this work, the real geometry of the beam screen has been replaced by an ellipse [see Fig. 2(b)].The pumping slots do not play a role in our simulations nor-we expect-in reality, because they are protected by the pumping slot shield (see Fig. 2), which prevents electrons from penetrating to the cold bore of the magnet and should act like the surface of the beam-screen proper (in particular taking into account the small cyclotron radius at 8.33-T field, and neglecting the slight difference in path length) [8].The incident cloud electrons heat the beam screen, for which only a limited cooling capacity is available.If the beam-screen heat load exceeds the available cooling the cold superconducting magnets of the LHC arcs, surrounding the beam pipe, quench; i.e., they will lose their superconducting state.Thereby, the electron cloud may limit the maximum permissible beam current of the LHC.The expected heat load does not only depend on the beam current, but also on the bunch spacing, bunch intensity, and the time-dependent surface properties of the beam screen.
Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License.Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
PHYSICAL REVIEW SPECIAL TOPICS -ACCELERATORS AND BEAMS 15, 051001 (2012) 1098-4402=12=15(5)=051001 (13) 051001-1 Ó 2012 American Physical Society Hence, one motivation for this report is that the electroncloud heat load can limit the performance of the present LHC.Updated predictions are presented on this possible limitation.In addition, we assess the heat load from electron cloud for the various LHC upgrade scenarios proposed [9] to unveil potential merits and disadvantages of the different upgrade paths under consideration with regard to electron cloud.

II. SECONDARY-EMISSION MODEL
The secondary-emission yield tot describes the average number of secondary electrons emitted per incident electron.It is a function of the energy of the primary incident electron and of its angle of incidence.The present model, employed in our simulation, describes the total secondaryemission yield as the sum of a true secondary-electron component ( true ) and an elastic reflection component ( elast ): where E p is the incident-electron energy and the angle of incidence (with ¼ 0 referring to perpendicular impact).We note that the true secondary yield is taken to depend on , while for the reflectivity part the approximation of being independent of is assumed.The coefficient R designates the probability for an elastic reflection in the limit of zero primary energy (0 < R < 1).
Figure 3 illustrates the model dependence of the secondary-emission yield, and its two components, on the parameters Ã max , also called ''SEY,'' and " Ã max , as well as on the angle of incidence .Other earlier simulations had also included a ''rediffused'' secondary-electron component [15], which is ignored in our model.Simulations with different secondary-emission models, e.g., ones with and without rediffused electrons, have been compared by Bellodi [16].
The emission angles of the true secondaries are distributed according to dN=d / cos e , where is the solid angle and e the emission angle with respect to the surface normal.The initial energy distribution of the secondary electrons is taken to be a half Gaussian (centered at 0) with an rms spread of 5 eV.For the elastically scattered electrons, conservation of the electron kinetic energy and a perfect specular reflection at the surface are assumed; i.e., no energy is absorbed by the wall since ECLOUD calculates the heat load as the difference in the incoming and outgoing products of macroparticle charge and energy.The electron conserves its initial energy, and the momentum vector is reflected.The reflectivity value R affects the simulated heat load, as it changes the average survival time of electrons in the beam pipe.
The electron cloud itself (by electron bombardment) after some time-days-will lead to a reduction of the maximum secondary-emission yield ( Ã max ) in time, from about 1.9 to a final value of 1.1, a process which is known as ''scrubbing'' [6,12,14].

III. PHOTOEMISSION MODEL
Photoemission from synchrotron radiation provides a copious source of primary electrons.The assumed creation rate of primary photoelectrons, of order 0:6-1:2 Â 10 À3 per proton and per meter, corresponds to the computed synchrotron radiation flux in the arcs and the photoelectron generation rate, n 0 e [ðe=pÞ=m], inferred from measurements with test beams on prototype chambers before or after surface scrubbing.Closely related, the measured photoelectron yield per absorbed photon and per meter, Y Ã , is about 5.0% and 2.5% at a maximum secondary-emission yield Ã max of 1.9 and 1.1, respectively [17,18].During scrubbing, the parameter " Ã max also changes, from 249 to 230 eV [19][20][21].We linearly interpolate n 0 e [ðe=pÞ=m] and " Ã max as a function of Ã max , from the quoted pairs of values, as is illustrated in Table I, which shows a list of surface parameter combinations assumed in our simulations.
About 80% of the incident photon flux produces photoelectrons at the primary impact point on the horizontally outward side of the vacuum chamber, within a narrow cone of rms angle 11.25 .Inside the strong 8.33-T magnet field of the LHC dipole magnets, these electrons cannot approach the proton beam and do not contribute to the further electron-cloud buildup.We assume that the other 20% of the flux produces photoelectrons distributed azimuthally according to a cos 2 distribution [3,22], where denotes the azimuthal angle with respect to the horizontal plane, spanned from the primary impact point of the synchrotron radiation.These 20% of all photoelectrons correspond to the fraction of the photon flux which is diffusely reflected from the sawtooth surface impressed on the outer side of the beam screen, inside the test volume from [3,22].A single value for the quantum efficiency for photoelectron production is assumed for all absorbed photons.The initial angular distribution of the newly generated primary photoelectrons is assumed to be uniform in the two spherical coordinates and defined with respect to the surface normal.The energy distribution of the emitted 3. The secondary-emission yield tot plus its true and elastic components as a function of the incident primary electron energy E p , for perpendicular incidence and incidence at an angle of =2, for two different value pairs of Ã max and " Ã max (i.e., the incident primary electron energy at which the secondary yield assumes its maximum value), and with an elastic reflectivity R ¼ 1, according to the model described in the text.photoelectrons is modeled as a truncated Gaussian centered at 7 eV with a standard deviation of 5 eV.

IV. MODEL UPDATES
Although simulation results for electron-cloud induced heat load have been reported for the LHC in 1998-2000 [2,3] and for possible LHC upgrades in 2004 [23], an update is in order since about 2005 [24], for a number of reasons: (i) the LHC cooling capacity available at 4.6-20 K has been revised [25], leading to an increase in the cooling capacity at high beam current, compared with earlier estimates; (ii) the dependence of the impact energy at which the secondary-emission yield assumes its maximum value, " Ã max , on the angle of incidence has been corrected in the simulation model [11,16]; and (iii) a benchmarking of simulations against beam observations in the Super Proton Synchrotron (SPS) [26] has led to a revision of the reflection probability R from the wall for low-energetic electrons from formerly R % 1 [14] to R % 0:5 [26,27].The second and third points in this list lower the simulated heat load by typically 40%-50% each.
Figure 4 shows the spare cooling capacity for electron cloud as a function of bunch population for the nominal LHC beam with 25-ns spacing and a Ã of 0.55 m, assuming a nominal gas pressure and a separate cryoplant for the rf cavities [25].The cooling capacity decreases with bunch intensity due to heat load from synchrotron radiation, image currents, and, at high luminosity, collision debris impinging on cold magnets in the interaction regions (IRs).Contingency is a factor which has been used in the design of the LHC cryogenics systems to account for uncertainties in the estimated values of static heat loads and cooling capacity at various levels of temperature [28].At zero bunch intensity the cooling capacity is limited locally by the hydraulic impedance of the beam-screen cooling capillaries.
For all upgrade scenarios with beam parameters different from nominal, the spare cooling capacity remaining for electron-cloud induced heat loads must be recomputed, taking into account that the heat load from synchrotron radiation grows in proportion to the total beam current, and that the contribution from image currents scales linearly with the number of bunches, as the square of the bunch charge, and with the inverse 3=2 power of the rms bunch length (for a Gaussian bunch).We neglect the additional dependence of the image-current heating on the longitudinal bunch shape.The cryogenics heat load related to luminosity scales directly with the luminosity, and, e.g., depends on Ã , which is assumed to be reduced to 0.08 and 0.25 m, respectively, for the various upgrade scenarios.The Ã dependence occurs because we subtract the heat load from luminosity debris, as in the present LHC the cryoplant is shared between arc and interaction regions.We have also subtracted from the total capacity the heat load due to all other known sources, so that the residual capacity (see Fig. 4) is taken to define the capacity available for electron cloud.From the numbers, it is evident that any LHC luminosity upgrade will require separate cryoplants for the two high-luminosity interaction regions, in order to preserve sufficient cooling capacity for the arcs.
The changes in the angular dependence of the secondary-emission yield curve and in the low-energy electron reflectivity are illustrated in Fig. 5. Earlier versions of the ECLOUD code, prior to 2005, hadmistakenly-featured the opposite sign in the angular dependence in Eq. ( 4), which made " max ðÞ decrease with , rather than increase.The corrected dependence corresponds to one inferred by Furman [10] from laboratory data for some candidate PEP-II vacuum-chamber materials [11].
FIG. 5.The secondary-emission yield tot for perpendicular incidence (unchanged) and for incidence at an angle of =4 according to old [23] and new models [24], with Ã max ¼ 1:3 and " max ¼ 234:7.The lower probability of elastic electron reflection in the limit of zero primary electron energy has been discussed in [26], where a value of R significantly lower than 1 was inferred from an SPS experiment by benchmarking simulations and measurements of electron-cloud buildup for one and two bunch trains at two different bunch-train spacings.Direct laboratory measurements of secondary emission can provide more precise values of the reflectivity R than the benchmarking of simulations and beam observations, which involves several additional uncertainties.For prototypes of the LHC beam pipe, laboratory measurements of the secondary-emission yield and the associated secondary energy spectra were performed in various states of surface conditioning.From such measurements on a copper surface which had been treated and cleaned according to the LHC vacuum-chamber recipe, Furman and Pivi [27] fitted an electron reflectivity of 0.496, i.e., close to 0.5.Consistently with both the SPS benchmarking and the Furman-Pivi results for a prototype LHC beam-screen chamber, the reflection probability assumed in our simulations was reduced from 1.0 to 0.5.

V. ARC HEAT LOAD
Figure 6 shows the simulated heat load per unit length as a function of bunch population for a quadrupole, a dipole, and a field-free region.The simulation was done for two nominal trains of 72 bunches separated by a gap of 8 missing bunches (225 ns).The heat load is higher in the field-free region than in a dipole.Regarding the quadrupoles, the heat load here can be higher or smaller than for the other sections, depending on the bunch population.In particular, for bunch population increasing beyond about 1:2 Â 10 11 the heat load in the quadrupoles decreases, while in the dipoles it approximately saturates.
A qualitative explanation of the differences between the different field regions is as follows.Inside the high-field dipoles, aside from a narrow cyclotron motion, electrons move essentially vertically.Strong multipacting happens at a certain horizontal distance from the beam.The distance where multipacting occurs is defined by the condition that during a bunch passage an electron located at the chamber wall receives a vertical momentum (and associated total kinetic energy) that corresponds to the impact energy " Ã max for which the secondary-emission yield is maximum.Above a certain bunch intensity, of about 1 Â 10 11 , for further increasing bunch charge the resulting vertical ''stripes'' of multipacting electrons move horizontally outward.In this regime, only the location of the multipacting varies, while the electron energy gain and the electron amplification by multipacting are roughly independent of the bunch charge.On the other hand, in a field-free region, without any magnetic confinement, all primary photoelectrons are accelerated in the field of a passing bunch.Here, the higher the bunch charge the higher is the electron energy gain.In the absence of a magnetic field, and for max ¼ 1:4, the acceleration of primary photoelectrons is the dominant contribution to the heat load, and the amplification through multipacting is only a secondary effect.Finally, in case of the quadrupole a large part of the primary photoelectrons is kept at the outer chamber wall by a strong magnetic field, exactly as in the case of the dipole magnet.At low and moderate charge, roughly up to the nominal LHC bunch population of 10 11 protons, in a quadrupole magnet the multipacting happens across the chamber diagonals where the magnetic field is close to zero.For further increasing bunch charge, during a bunch passage the primary or secondary electrons (initially close to the chamber wall) acquire an energy larger than " Ã max , so that the secondary-emission yield is decreasing and, as a result, the net heat load is being reduced as well.In other words, for the quadrupoles, at higher bunch charges there no longer is a region of strong multipacting, which is different from the dipole case.
An LHC arc half cell has a length of 53.45 m and it comprises 42.9 m of dipoles, 6.433 m of field-free region, and 4.119 m of quadrupoles.Taking this into account, the average heat load per meter in the LHC arcs can be calculated as a weighted average.The effect on the simulated arc heat load of the two changes to the model of the secondary-emission yield (angular dependence of " max , and the values of the elastic reflection probability), described in the previous sections, is illustrated in Fig. 7 together with the updated cooling capacity.The local limit of the cooling capacity applies over a length of 53.45 m (half an optical LHC cell) corresponding to the length of a beam-screen cooling loop [28].
In the simulations reported in this paper, we are assuming that the elastic reflection probability is independent of the angle of incidence.In another study by Furman and  Pivi [27], the authors had concluded that, while the true secondary yield for copper varied by 66% for incident angles between 0 and 90 degrees, the elastic yield was not completely independent of the angle, but it varied by a smaller factor of about 26%.We have studied the sensitivity of our results with respect to variations of R in order to estimate the potential effect of a possible angular dependence.

VI. LHC HIGH-LUMINOSITY SCENARIOS
The LHC luminosity for round Gaussian beams is given by [9,29] with where the IP rms divergence, and the minimum full crossing angle is derived from [30,31].In the above expressions, r and r denote the relativistic factors, f rev designates the revolution frequency, n b the number of bunches, N p the number of protons per bunch, Ã the beta function at the IP, " N the rms normalized transverse emittance, F a geometric form factor, and z the rms bunch length.We have also introduced the normalized parameters kb ¼ k b =k b;0 with k b;0 ¼ 2808 the nominal value, Ñp ¼ N p =N p;0 with N p;0 ¼ 1:15 Â 10 11 , and "N ¼ " N =" N;0 with " N;0 ¼ 3:75 m.
The value of c that allows a sufficient dynamic aperture is estimated from simulations [30].For the nominal LHC parameters, the coefficient a is close to 10 and b about 1.In the parameter range considered, the hourglass effect is always negligible and, therefore, not included in the above luminosity formula.In the simplifying case of two IPs with alternate crossing (one crossing in the horizontal plane and one in the vertical), the total beam-beam tune shift assumes the simple form [29] where r p is the classical proton radius.This tune shift is proportional to the brightness N b =" N and, up to a constant coefficient, it equals the last factor in the luminosity equation ( 6).
The beam-beam tune shift decreases with the crossing angle, in the same way as the luminosity, while the ''beambeam limit'' (an empirically found upper bound on the tune shift ÁQ bb ) is assumed to be independent of it, which might not be entirely correct, since in electron-positron colliders the beam-beam limit has been seen to depend on the crossing angle [32,33].
To increase the luminosity, one can raise the bunch population N b and decrease È p until the beam-beam limit is reached.To go further, at the beam-beam limit one must increase È p , by enlarging the bunch length or crossing angle, or the emittance " N , together with the bunch population.An additional factor ffiffiffi 2 p can be gained in the luminosity, for constant bunch charge and constant beam-beam tune shift, by transforming the longitudinal bunch profile from a Gaussian to a uniform, rectangular shape [34].The underlying reason is that the total beam-beam tune shift depends on the peak charge density only [29,34].
The nominal LHC luminosity is 10 34 cm À2 s À1 .In order to increase the nominal luminosity by a factor of 5-10, several high-luminosity-upgrade scenarios for the LHC have been proposed.They are briefly presented in the following.Key performance parameters for the nominal LHC and three upgrade scenarios are summarized in Table II along with an ''alternative nominal'' parameter set.The bunch patterns for the nominal LHC and for various upgrade scenarios are illustrated in Fig. 8.

A. Early separation or full crab crossing scheme
These schemes aim at reducing Ã to the smallest value permitted by magnet technology and optics, while minimizing the Piwinski angle.The number of bunches is equal to nominal (2808 per beam), and the bunch population corresponds to the so-called ''ultimate'' LHC parameters.The early separation (ES) scheme requires the installation of moderate dipole field magnets as close as possible to the interaction point [9].This scenario involves the installation of new hardware inside the ATLAS and CMS detectors, as well as the possible use of crab cavities which can reduce FIG. 7. Simulated average arc heat load per meter for Ã max ¼ 1:3, " Ã max ¼ 235 eV illustrating the effect of model changes to the angular dependence of " max and to the elastic reflection probability, for Ã max ¼ 1:3 and " Ã max ¼ 235 eV, as well as the old [23] and updated predictions [24] for the available cooling capacity.

B. Large Piwinski angle scheme
In the large Piwinski angle (LPA) scheme, the Piwinski angle is intentionally increased, and the longitudinal bunch profile made quasirectangular, together with a large increase in the bunch population.To confine the total beam current, the spacing between bunches is doubled, that is fewer more intense and longer longitudinal-plane bunches are collided at a large Piwinski angle.The beta function at the interaction point is decreased by about a factor of 2 compared with the nominal LHC, but taken to be significantly larger than for the early separation or full crab crossing schemes.The LPA scheme permits one to increase the bunch intensity in such a way that the beam-beam tune shift remains constant.Advantages of this scenario are the absence of accelerator elements inside the detectors and the fact that no crab cavities are needed.

C. LHCb compatibility scheme
An upgrade to the LHCb experiment is planned in order to exploit luminosity around 2 Â 10 33 cm À2 s À1 or 2% of the (upgraded) luminosity delivered to the ATLAS and CMS experiments.The LHCb detector is special because of its asymmetric position around the LHC circumference.This scenario is in essence the LPA scheme, but with ''satellite'' bunches between the main bunches as illustrated in Fig. 8. Low-intensity satellites would collide in LHCb with the main bunches of the other particle beam in the standard time intervals of 25 ns.The intensity of the small satellite bunches is chosen so as to yield the (upgraded) LHCb target luminosity of 2 Â 10 33 cm À2 s À1 at Ã ¼ 1 m, neglecting the small effect of the crossing angle and assuming that all 2808 bunch pairs collide in LHCb.The satellite bunch intensity N b;sat is, therefore, scaled inversely with the main-bunch intensity N b according to N b;sat ¼ 1:1 Â 10 10 Â 10 11 =N b .

VII. SIMULATION METHODOLOGY
The simulations reported in the following have been performed using the program ECLOUD version 4.b [37,38] to compute electron-cloud buildup and the resulting heat load in the LHC arc.As described above, the heat load per arc unit length is found as a weighted average of three independent simulations for dipoles, field-free regions, and We have conducted eight sets of simulations, the characteristics of which are listed in Table III.The physical parameters which were varied are: the spacing between bunches, the bunch profile, the number of protons per bunch, the secondary-emission yield, and the bunch length.The parameters used in the calculations are listed in Table I.The filling pattern for simulations with 25-ns bunch spacing was two trains of 72 bunches separated by 8 missing bunches; for simulations of 50-ns bunch spacing we considered two trains of 36 bunches separated by 4 missing bunches.Heat loads quoted always refer to the average over the two trains.
The simulation set A corresponds to the nominal LHC values.The set B assumes the same LHC design parameters except for a larger bunch spacing of 50 ns instead of the nominal 25 ns.
The set C represents an LHC luminosity upgrade of 50 ns bunch spacing where the number of particles per bunch N b is increased.The set D is another highluminosity LHC with 50 ns bunch spacing where we consider longer bunches with a flat bunch profile, instead of the nominal Gaussian bunches.
The set E is a special case, with main bunches spaced at 50 ns, but with additional satellite bunches at 25-ns distance between the main bunches.As mentioned, this scheme has been proposed to ensure compatibility with the LHCb experiment, so as to have collisions in LHCb, ATLAS, and CMS at the same time, but at a variable reduced luminosity in LHCb.The set F has the same characteristics as set E, but without satellite bunches.The results for set F, together with set E, allow determining the additional heat load due to the satellites.
Finally, the sets G and H represent a study of the heat load as a function of the bunch spacing for a longer flat and shorter Gaussian bunch profile, respectively.These two sets of simulations were performed to explore the heatload dependence on bunch profiles, considered for the ES/ FCC and LPA upgrade schemes, respectively.

A. Simulation results for the nominal LHC
According to the methodology outlined in the previous section, the average heat load for the simulations sets A, B, C, and D is presented in Figs.9-12.The surface parameters are those listed in Table I.
As illustrated in Fig. 9 the heat load depends on the maximum secondary-emission yield ( Ã max ) and on the number of particles per bunch N b .For certain combinations of Ã max and N b , the average heat load stays within the available cooling capacity, but for higher values of either Ã max or N b the average heat is well above the coolingcapacity limit.In particular, for reaching the nominal LHC bunch intensity of N b ¼ 1:15 Â 10 11 with 25 ns spacing (set A) the maximum secondary-emission yield Ã max should stay below 1.4, which should be readily achieved [6], while for reaching the ultimate bunch intensity of 1:7 Â 10 11 protons, Ã max must be smaller than 1.3, which is close to the edge of what appears possible with beam scrubbing.These results are consistent with the previous findings of Furman and Chaplin [21], where, for equivalent model assumptions (i.e. when neglecting the rediffused FIG. 9. Average heat load for the simulation set A. The simulation data for set B (50 ns spacing) are compiled in Fig. 10.The average heat load is well below the cooling capacity for any Ã max value and N b explored.In fact, the maximum simulated value of the heat load is 0:25 W=m for a Ã max ¼ 1:7 and N b ¼ 1:8 Â 10 11 , which is more than a factor 2 below the available cooling capacity of 0:55 W=m at that point.

B. Simulation results for various LHC upgrade schemes
1. Early separation or full crab crossing scheme Figure 9 shows that with 25-ns spacing, for the proposed ES/FCC high-luminosity scheme, the maximum acceptable arc heat load is exceeded not far above N b % 2 Â 10 11 , with dedicated IR cryoplants.Without such dedicated cryoplants not even nominal bunch intensities would be possible at the luminosity expected for the assumed Ã % 8 cm.In Fig. 11, we present results for twice the bunch spacing; i.e., 50 ns, up to much higher bunch intensity.Without IR cryoplant upgrade, the heat load now exceeds the cooling capacity available (cooling capacity-high-luminosity line) at a bunch intensity of 1:5 Â 10 11 , almost independently of Ã max .On the other hand, with the appropriate cryogenics upgrade (separate IR cyoplants: cooling capacity-low luminosity line) the heat load remains acceptable up to at least 4:5 Â 10 11 protons per bunch for Ã max < 1:5 and until 4:0 Â 10 11 protons per bunch for Ã max 1:7.

Large Piwinski angle scheme
The results for set D are shown in Fig. 12.This scheme differs from the previous one by its longer and flat rather than Gaussian bunch profile.The heat-load results are similar to, and even more favorable than for, the previous case of set C. With separate IR cryoplants, the electroncloud induced heat should not be a problem until well above 4 Â 10 11 protons per bunch.

LHCb compatibility scheme
In Fig. 13 we present the average heat load for simulation sets E and F; i.e., for the LHCb satellite scenario and the same scheme without satellite, at various values of Ã max .The satellite bunches increase the heat load, but at high main-bunch intensities the additional heat load is small and barely affects the (high) value of the maximum bunch intensity at which the cooling-capacity limit is reached (compare Fig. 12).The nonmonotonic dependence on the main-bunch intensity N b is expected, since the satellite bunch intensity is varied inversely to N b and it has been shown earlier (albeit for constant main-bunch intensity) that there exists an optimum satellite bunch intensity at which electrons are most efficiently removed from the beam pipe, and for which the electron-cloud heat load is consequently minimized [39].

Variation of bunch spacing
Simulation sets G and H explore the detailed dependence of the heat load on the bunch spacing, and on the longitudinal bunch shape.Results were obtained from the scanning of the bunch spacing for two types of bunch profiles: nominal Gaussian and longer flat bunches.Figure 14 shows the results for Ã max ¼ 1:1 and both types of bunch profiles.Even at this low value of Ã max , the heat load strongly varies with the bunch spacing.The heat load is slightly higher for the Gaussian bunch profile of shorter, nominal bunch length.Finally, Fig. 15 presents the results for Ã max ¼ 1:7, where again the heat load is higher for the nominal Gaussian profile than for the flat one.It is clear that increasing the bunch spacing from 25 to 50 ns is a very efficient means for mitigating electron-cloud effects in the LHC and its upgrades.

C. Uncertainty of the simulated heat load
In view of the importance and possible uncertainty of the value of the low-energy electron reflectivity R, we have investigated the sensitivity of the results to variations in the value of R. Some pertinent results are presented in Fig. 16.For variations in R of AE0:1, the simulated heat load changes by less than 20%.We have also varied the number of primary macroparticles per bunch, i.e., the number of macroelectrons that are generated at each bunch passage in the beam line section under consideration, while keeping the total charge of all these macroelectrons constant.Example results are shown in Fig. 17.From these studies we infer that, for given values of Ã max and R the uncertainty in the absolute heat-load value is of the order of 10%-20%.

IX. CONCLUSIONS
For the nominal LHC with 25-ns bunch spacing at top energy, the maximum secondary-emission yield should be below 1.4 in order to reach the nominal bunch intensity N b ¼ 1:15 Â 10 11 with acceptable heat load.If the maximum Ã max is higher, between 1.4 and 1.5, bunch populations around 10 11 are still feasible.
For 50-ns bunch spacing the electron-cloud buildup is much weaker, and the ''nominal'' LHC beam-at this, twice the nominal bunch spacing-gives rise to heat loads which remain acceptable up to bunch intensities of 2 Â 10 11 , without any upgrade to the LHC cryogenics infrastructure.
Regarding the proposed LHC high-luminosity-upgrade schemes, and here always assuming separate new cryoplants for the higher-luminosity interaction regions, with the ES/FCC schemes at nominal 25-ns bunch spacing the bunch population will be limited to about 2 Â 10 11 protons.The heat load for a modified ES/FCC beam at twice the proposed spacing; i.e., 50 ns, remains acceptable up to 4:5 Â 10 11 protons per bunch.A similar result is obtained with the LPA scheme, for which 50 ns represents the nominal spacing.
With respect to the compatibility scheme for the LHCb experiment, we find that at a maximum Ã max ( Ã max ) of 1.1 the heat load for the LPA beam with and without satellites is almost the same.At larger Ã max values the satellite scenario shows a higher heat load than the nominal scheme.With satellite bunches present, the total heat load decreases to a minimum at main-bunch intensities of a few 10 11 , and the difference due to the satellites further shrinks as the limit of the cooling-capacity limit is approached at even higher bunch intensities.In particular, we can conclude that the scheme with satellite bunches would deliver the desired luminosity to the LHCb experiment in a controlled manner, while confining the additional heat load to acceptable values.
The electron-cloud heat load in the LHC strongly depends on the bunch spacing.Operation with a bunch spacing of 50 ns instead of 25 ns looks ''safe'' up to bunch populations above 4 Â 10 11 , e.g., roughly 4 times the nominal.For any bunch spacing, additional margins, both in the predicted heat load and especially in the available cooling capacity, can be obtained by operating with longer and flat bunches, instead of the nominal Gaussian ones.

FIG. 4 .
FIG.4.Spare cooling capacity for electron cloud per meter and per beam aperture in the LHC arc as a function of bunch population.The old assumption used until 2004[23] is compared with an updated cryogenic assessment from 2005[24,25].

FIG. 6 .
FIG.6.Simulated heat load in the LHC as a function of bunch population and different sections, for 25-ns bunch spacing.

FIG. 15 .
FIG. 15.Average heat load in function of the bunch spacing and a Ã max ¼ 1:7 for (a) flat bunch profile and (b) nominal Gaussian profile.

TABLE II .
(1)ameters for the(1)nominal LHC and (2) an alternative nominal[asterisk]with half the number of bunches and ultimate bunch intensity compared with those for two upgrade scenarios with (3) more strongly focused ultimate bunches at 25-ns spacing with either early separation and crab cavities [ES] or full crab crossing [FCC], and (4) longer intense flat bunches at 50-ns spacing in a regime of large Piwinski angle[LPA].The numbers refer to the performance without luminosity leveling.T ta denotes the turnaround time between successive physics runs.The normalized transverse rms emittance is " ¼ 3:75 m in all cases.

TABLE III .
Summary of simulation sets.