1. The Theory of Charged Particle Energy Loss and Multiple Scattering in Materials and its Application to Muons in Liquid Molecular Hydrogen W W M Allison, Oxford Presented at Oxford 20 th January 2004

2. 20 Jan 2004Oxford Seminar2 From: %%%%%%%%%% Sent: 10 December 2003 02:20 To: %%%%%% Subject: [Mice-sofware] Muon multiple Coulomb scattering Hi %%%%, The MICE proposal to RAL (January 10, 2003), Chapter 2 "Cooling", contains expression (2.1) for derivative of normalized transverse emittance, de_n/ds. It was mentioned there that it is an approximate expression. A member of the muon collaboration, Sergei Striganov, recently completed a stage of his work on muon multiple scattering and showed that a more correct expression can be derived. He performed a lot of comparisons with available experimental data to prove it. Fortunately for MICE, for 200-MeV muon multiple scattering on hydrogen the predicted average rms scattering angle is reduced by 30-40% when compared to the traditional expression with radiation length by Rossi. It means less heating due to m.c.s. and that's good. In such a case the expression for equilibrium emittance (2.2) from the same proposal to RAL can be reduced by the same 30-40%. For a single hydrogen absorber the effect is, of course, less significant. I think that Sergei will present his results soon and it makes sense to include the updated multiple scattering law in G4MICE to have it in addition to the traditional m.c.s. simulation scheme. ! !

3. 20 Jan 2004Oxford Seminar3 We are interested in the distribution in transverse momentum and energy transfer as a result of a monochromatic beam of momentum P passing through an absorber of thickness with target atom density N. This depends on the cross section per target atom in the absorber. Thus the prob. per metre of a single collision involving energy transfer between E and E+ΔE AND transverse momentum transfer between p t and p t +Δp t is However the cross section is large so that multiple collisions occur.

4. 20 Jan 2004Oxford Seminar4 Two questions: 1.What is the cross section? 2.How do the single collisions combine to give the distributions in net energy loss and transverse-momentum transfer? We will answer the questions for the particular case of muons in liquid hydrogen, although the analysis could be extended to other materials.

5. 20 Jan 2004Oxford Seminar5 Single collision. Double differential cross section for energy and transverse mtm. transfers, E and p T : Input data (Low energy) Photoabsorption cross section of medium Density & refractive index Incident particle momentum, P Incident particle mass (muon) Overview Input theory Maxwell’s equations Causality and dispersion Dipole approx. Oscillator strength sum rule Point charge scattering with rel. recoil kinematics & i) hi-Q 2 μ-e scattering (Dirac) ii) lo-Q 2 H atomic form factor (exact H wave fn.) iii) hi-Q 2 μ-p scattering (Rosenbluth) Radiative energy loss (Bremsstrahlung) all known recent data for H 2 Database (by MC or integ.) for energy and transverse momentum loss in thin absorbers, including correlations and non-gaussian tails [const  and pathlength] Cooling in interesting geometries? MC tracking of muons in general hydrogen absorbers

6. 20 Jan 2004Oxford Seminar6 Plan of talk 1.Theory of the double differential cross section 2.The photoabsorption data input for atomic and molecular Hydrogen 3.The Energy Loss and Multiple Scattering in thin absorbers 4.Energy Loss and Multiple Scattering distributions for general absorbers of H 2 5.Estimation of systematic uncertainties and errors 6.Correlations between Energy Loss and Multiple Scattering distributions 7.Conclusions

7. 20 Jan 2004Oxford Seminar7 It is just an inelastic scattering process... σ as a function of (Q 2,ν) or (Q 2,x) or (p T, ν)... First the atomic and electron-constituent coulomb part. Second the nuclear-constituent coulomb part The two are widely separated and there is no interference between them. [In each collision there is an accompanying radiative energy loss. This can be calculated in first order Pert Theory from the charge velocity discontinuity (see Jackson for example). This Bremsstrahlung is small.] 1. Theory of the double differential cross section

8. 20 Jan 2004Oxford Seminar8  In a collision 3-mtm (P T,P L ) is exchanged and also energy E  The interaction is single photon exchange  The 4-momentum transfer (in m -2 ):  The μ mass is conserved: This leads to the relationbetween the E and P transfer  If the collision is elastic with a stationary constituent of mass M, there is a similar relation: (This is like the previous relation with β=0, γ=1, ω  - ω, μ  M)  Such constituent scattering gives rise to the familiar deep inelastic constraint: Kinematics

9. 20 Jan 2004Oxford Seminar9 For a discussion of the energy-loss part with non-relativistic electron recoil see Allison & Cobb, Ann. Rev. Nucl. Part. Sci 1980 The longitudinal force F responsible for slowing down the particle in the medium is the longitudinal electric field E pulling on the charge e F = eE where the field E is evaluated at time t and r =  ct where the charge is. By definition this force is the energy gradient and thus the mean rate of energy change with distance (“dE/dx”) is the force itself Scattering by the atom as a whole and its constituent electrons What is the value of this electric field?

10. 20 Jan 2004Oxford Seminar10 From the solution of Maxwell’s Equations for the moving point charge in a medium with dielectric permittivity ε and magnetic permeability μ, this field is: Everything here is known except the (complex) material response functions ε(k,ω) and μ(k,ω) where k is the wavenumber (or transverse momentum transfer) and ω is the angular frequency (or energy transfer), both of which are integrated over. Integrating over  this gives...

11. 20 Jan 2004Oxford Seminar11 However we can also write down the mean energy loss is due to the average effect of collisions with probability per unit distance N dσ for a target density N. where  1 and  2 are the real and imaginary parts of ε(k,ω). We may equate integrands and deduce the cross section for collisions with transverse momentum transfer  k T and longitudinal momentum transfer  k L

12. 20 Jan 2004Oxford Seminar12 The first term describes collisions with the whole atom in the Dipole Approximation (wavelength >>atomic dimensions). The second describes collisions with constituent electrons assumed free and stationary for those that are free at frequency ω. The two are tied together by the Thomas-Reiche-Kuhn Sum Rule.  1 is given in terms of  2 by causality, the Kramers Kronig Relations [see J D Jackson]. So all we need is  2 This is given in terms of the low energy atomic/molecular photoabsorption cross section σ(ω) for free photons and electron mass m, see Allison & Cobb, Ann Rev Nucl Part Sci (1980)

13. 20 Jan 2004Oxford Seminar13 The resulting differential cross section covers both collision with an atom as a whole, and with constituent electrons at higher k 2. This already includes Cherenkov radiation, ionisation, excitation, density effect, non-relativistic delta ray production. To extend this formalism (Allison/Cobb) to relativistic electron recoil, we simply replace the non-relativistic kinematic condition by its relativistic form with the 4-momentum transfer  Q given by At high Q 2 this cross section becomes the relativistic Rutherford form for spinless point charges:

14. 20 Jan 2004Oxford Seminar14 As such it describes Deep Inelastic Scattering from atoms with stationary constituent spinless electrons for which Near the kinematic upper limit in Q 2 there is a modest factor which describes the contribution of magnetic scattering. This depends on the mass, spin and structure of the incident charge and target. For muon-electron scattering (“Dirac”) this factor is known to be where θ is the scattering angle in the μ-e CM. For our purposes we wish to express this inelastic cross section in terms of p L (or E) and p T rather than x and Q 2. Let us have a look…

15. 20 Jan 2004Oxford Seminar15 Illustration: Calculated cross section for 500MeV/c  in Argon. Note that this is a log-log-log plot log k L 2 18 17 7 log k T whole atom, low Q 2 (dipole region) electron, at high Q 2 electron, μ backwards in CM nuclear small angle scattering (suppressed by screening) nuclear backward scattering in CM (suppressed by nuclear form factor) Log p L or energy transfer (16 decades) Log p T transfer (10 decades) Log cross section (30 decades)

16. 20 Jan 2004Oxford Seminar16 Because nuclear masses M are 000’s times larger than m the regions of constituent scattering do not overlap the electron scattering region. The kinematic condition for collision with a nucleus of mass M is The Rutherford cross section is still for a nuclear charge Z with a charge distribution modified by F(Q 2 ). At low Q 2 the nuclear charge is screened by atomic electrons. Q min 2  1/a 0 2  10 20 m - 2 In the case of Hydrogen the electron wavefunction is particularly well known. At high Q 2 the finite nuclear size gives Q max 2  1/r 0 2  10 30 m -2. At high Q 2 there are also magnetic effects, depending on the magnetic moment of the nucleus as well as that of the muon. In the case of Hydrogen this is given by Rosenbluth Scattering which is well known and includes the finite proton size as well (see Perkins, 3rd edn.). Nuclear constituent coulomb scattering

17. 20 Jan 2004Oxford Seminar17 log 10 Q 2 m -2 Here lies the source of the statistical problem: dσ/dQ 2 varies over 20 orders of magnitude! Further, the contribution to energy loss, for example, goes like F(Q 2 ) prop. to area under this curve, which is log dependent on the max/min Q 2. 1. the importance of very rare collisions at high Q 2 is never negligible in the computation of even low moments, eg mean energy loss, tranverse momentum 2. higher moments (RMS, errors etc) are even more dependent on them, and tend to be statistically unstable. 3. however a factor 2 error in max/min Q only changes the mean or area by 2%. Rosenbluth form factor atomic H wave fn ffactor screening by electrons at 10 -10 m proton structure at 10 -15 m Nuclear formfactor F(Q 2 ) vs. log 10 Q 2

18. 20 Jan 2004Oxford Seminar18 2. Data input for atomic and molecular hydrogen Particularly well known for H and H 2. New data compilation “Atomic and Molecular Photoabsorption”, J Berkowitz, Academic Press (2002). Atomic H photoabsorption cross section (mostly theory) Molecular H photoabsorption cross section (theory and experiment) 10 100 1000eV m 2 per atom We need the Photoabsorption cross section.

19. 20 Jan 2004Oxford Seminar19 Dielectric Permittivity Experimental value 1.236 Mean Ionisation Potential (not used!) Calculated valuesunshiftedwith shift -0.8eVunshiftedwith shift -0.8eV Molecular H 2 1.2141.23619.2218.59 Atomic H1.35315.01 But that is for low density Hydrogen. Condensed Matter effects a) broaden the binding energies of discrete lines, and b) reduce their binding energies (essentially because e  e/√ε(ω) ). Data match the observed low frequency permittivity with a shift of -0.8eV A width of 0.2eV is used for discrete lines. The uncertainty in these effects is taken into account in calculating the systematic error in the cross section.

20. 20 Jan 2004Oxford Seminar20 Thence the real and imaginary part of ε(ω) for on-mass-shell photons. For molecular Hydrogen at 0.0708 g cm -3 : photon energy eV ε1 ε2

21. 20 Jan 2004Oxford Seminar21 Thomas Fermi Atomic H electron screen Saxon Woods with & without Mott spin factor Rosenbluth with/without spin factor Hydrogen nuclear formfactors (magnetic factors for 200MeV/c μ) log10(Q 2 )

22. 20 Jan 2004Oxford Seminar22  Generally continuity requirement determines the evolution with time of a general probability density of particles of mass m in phase space, ρ(P,r,t), in a region where the target density is N(r).  The probability current density is where the velocity  The continuity condition modified by scattering gives the transport equation:  Practical problems are solved by considering the “point spread function” in momentum space for an incident monochromatic beam as it traverses a small thickness of target. 3. Energy Loss and Multiple Scattering in thin H2 absorbers (ELMSA)

23. 20 Jan 2004Oxford Seminar23  The thickness must be small enough that, (although there may be many collisions), the cross section does not change significantly and the pathlength within is not extended. [typically we use 1mm in liquid H 2 at 200MeV/c but much smaller and larger values at lower and higher momenta respectively]  The 3D probability distribution in momentum transfer as a result of passage through a thin absorber can be calculated from the cross section either - by numerically folding the effect of collisions in thin layers, or - by Montecarlo simulation of the collisions. In this work we use the latter.  We represent each class of collision in thickness by its respective element of probability. Thus the chance of a collision with transverse momentum between  k T and  (k T +Δk T ) and longitudinal momentum between  k L and  (k L +Δk L ) is:

24. 20 Jan 2004Oxford Seminar24  The size of the cells is chosen so that the fractional range of k covered is small, 2% or less. Typically there are 5-10 10 4 such cells.  In considering multiple collisions the effect of longitudinal collisions (or energy loss) simply add  The Mean Energy Loss can be calculated as without MC.  Other estimators and distributions for thin absorbers are calculated from the cross section by the MC program ELMSA.  In considering scattering transverse momentum changes have to be combined with uncorrelated random azimuths to give “2-D P T ”  We also consider the 1-D projected momentum transfer, P x  (A further MC program, ELMSB, tracks through thick absorbers using a database generated by ELMSA. This allows the cross section and projected pathlength to change as a result of collisions within the absorber.)

25. 20 Jan 2004Oxford Seminar25  The database generated by EMSA contains 10 5 traversals of different “thin” thicknesses (from a few μm to 10cm) and different momenta (5MeV/c to 50GeV/c) allowing interpolation for the momentum of interest.  In a given thickness of material some collisions occur rarely; others will occur so many times that fluctuations in their occurrence are less important - time spent montecarlo-ing all of them is unnecessary. An order of magnitude in calculation time is saved by mixing folding and generating techniques.  However the probability of cells varies over tens of orders of magnitude. There are always a majority of collisions that are rare. Consolidating elements for display purposes we can look at the cross section on different scales...

26. 20 Jan 2004Oxford Seminar26 Cross section of 200 MeV/c muons in liquid H 2 2 GeV/c Horiz axis Log Energy transfer from 3E-08 to 3E09 eV Vert axis Log Transverse momentum transfer from 1E01 to 2E09 eV/c High P T resonance region. Possible multipole contributions P T -independent atomic scattering region (Dipole Approximation)

27. 20 Jan 2004Oxford Seminar27 Study of the contribution of different mechanisms Muon momentum, MeV/c200200020000 Collisions, 10 6 m -1 0.9700.8930.892 Mean dEdx, MeV cm 2 g -1 (calc. from cross section, not MC) all mechanisms4.3024.2724.896 nuclear recoil0.002 0.012 electron recoil2.1802.3912.997 resonance, continuum1.1080.9710.973 resonance, discrete0.9940.8350.834 cherenkov0.0180.0710.072 bremsstrahlung0.0000.0010.019 Projected P T in 10mm of liquid H 2, RMS98, MeV/c all mechanisms0.39410.35730.3583 nuclear recoil0.26750.23160.2309 electron recoil0.26520.24440.2414 resonance continuum0.06060.05400.0538 resonance discrete0.05490.04890.0487 Note: Half the scattering is due to constituent electrons Note: Half the energy loss is due to constituent electrons

28. 20 Jan 2004Oxford Seminar28 Cross section of 200 MeV/c muons in liquid H 2 (just below threshold) Horiz axis Long. momentum transfer on linear scale from 0 to 20 eV/c Vert axis Trans. momentum transfer on linear scale from 0 to 20 eV/c 2 GeV/c... Cherenkov Radiation already included PTPT P L, also freq.

29. 20 Jan 2004Oxford Seminar29 Bremsstrahlung energy loss, MeV cm 2 g -1 MomentumMean total Brem, cf 4.2-4.9 Contribution by primary collision type GeV/cnuclear constituent electron constituent atomic resonance 20.001 30.0030.0020.001 60.0060.003 130.0120.0070.005 260.0240.0130.011 510.0510.0260.0240.001 1020.1050.052 0.001  negligible contribution from atomic collisions at any energy  combined effect rises to 1% of dedx at 50 GeV/c  at lower energy nuclear collisions are more effective than electron ones, because of the larger value of max Q 2  at higher energy the nuclear form factor reverses the relative contributions

30. 20 Jan 2004Oxford Seminar30 ELMS total cross section as number of collisions/10 6 per metre against log 10 P, with muon P in MeV/c (note, this is not a Monte Carlo result) Non relativistic regionRelativistic region. Note suppressed zero log 10 P

31. 20 Jan 2004Oxford Seminar31 Sp. Mean Energy Loss Muon in liquid H 2 MeV cm 2 g -1 vs. log 10 P where P is muon momentum Curve = Bethe Bloch with Mean Ionisation Potential = 18.59eV as determined from photoabsorption spectrum of H2 Points = ELMS using the same photoabsorption spectrum (note, this is not a Monte Carlo result) log 10 P

32. 20 Jan 2004Oxford Seminar32 Sp. Mean Energy Loss Muon in liquid H 2 MeV cm 2 g -1 vs. log 10 P where P is muon momentum Curve = Bethe Bloch with Mean Ionisation Potential = 18.59eV as determined from photoabsorption spectrum of H2 Points = ELMS using the same photoabsorption spectrum (note, this is not a Monte Carlo result) log 10 P

33. 20 Jan 2004Oxford Seminar33 4. Energy loss and multiple scattering distribut- ions for general absorbers of H 2 (ELMSB) Momentum GeV/c0.10.20.41.02.04.010.020.0stat err Collisions/10 5 1.4550.9700.9070.8940.893 0.892 Mean dEdx (σ)6.3344.3033.9674.0914.2724.4624.7074.896 Mean dEdx (MC)6.2224.2933.9644.1064.2944.4714.7154.8600.3% Median dEdx6.4504.0933.6063.4943.4703.4733.4863.4820.1% 90%ile dEdx7.1655.1344.9014.9635.0045.0365.0685.0530.3% 99%ile dEdx7.9547.2239.80114.3916.8218.3119.419.3 RMS dEdx0.470.721.33.05.69.728.732.9? Mean 2D-Pt2.0901.6831.5721.5641.5731.5831.5941.597~1% RMS proj Px/y1.7971.4661.3581.3831.5311.3981.4431.482? RMS98 proj Px/y1.6451.3231.2391.2331.2351.2411.2471.2460.3% 10 5 muons tracked through 10cm liquid H 2 absorber

34. 20 Jan 2004Oxford Seminar34 1mm2mm4mm1cm2cm4cm10cm20cm40cm1m 100MeV/c.169.237.341.546.7701.0881.7962.530 200MeV/c.130.189.284.447.7361.0291.4672.2923.3004.408 400MeV/c.130.180.252.405.624.8801.3591.9142.7144.275 1GeV/c.219.410.432.616.8721.3831.9532.7514.313 2GeV/c.133.191.270.419.605.8641.5312.0563.0904.554 4GeV/c.135.191.269.427.615.8781.3981.9732.7954.408 10GeV/c.147.208.294.466.673.9521.4442.0422.8874.562 ELMS values of RMS 1-D projected P T (MeV/c) Consider any column Numbers fluctuating unstably by several %, even with these statistics! Samples of 10 5 muons for a variety of thicknesses and momenta.

35. 20 Jan 2004Oxford Seminar35 We follow the statistical procedure suggested by PDG: Discard largest 2% (projected) scatters and fit the RMS using the remainder: - calculate the RMS of the rest, - correct by a factor such that a normal dist. gives the correct RMS by this procedure (divide by 0.9346). Call this estimator “RMS98” PDG quote a formula for RMS98 in terms of the radiation length They quote an error of 11% for In comparing ELMS with PDG for H we use X 0 = 61.28 g cm -2 The next table shows ELMS values for RMS98 are smooth at the level of 1% and gives values for this Multiple Scattering estimator 1-5% higher than PDG. Actually PDG divide by the momentum to get the RMS98 angle. The blue figures have been corrected by a few % on this score.

36. 20 Jan 2004Oxford Seminar36 1mm2mm4mm1cm2cm4cm10cm20cm40cm1m 100MeV/c.144 1.03.210 1.02.304 1.01.497 1.00.713 0.98 1.026 0.99 1.644 1.00 200MeV/c.105 0.97.159 1.00.241 1.03.395 1.02.567 1.00.815 0.99 1.324 0.98 1.879 0.98 400MeV/c.100 1.01.147 1.01.218 1.02.364 1.02.529 1.02.763 1.00 1.239 1.00 1.774 0.99 2.512 0.98 1GeV/c.100 1.04.147 1.04.216 1.04.357 1.04.524 1.04.758 1.02 1.233 1.01 1.773 1.00 2.529 0.98 4.042 0.99 2GeV/c.101 1.05.147 1.04.216 1.04.357 1.04.521 1.04.758 1.03 1.235 1.02 1.788 1.01 2.564 0.99 4.134 0.99 4GeV/c.100 1.05.148 1.05.217 1.05.358 1.04.522 1.04.760 1.03 1.243 1.02 1.794 1.01 2.592 1.00 4.184 1.00 10GeV/c.100 1.05.148 1.05.217 1.05.357 1.04.524 1.04.764 1.04 1.247 1.03 1.807 1.02 2.614 1.01 4.265 1.00 ELMS values of RMS98 projected Pt (MeV/c) and ratio to PDG with X 0 = 61.28 g cm -2 (corrected for energy loss)

37. 20 Jan 2004Oxford Seminar37 Conclusion thus far: MS is underestimated by PDG by 0-5%. Of course we need to examine the distributions in scattering and energy loss themselves, not just one or two moments...

38. 20 Jan 2004Oxford Seminar38 Energy-loss spectra for different momenta and absorber thickesses

39. 20 Jan 2004Oxford Seminar39... and 2-D transverse momentum transfer spectra for same

40. 20 Jan 2004Oxford Seminar40 Comparisons with GEANT by Simon Holmes Version GEANT 4.5.2 Patch 02, released 3 October 2003 There is a new version 4.6 released 12 December 2003 quoting changes: “Multiple-scattering: New Tuning of multiple scattering model Fixed problems for width and tails of angular distributions. Fixed numerical error for small stepsize in G4MscModel (z sampling). Bugfix in G4VMultipleScattering::AlongStepDoIt() and added check truestep <= range in G4MscModel. Set highKinEnergy back to 100 TeV for multiple scattering. Set number of table bins to 120 for multiple scattering. ” But Multiple Scattering and Energy Loss are still separate “processors”!

41. 20 Jan 2004Oxford Seminar41 ELMS 10 5 incident muons Momentum 200MeV/c Thickness 10cm Linear plot of projected transverse momentum transfer, MeV/c Normal distribution with same variance as ELMS for the least 98% GEANT

42. 20 Jan 2004Oxford Seminar42 ELMS 10 5 incident muons Momentum 200MeV/c Thickness 10cm Log plot of projected transverse momentum transfer, MeV/c GEANT Normal distribution with same variance as ELMS for the least 98%

43. 20 Jan 2004Oxford Seminar43 ELMS GEANT 10 5 incident muons Momentum 200MeV/c Thickness 10cm Linear plot of energy transfer, MeV

44. 20 Jan 2004Oxford Seminar44 ELMS GEANT 10 5 incident muons Momentum 200MeV/c Thickness 10cm Log plot of energy transfer, MeV

45. 20 Jan 2004Oxford Seminar45 Comparison of Elms and GEANT (4.5.2 Patch 02, released 3 October 2003) 10 5 muons 200MeV/c passing through 10cm LH2, ρ = 0.0708 g cm -3 ELMSGEANT autoratio ELMS/GEANTGEANT 1mmGEANT 2mm Mean dE, MeV3.0463.0930.9853.0893.090 Median dE2.8882.9280.9862.9242.926 90%ile dE3.6233.7150.9753.7033.707 99%ile dE5.0995.3150.9595.3215.356 Mean Pt, MeV/c1.6811.7740.9481.5851.640 RMS98 proj.1.3211.4050.9411.2641.306 Statistical errors <1%, but somewhat more for “dedx 99%ile” GEANT overestimates energy loss by 2% GEANT (auto stepsize) overestimates P T by 5-6%, although predictions vary up to 10% depending on stepsize

46. 20 Jan 2004Oxford Seminar46 Comparison of Atomic H and Molecular H 2 The different binding changes the energy loss but not the scattering

47. 20 Jan 2004Oxford Seminar47 5. Estimation of systematic uncertainties and errors Look at Energy Loss and scattering of 200 MeV/c muons in 10 cm liquid H 2. Simulate with variations in the cross section. How much difference do they make? Percentage changes to ELMS values due to variations TheoryPhotoabsorption dataForm factor & spin multipoleHalved shift, -0.4eV Doubled linewidth 0.4eV Mott (  spinless) Thomas Fermi Saxon Woods Rutherford (e  spinless) Collisions m -1 +2.5-1.500+5.200 Specific Energy Loss, MeV g cm -2 Tabulated+1.9+0.10000+4.5 Projected Transverse Momentum Transfer, MeV/c RMS98+1.5+0.5+0.9-1.2+1.3-0.2+1.7

48. 20 Jan 2004Oxford Seminar48  Rutherford, Thomas-Fermi, Mott, Saxon-Woods modifications are interesting but untenable.  Modification of the condensed-matter shift and broadening effect create small changes which reflect some systematic error, probably less than 1%  Uncertainties are dominated by the unknown contribution of multipole excitation in the highest Q 2 part of the resonance region. This has been estimated crudely by increasing the cross section there by 25%, ie by factor 1+0.25*(Q 2 a 0 2 )  We believe that the true systematic errors in ELMS are less than 2% for energy loss and less than 1.5% for scattering. However statistical errors often dominate.

49. 20 Jan 2004Oxford Seminar49 6. Correlations between the energy loss and scattering distributions For a single incident momentum and absorber thickness let us look at the P T distributions for each of ten energy- loss deciles. First, 200 MeV/c 1mm absorber 10 5 muons, 10 4 per decile [Or plot the energy loss in each of ten 3D-P T deciles] decile? 0-10%, 10-20%, 20-30%,etc of distribution

50. 20 Jan 2004Oxford Seminar50 Projected Pt dist. for tracks selected by energy loss decile 200 MeV/c 1mm

51. 20 Jan 2004Oxford Seminar51 Specific dedx dist. for tracks selected by absolute Pt 200 MeV/c 1mm

52. 20 Jan 2004Oxford Seminar52 Comparing first and last decile... specific dedx and Pt correlations 1mm H2, 200MeV/c

53. 20 Jan 2004Oxford Seminar53 The same for higher momentum and thicker... 1m H2, 20GeV/c

54. 20 Jan 2004Oxford Seminar54 Correlations arise because  If the particle scatters through a significant angle the tracklength in the absorber is increased, leading to greater energy loss (and more scattering). This effect is negligible.  In the non relativistic region, if the particle loses energy, the scattering cross section increases. This gives rise in absorbers of a certain thickness to some secondary correlation, even with simulations like GEANT with separate processors. This effect is quite small.  Cross section correlations included in ELMS give rise to primary effects. These are due to constituent scattering with electrons. These are responsible for half the scattering in Hydrogen; thus correlations are more significant than in other elements. This effect is not small.

55. 20 Jan 2004Oxford Seminar55 Nuclear target: Negligible energy loss, so no correlation Electron target: Complete correlation between energy loss and scattering Resonant atomic target: No correlation in Dipole approximation Cross section for Hydrogen, log P T vs. log energy transfer

56. 20 Jan 2004Oxford Seminar56 200 MeV/c 1mm of LH2 ELMS Mean 2-D transverse momentum transfer, MeV/c (y) vs. mean energy transfer, MeV (x) for each energy transfer decile mean energy transfer, MeV Mean 2-D transverse momentum transfer, MeV/c

57. 20 Jan 2004Oxford Seminar57 ELMS GEANT mean energy transfer, MeV Mean 2-D transverse momentum transfer, MeV/c 200 MeV/c 10cm Mean 2D P T transfer MeV/c (y) vs. mean energy transfer, MeV (x) for each energy transfer decile

58. 20 Jan 2004Oxford Seminar58 Momentum Thickness200MeV/c2GeV/c20GeV/c 1mm0.300.460.47 10mm0.190.340.48 100mm0.190.300.47 1m0.160.140.44 10m0.090.15 100m0.18 Putting numbers on the correlations Values of dimensionless correlation parameter Electron constituent scattering is ~100% correlated between P T and energy loss. It is responsible for half the scattering and half the energy loss. So we should expect a correlation of 50%, falling due to the effect of random azimuths.

59. 20 Jan 2004Oxford Seminar59 7. Conclusions 1.The cross section for energy and momentum transfer collisions in matter can be derived with some rigour 2.The only significant uncertainty is the contribution of multipole excitation at the highest Q 2 in the resonance region. This contributes perhaps 1.5% and 2% systematic error to scattering and energy loss estimates respectively. 3.Statistical effects remain slippery to handle. Distributions of variables, not their simple means and standard deviations, are required to answer serious questions. These can be simulated. 4.Collisions with constituent electrons generate correlations between scattering and energy loss. The effect is most pronounced for Hydrogen and is ignored in usual simulations. It is likely that these correlations will be beneficial to Ionisation Cooling. The effect of this remains to be studied. 5.These calculations could be extended to other materials (for which the effect of correlations would be smaller). 6.Comparison with MUSCAT data will be interesting but may not have the energy resolution to reveal correlations.

60. 20 Jan 2004Oxford Seminar60 Nuclear target: Negligible energy loss, so no correlation Electron target: Complete correlation between energy loss and scattering Resonant atomic target: No correlation in Dipole approximation Cross section for Hydrogen, log P T vs. log energy transfer