LHC Project ote 45 June 0, 004 Elia.Metral@cern.ch Andre.Verdier@cern.ch Emittance limitation due to collective effect for the TOTEM beam E. Métral and A. Verdier, AB-ABP, CER Keyword: TOTEM, collective effect, emittance limitation. Summary Obervation of mall-angle cattering and Landau damping of collective intabilitie are partly conflicting requirement becaue of the mall value of the tranvere beam emittance. The phyic requet et a imum value for the beam emittance. The beam tability require a imum value. A method to find good et of beam parameter i given.. Introduction The TOTEM experiment aim mainly at meauring the total proton-proton cro ection at the LHC []. The method ued i luoity independent. It require imultaneou meaurement of elatic cattering at low momentum tranfer and total inelatic rate. It aim at an abolute error of about mb. The interface between TOTEM and the LHC machine i quite intricate ince a pecial, high-beta (* = 540 m), optic i required to oberve mallangle cattering. Beide, detector, called Roman Pot (RP), it cloe to the beam axi. They are intalled about 0 m from the colliion point. The requeted luoity i in the range of 0 8 cm - -, the actual value being not critical. The impedance of the RP doe not ignificantly affect the beam tability, but a potential problem i aociated with the electromagnetic power depoited. The problem i not when the detector i in the I-poition. It i when it i in the OFF-poition, a a cavity i then created. A mechanical way of hielding it ha been found []. However, TOTEM beam may uffer from the reitive-wall intability induced by the collimator impedance at top energy becaue of the mall emittance neceary for the meaurement []. The requet for the TOTEM experiment are briefly reviewed in Section. Stability diagram with imum available octupolar trength at top energy and coherent tune hift for the mot untable coupled-bunch mode number and head-tail mode 0 are then given and dicued in Section, for different et of beam parameter. Finally, a method to find good et of beam parameter i given in Section 4. Thi i an internal CER publication and doe not necearily reflect the view of the LHC project management.
. Phyic requet The meaurement of the total proton-proton cro ection requet an efficient detection of cattered proton with a momentum tranfer quared of t 5 0 (GeV/ c). In order to have a ufficiently high acceptance of the detector in the relevant range t > 5 0 (GeV / c), the value of the imum detectable momentum tranfer quared ha to be of the order of t 0 (GeV / c) []. For t = 0 (GeV / c) and the beam momentum p = 7000 GeV/c, the aociated cattering angle i given by θ t 0.00 = = 4.5 µrad. () p 7000 = Thi imum angle θ i computed a follow. The RP are placed tangent to the beam halo at a location where the trajectory of the cattered proton ha a imum amplitude. The bottom of the RP ha a certain thickne d a hown on Fig.. The bottom of the pot mut be at a ditance from the beam centre of at leat 0 σ, where σ i the rm tranvere beam ize at the location of the RP, in order not to intercept the proton in the econdary beam halo. Thi aume that the primary collimator are et at a ditance from the beam centre of 7 σ (the econdary halo extend then up to.4 7 0 σ). In principle the primary collimator could be et at a ditance from the beam centre of 6 σ, but we kept a afety margin of σ. Figure : Rough ketch for the TOTEM experiment. The imum cattering angle which can be detected i then given by θ. L eff = 0 σ + d, where L eff = 7 m i the matrix element which tranform the angle at the interaction point (IP) in tranvere diplacement at the RP. For an arbitrary etting of the primary collimator, the normalied rm tranvere beam emittance ε = l γ σ / mut atify accordingly ε ε l γ = L eff ϑ h n d, () where = 48 m i the betatron function at the RP, l and γ the relativitic velocity and ma factor, d mm, h =.4 the gemetrical factor of the econdary halo (if the gap of the
primary collimator i G = n σ, the econdary halo extend up to h n σ) [4]. A plot of the gap of the collimator i repreented in Fig.. e @mmd.5.5.5 ( ε ) t = 0. 00 6 8 0 4 6 n = G ÅÅÅÅÅÅÅ ε v. Figure : Plot of ε (due to phyic requet) v. the gap of the collimator (in beam σ). The point above the curve are forbidden.. Stability diagram and coherent tune hift The different cenario which have been tudied firt are ummaried in Table. The gap G 0 of the collimator correpond to the one for the noal LHC beam [], i.e. it i equal to 6 σ of the beam with a normalied rm tranvere emittance ε =.75 µm. The gap i already very mall in normal operation, and need to be further reduced for the TOTEM experiment in order to protect the RP. Table : Different et of beam parameter tudied. umber of proton per bunch b 0 0 umber of equi-paced bunche M 4 or 56 ormalied rm tranvere emittance ε µm Gap of the collimator G G 0 (, ½, ¼) (G 0 i the gap for the noal LHC beam []) ( ε ) t = 0. 00 Stability diagram correponding to a quai-parabolic ditribution function [5] are plotted in Fig. for the cae b = 0 0 p/b, G = G 0, ε = µm, and ε =. µm. The beam i table if the coherent tune hift for the mot untable coupled-bunch mode number and head-tail mode 0 lie inide the table region (below the curve). It i untable otherwie. The coherent tune hift have been computed uing Sacherer formalim [6], taking into account the inductive-bypa effect for the reitive-wall impedance of the collimator [7]. ote that the two tability diagram correpond to poitive (a > 0) or negative (a < 0) detuning according to the ign of the octupole current, and that thee curve cale linearly with the tranvere beam emittance. Furthermore, the coordinate of the dot which repreent the coherent tune hift, varie linearly with the number of proton per bunch. The real part of the coherent tune hift (which i related to the imaginary part of the coupling impedance) cale with ~G -, wherea the intability rie-time (which i related to the real part of the coupling impedance) cale with ~G. The reult are that all the cae of Table are untable, i.e. the coherent tune hift for the mot untable coupled-bunch mode number and head-tail mode 0 lie outide the table region, with intability rie-time between ~0 and 00 (ee Table ). Thi i due to the almot ingle-bunch effect of the inductive bypa in the reitive-wall impedance from the
collimator []. Thee reult could have been in fact anticipated from the reult for the noal LHC beam, where b =.5 0 p/b and ε =.75 µm, and which i untable []. Table : Intability rie-time for the different cenario. Rie-time [] G = G 0 G = G 0 ½ G = G 0 ¼ M = 4 68 9 64 M = 56 4 9 9 G = G 0 Vertical plane 0.00004 0.0000 0.0000 56 bunche 4 bunche 0.0000-0.000-0.0005-0.000-0.00005 0.00005 0.00004 0.0000 0.0000 0.0000-0.000-0.0005-0.000-0.00005 Figure : Stability diagram with imum available octupolar trength and coherent tune hift for the mot untable coupled-bunch mode number and head-tail mode 0, and for the cae b = 0 0 p/b, G = G 0, ε = µm (upper untable), and ε =. µm (lower table). The horizontal and vertical axe are the real and (u) imaginary part of the tune hift repectively. 4
The TOTEM beam of Table have the ame brightne a the noal LHC beam, a both intenity and emitance are divided by ~4. Therefore, the real part of the coherent tune hift i ~4 time maller, but o i the tability diagram. Eventually the beam tability i almot the ame, except that now the intability rie-time are much longer (due to the almot inglebunch effect of the inductive bypa). Compromie are propoed in Table, taking 0% afety margin for the coherent tune hift to be inide the table region. Table : Poible compromie. G = G 0 b = 0 0 p/b ε =. µm G at 0.6 σ b = 0 0 p/b ε = 0.8 µm G at σ b = 0 0 p/b ε = 0.4 µm G at 8.4 σ G = G 0 ½ b = 0 0 p/b ε = 8.8 µm G at σ b = 0 0 p/b ε = 5.9 µm G at.4 σ b = 0 0 p/b ε =.9 µm G at.4 σ Starting with the good et of beam parameter (for beam tability conideration) b0 = 0 0 p/b, ε 0 =. µm and G 0 = 0.6 σ 0, it can be deduced that the following condition can be ued to find another good et of beam parameter, b b0. () ε G ε G It i extremely attractive to conider cae where the beam brightne i maller than the noal one in order to reduce the ditance of the RP to the beam. For intance for the cae b = 0 0 p/b and ε =. µm, the collimator gap et at the noal value G 0 correpond to 0.6 σ, which i too large a 7 σ i ufficient. If the beam emittance i increaed to.97 µm, Eq. () i fulfilled with G = 7 σ. In thi cae t =.7 0 (GeV / c), which i atifactory ince t ha to be of the order of 0 (GeV / c) (ee Section ). It i poible to reduce further t by decreaing both the emittance and the bunch intenity a dicued below. 4. Dicuion and concluion There i a trade-off between mall-angle cattering obervation and beam tabiliation by Landau damping. On the one hand, obervation of mall-angle cattering et a imum value for the tranvere beam emittance, i.e. Eq. () ha to be fulfilled. On the other hand, beam tability require a imum value for the tranvere beam emittance, i.e. Eq. () ha to be atified. The et of beam parameter ( b0 = 0 0 p/b, ε 0 =. µm and G 0 = n 0 σ 0, with n 0 = 0.6) atifie both condition. Thi i therefore, a good et of beam parameter for the TOTEM experiment. The other good et of beam parameter can be found a follow. oting that the gap G i related to the emittance through 0 0 5
n ε G = G 0, (4) n ε the condition of Eq. () and () can be put together in the following equation 0 0 ε = ε 0 b b0 /5 n 0 n 6/5 ε ε γ l = L eff ϑ h n d. (5) Thee condition are plotted in Fig. 4 and 5 for two different number of proton per bunch, for t = 0 (GeV/ and t = 0 (GeV/ repectively. c) c) (a) (b) e @mmd.5.5 e @mmd.5.5 ε 6 8 0 4 6 n = G ÅÅÅÅÅÅÅ ε ε ε 6 8 0 4 6 n = G ÅÅÅÅÅÅÅ Figure 4: Minimum (due to beam tability condition) and imum (due to phyic requet) normalied rm tranvere beam emittance v. the gap of the collimator (in beam σ) for t = 0 (GeV / and (a) b = 5 0 9 p/b, (b) b = 0 0 p/b. c) If the RP have to be protected againt the econdary halo, it i important to know the value of the emittance aociated with a collimator gap equal to 7 σ a it provide the mallet value of θ. Thi correpond to the point aociated with n = 7 on Fig. 4 and 5. Thee point are plotted on Fig. 6 a a function of the number of proton per bunch. 6
(a) (b) e @mmd.5.5.5 e @mmd.5.5.5 ε ε 6 8 0 4 6 n = G ÅÅÅÅÅÅÅ ε ε 6 8 0 4 6 n = G ÅÅÅÅÅÅÅ Figure 5: Minimum (due to beam tability condition) and imum (due to phyic requet) normalied rm tranvere beam emittance v. the gap of the collimator (in beam σ) for t = 0 (GeV / and (a) b = 5 0 9 p/b, (b) b = 0 0 p/b. c) e @mmd 4.5.5.5 ε ( ε ) t = 0. 00 4 6 8 0 b @µ0 0 D ( ε ) t = 0. 00 Figure 6: Minimum (due to beam tability condition) and imum (due to phyic requet) normalied rm tranvere beam emittance v. the number of proton per bunch for the gap of the collimator G = 7 σ. In concluion, the mallet value of ε = 0.96 µm and G = 7 σ, and i given by θ achievable i obtained for b = 5 0 9 p/b, 7
ε h n d + l γ 0 ϑ = = 4.67 µrad, (6) L eff which yield t = 0 (GeV / c). Finally, it hould be noticed that the beam tability from Landau damping ha been evaluated up to thi point for the cae of a quai-parabolic ditribution function, which extend up to. σ in tranvere pace [5]. Thi ditribution function underetimate the beam tability if the tranvere beam profile extend up to 6 σ, a it i foreeen to be the cae in the LHC at top energy with the noal collimator etting. The Gauian ditribution extend to infinity in tranvere pace and thu overetimate the beam tability. In Ref. [8], the beam tability ha been analyzed for the ditribution function conitent with the collimator etting at top energy, i.e. extending up to 6 σ in tranvere pace. The reult i that a factor of ~ i gained for the real part of the coherent tune hift compared to the cae with the quaiparabolic ditribution function. Thi gain can be ued either to reduce the value of the emittance by the ame factor, the correponding momentum tranfer quared then become t = 0.7 0 (GeV/ c), or to increae the beam intenity. For ε = µm, the number of proton per bunch can be increaed to b =.04 0 0 p/b. The correponding momentum tranfer quared i t =. 0 (GeV / c). The cae of a ditribution extending up to 6 σ but with more populated tail than the Gauian ditribution ha alo been conidered (a thi may be the cae in reality in proton machine, where everal diffuive mechanim can take place, in particular during beam acceleration) and revealed that a factor of ~4 i gained in thi cae for the real part of the coherent tune hift compared to the cae with the quai-parabolic ditribution function. Thi gain can be ued either to reduce the value of the emittance by the ame factor 4, the correponding momentum tranfer quared then become t = 0 (GeV / c) (i.e. cloe to the Coulomb region [9]), or to increae the beam intenity. For ε = µm, the number of proton per bunch can be increaed to b =.08 0 0 p/b (the correponding momentum tranfer quared i the ame a before, i.e. t =. 0 (GeV / c) ). However, the new tability diagram of Ref. [8] hould be ued with care for beam tability prediction, a the preence or not of the high-amplitude tail in the ditribution can ubtantially affect the amount of Landau damping. Thi i why the analyi ha been made in thi note conidering the conervative quai-parabolic ditribution function, which doe not take into account the beneficial effect on Landau damping of the particle with tranvere amplitude larger than. σ. Furthermore, reducing the normalied rm tranvere beam emittance down to ~0.4 µm (for the cae where t = 0 (GeV / c) ) i far from being eay. 5. Reference [] TOTEM Technical Deign Report, CER-LHCC-004-00, January 7, 004. [] P. Jarron, E. Métral, F. Ruggiero, Minute of a meeting on electromagnetic hielding of the Roman Pot for TOTEM, January 0, 00. [] LHC Deign Report (Chapter 5), to be publihed. [4] J.B. Jeanneret, Private communication (004). 8
[5] J. Scott Berg and F. Ruggiero, Landau Damping with Two-Dimenional Betatron Tune Spread, CER SL-AP-96-7 (AP), 996. [6] F. Sacherer, Tranvere Bunched-Beam Intability-Theory, Proc. 9 th Int. Conf. on High Energy Accelerator, Stanford 974 (COF 7405, US Atomic Energy Commiion, Wahington D.C., 974), p. 47. [7] L. Vo, The Tranvere Impedance of a Cylindrical Pipe with Arbitrary Surface Impedance, CER-AB-00-005 ABP, 00. [8] E. Métral and A. Verdier, Stability Diagram for Landau Damping with a Beam Collimated at an Arbitrary umber of Sigma, CER-AB-004-09-ABP, 004. [9] ATLAS forward detector for luoity meaurement and monitoring, Letter of intent, CER/LHCC/004-00. 9