Analytical calculation of the power dissipated in the LHC liner. Stefano De Santis - LBNL and Andrea Mostacci - CERN

Transcription

1 Analytical calculation of the powe dissipated in the LHC line Stefano De Santis - LBNL and Andea Mostacci - CERN

2 Contents What is the Modified Bethe s Diffaction Theoy? Some inteesting consequences of the expessions found. Powe dissipation in the LHC line.

3 LHC Beam Pipe

4 Equivalent dipole (and quadupole) moments of an apetue in the vacuum chambe The electomagnetic field can be expessed as a sum of modes: E (,ω ) = c n E n (,ω ), H (,ω ) = c n H n (,ω ) The expansion coefficients can be obtained though equivalence and ecipocity theoems: c n = 1 H n J n ds Bewae: when dealing with esonating modes the above fomulas ae diffeent Expanding the magnetic field in a Taylo seies about the oigin H n ( ) = H n (0)+ u u H n + v H n u=v=0 v u=v= H n (0)+ H n u=v=0 c n = jω µ H n M E n P + µ H n Q t

5 Bethe s Diffaction Theoy (BDT) Longitudinal impedance: Loss facto: Z(ω ) = 1 q + E d l k σ = 1 π + 0 Re[Z(ω )]e (ωσ z/c) dω The scatteed electomagnetic field is that adiated by the equivalent dipole moments: E = f M, P ( ) The equivalent dipole moments ae calculated fom the apetue static polaizabilities (geometic paametes) and the incident field: M = t α m H 0, P = ε t α e E 0

6 Modified BDT Bethe s diffaction theoy gives a good appoximation fo the imaginay pat of the impedance only. Real impedance and thus loss facto ae null; enegy is not conseved. In the modified vesion of the theoy [Collins] adiation eaction fields ae intoduced in the dipole moments calculation: M = t α m ( H 0 + H s ), P = ε t α e ( E 0 + E s ) The eaction fields dipend on the polaizability tensos, so that the dipole moments components ae obtained solving a linea system: M [] S = P t α m H 0 ε α e E 0 The coefficients matix [S] is a function of the modes chosen to epesent the electomagnetic fields. Its expession is paticulaly simple in the low fequency appoximation, when only one popagating mode is used. The tems outside the pincipal diagonal epesent coupling between apetues.

7 BDT vs. Modified BDT Fo a single hole the modified theoy adds up a fequency dependant tem in the dipole moments expession. Fo example at low fequency in a coaxial beam pipe: M ϕ = 1 + j α m H 0ϕ, P α m ω / c = 4πb ln( d / b) εα e E 0 α 1 + j e ω / c 4πb ln( d / b) whee the oiginal theoy would only give: M ϕ = α m H 0ϕ, P = εα e E 0 In the pesence of multiple holes the diffeence is even moe appaent since the modified theoy takes into account the coupling between holes that the oiginal one disegads entiely.

8 Diffeential Modified BDT The Modified BDT allows to calculate the inteaction between multiple apetues (if they ae distant enough), but it cannot be used when the apetue dimensions ae lage than the wavelength. If this is the case, one can esot to use dynamic polaizabilities, that do not depend on the apetue geomety only and equies complex calculations, o subdivide the apetue in infinitesimal elements still satisfying the equiements fo using the BDT and then take into account thei inteaction. The dipole moments ae eplaced by diffeential dipole moments d M = d t α m ( H 0 H s ), d P = εd t α e ( E 0 E s ) and integal equations take linea system place. In the case of a long naow slot: dm ϕ dz = α m L H 0ϕ j ωµh 0ϕ L/ L/ dm ϕ dξ e jk 0 z ξ dξ + j ωh 0ϕ e 0 L/ L/ sign(ξ z) dp dξ e jk 0 z ξ dξ dp dz = εα e L E 0 j ωµe 0 L/ L/ dp dξ e jk 0 z ξ dξ + j ωµh 0ϕ e 0 L/ L/ sign(ξ z) dm ϕ dξ e jk 0 z ξ dξ

9 As diect consequence of the expessions calculated, one can find both expected and not-so-expected esults: aveage distance between holes (cm) Loss facto vs. length fo a naow slot (solid line), its static appoximation (dashed line) compaed to MAFIA simulations (black diamonds) Loss facto fo 15 ound holes andomly spaced (b=0 mm, d=4 mm, R=6 mm, l=300 mm, sigma=5 cm). The loss facto is popotional to the numbe of holes squaed.

10 LHC line - Summay Impedance (beam dynamics) Powe load on cold boe (cyogenics) A complex popagation constant has been included in ou fomalism. This is suitable in the teatment of both lossy and peiodic loaded stuctues Main esults: Randomization of the slots position stongly deceases the peaks of the eal impedance. The imaginay pat and the loss facto ae unaffected. Dissipated powe pe unit length estimated aound 1 mw/m. Identified two diffeent egimes fo the powe loss pe unit length: Shot device Long device π Z P lin = 0 Q b c (α m + α e ) 18π 4 σ 3 S b b 4 D ln(d / b) L P = Z 0Q b c σ 3 S b π 8 +Γ(5/ 4) σω / c αd (α m + α e ) b 4 Dln(d / b)

11 Effect of ohmic losses Appox: distibuted dielectic losses b cold boe α () ω = ρ Z 0 ln( d b) b d µ ω beam line Analytical fomulae: Dipole moments Coupling impedance Loss Facto k(σ) Powe lost pe unit length P S P = c Q d S b k () σ L b bunch spacing L device length

12 Powe lost pe unit length P lin no attenuation P limit value P exact fomula mw/m P lin LHC paametes P P Length (m) L α = 4 Γ π 1 () 5 4 α ( ω ) c ωc Bunch cut-off angula fequency

13 Powe lost pe unit length The powe lost pe unit length is impotant in eal machines (LHC) to estimate the heating of the cold boe L α is ~80 m fo LHC Fo small length P is still function of the length (intefeence effects) Satuation effect fo long pefoated chambes (the theoy with no losses gives an infinite powe lost) In LHC we emain in the linea egime (small device lengths), since the coaxial stuctue is inteupted by many devices ( the dipole modules length is 14 m) The satuation value is an uppe limit useful in the poject Only a pat (~ 45%) is dissipated in the extenal wall (=d)

14 LHC beam sceen Aound LHC nominal values (t): P P 0 ( π T W) P0 Exp 1.75 W = 4 mw / m 1.5 mm 4 Cuves of constant powe pe unit length (mw/m) W (mm) t W slot width T wall thickness Fom: A. Mostacci and F. Ruggieo, LHC Poject Note T (mm)

15 Compaison with pevious studies F. Caspes, E. Jensen and F. Ruggieo, EPAC, Belin 9 Measuements on a m long model of LHC vacuum chambe Simplified theoy accounting only fowad TEM wave Electic Field adiated in coax. egion Souce Electic Field N α m + αe b ln( d b)λ Calculated tansmission coefficient is identical to what epoted in EPAC 9. Both theoetical values ae a facto below measuements.

16 LHC beam sceen The holes ae such that the slotted suface is 4.4 % of the total suface (fixed fom vacuum equiements) stong dependence on T and W The fomula is a simple analytic fit to see the behavio aound nominal values; it tells what happens if you educe T o incease W The nominal LHC is the geen point (P 1.1 mw/m) Conclusion: NO DANGER