Calculations of Energy Loss and Multiple Scattering (ELMS) in Molecular Hydrogen W W M Allison, Oxford Presented at NuFact02 Meeting, July 2002

3 July 2002ELMS calculations in Molecular Hydrogen 2  To show that Ionisation Cooling will work we must be able to simulate energy loss and scattering in media. To do this we have a choice we can use traditional calculations with their uncertainties, we can wait for MUSCAT, or we can calculate the phenomena afresh from first principles  We need to achieve precise and reliable distributions of P T transfer (scattering) and P L transfer (energy loss) from the muon to the medium, including all non gaussian tails and correlations.  The traditional methods of calculation date from days when data on media were poor, computers were rare and the priority was on quick back-of-the-envelope results using simple parameters, eg radiation length, mean ionisation potential etc. Hydrogen, the medium of most interest for cooling, is most difficult.  So we start again. Today we have good data on media properties, specifically of the low energy photoabsorption cross section, thanks to better calculations and data from synchrotron radiation sources.

3 July 2002ELMS calculations in Molecular Hydrogen 3 This is a report on some work in progress at Oxford  We have derived from first principles the cross section for transfers, P T and P L, between the medium and the muon in terms of the low energy photoabsorption cross section and the kinematics of basic atomic physics.  We have used this cross section to generate distributions in the energy loss and scattering (with their correlations) arising from multiple collisions in finite thicknesses of these materials.  Some preliminary results from this analysis (ELMS) are given, in particular for atomic and molecular Hydrogen at liquid density.  Some reservations, qualifications and need for further work are noted.

3 July 2002ELMS calculations in Molecular Hydrogen 4 Input data Photoabsorption cross section of the medium Density Incident particle momentum, P Incident particle mass (muon) Distributions in P t and energy loss - the ELMS program Input theory Maxwell’s equations and point charge scattering Causality Oscillator strength sum rule Dipole approximation Electron constituent scattering (Dirac) Nuclear constituent scattering (Rosenbluth) Atomic form factor (exact H wave function) Nuclear form factor (Rosenbluth) Double differential cross section for long. and trans. mtm. transfers, p L and p T : MC distributions in longitudinal and transverse momentum loss in thin absorbers, including correlations and non-gaussian tails Effect of general absorber thickness on 6-D phase space distributions including correlations and non- gaussian tails not done yet all known recent data for H 2

3 July 2002ELMS calculations in Molecular Hydrogen 5 Plan of talk 1.Theory of the double differential cross section 2.The photoabsorption data for atomic and molecular hydrogen 3.The ELMS Monte Carlo program 4.Energy loss and multiple scattering distributions for H and H 2 (preliminary). 5.Verification and estimation of systematic uncertainties 6.Further work 7.Preliminary conclusions

3 July 2002ELMS calculations in Molecular Hydrogen 6 First the atom and constituent electron part; then the nuclear coulomb part The atom and constituent electron part. (see ARNPS 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 the rate of work is force×velocity and thus the mean rate of energy change with distance is the force itself From the solution of Maxwell’s Equations for the moving point charge in a medium 1. Theory of the double differential cross section

3 July 2002ELMS calculations in Molecular Hydrogen 7 Integrating over  this gives This mean energy loss is due to the average effect of collisions with probability per unit distance N dσ for a target density N. The energy loss dE = - Σ  ω = - Σ  ω N dx dσ. Thus equating integrands where  1 and  2 are the real and imaginary parts of ε(k,ω).  1 is given in terms of  2 by the Kramers Kronig Relations [see J D Jackson]. So all we need is  2 which is given in terms of the photoabsorption cross section σ(ω). Following Allison & Cobb, Ann Rev Nucl Part Sci (1980) we may write where m is the electron mass.

3 July 2002ELMS calculations in Molecular Hydrogen 8 The resulting cross section covers both collision with an atom as a whole, and with constituent electrons at higher Q 2. Included are Cherenkov radiation, ionisation, excitation, density effect, relativistic delta Second there are the nuclear coulomb collisions. The Rutherford cross section is with The kinematic condition for collision with a nucleus of mass M is. The cross section may then be expressed in terms of p L and p T. The first term describes collisions with the whole atom; the second with constituent electrons. The two together satisfy the Thomas-Reiche-Kuhn Sum Rule. In ELMS this formula is corrected for relativistic electron recoil and magnetic scattering. in a different part of phase space so that there is no interference!

3 July 2002ELMS calculations in Molecular Hydrogen 9 Example: Calculated cross section for 500MeV/c  in Argon gas. Note that this is a Log-log-log plot - the cross section varies over 20 and more decades! log k L log k T whole atoms at low Q 2 (dipole region) electrons at high Q 2 electrons 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)

3 July 2002ELMS calculations in Molecular Hydrogen 10 50GeV/c  in Argon gas.... Yes, at larger angles PTPT P L or E 5GeV/c  in Argon gas.....Yes PTPT P L or E Zooming in to look for Cherenkov Radiation just below ionisation threshold MeV/c  in Argon gas.... No PTPT P L or E

3 July 2002ELMS calculations in Molecular Hydrogen Photoabsorption cross section for atomic and molecular hydrogen New data compilation “Atomic and Molecular Photoabsorption”, J Berkowitz, Academic Press (2002). Atomic H photoabsorption cross section (all theory) Molecular H photoabsorption cross section (theory and experiment) eV m 2 per atom

3 July 2002ELMS calculations in Molecular Hydrogen 12 The Rutherford cross section is modified at both high and low Q 2. Form factors corrections to electron constituent scattering For scattering from constituent electrons at low Q 2 there is a fraction g(  ) describing the proportion of electrons that are effectively free for energy transfers The remaining fraction (1 - g) are involved in atomic resonance scattering. The maximum Q 2 in scattering off constituent electrons is small as can be seen from the given formula. At maximum Q 2 there is pure point-like μ-e Dirac scattering. Form factor corrections to proton constituent scattering at high Q 2 These are due to the finite nuclear size and magnetic scattering. They are described together by Rosenbluth Scattering for a spin-½ incident muon. These high Q 2 corrections are not very important but we get them right anyway. Form factor corrections to proton constituent scattering at low Q 2 These are due to electron shielding. We use the exact atomic hydrogen wave function. The effect of these may be judged from the area under the formfactor curve plotted against log Q 2....

3 July 2002ELMS calculations in Molecular Hydrogen 13 Formfactors (squared) for Hydrogen. At high Q 2, Green = the Rosenbluth form factor. At low Q 2, Red = atomic hydrogenic wave function form factor log Q 2 m -2 nucleus screening by electrons at m proton structure effect at m F(Q 2 )

3 July 2002ELMS calculations in Molecular Hydrogen 14 The double differential cross section in molecular hydrogen

3 July 2002ELMS calculations in Molecular Hydrogen 15 To calculate the distribution of long. mtm. and transverse mtm. transfer due to the many collisions in a finite absorber thickness. (In Hydrogen there are about 10 6 collisions per m.) We cut the problem up into elements of probability. Thus the chance of a collision with transverse momentum between  k T and  (k T +dk T ) and longitudinal momentum between  k L and  (k L +dk L ) is: (The size of the cells is chosen so that the fractional range of k covered is small.) The value of the different P per metre vary over many orders of magnitude. In a given thickness of material some occur rarely or a small number of times; others will occur so many times that fluctuations in their occurrence are less important and time spent montecarloing all of them is unnecessary. Two orders of magnitude in calculation time can be saved by mixing folding and generating techniques. The method has been rigorously checked. 3. The ELMS Monte Carlo Program

3 July 2002ELMS calculations in Molecular Hydrogen 16 Refractive index Mean dE/dx RMS projected θ MS (98% lowest) MeV cm 2 g -1 mrad Molecular Hydrogen Atomic Hydrogen Input: ELMS calculation of 10 5 traversals of 180 MeV/c muons. Absorber 0.5m liquid H 2, density = g cm -3. [bubble chamber value] Result: 4. Energy loss and multiple scattering distributions for H and H 2 (preliminary). Actual value at this density (bubble chamber data) Expected value (PDG) mrad with radiation length g cm 2 Preliminary comp. with range/mtm relation in H 2 bubble chambers. OK

3 July 2002ELMS calculations in Molecular Hydrogen 17 element mm Values from10 5 ELMS samplesPDG book values RMS deviation P l MeV/c RMS deviatio n 2D P t MeV/c Correl- ation (MeV/c) 2 RMS98 proj ang RMS 98 /X 0 Ratio ELMS/ PDG mrad Molecular H Lithium Beryllium Carbon Aluminium Iron Lithium Beryllium Carbon Aluminium Iron Results for some other elements. Muons at 180MeV/c. Most elements agree to 2-3% - except hydrogen

3 July 2002ELMS calculations in Molecular Hydrogen MeV/c muons in 500mm molecular H at liquid density samples. Red curve = normal dist Left plots projected p t, right plots dE/dx or p L.

3 July 2002ELMS calculations in Molecular Hydrogen 19 Contributions of each component of cross section. 180MeV/c muon. Liquid Molecular Hydrogen. Molecular Hydrogen 500mmAtomic Hydrogen 500mm Mean Pt Mean dEdx correl- ation RMS98 ang mean Pt mean dEdx correl- ation RMS98 ang MeV/c MeV cm 2 g -1 (MeV/c) 2 mradMeV/c MeV cm 2 g -1 (MeV/c) 2 mrad Cherenkov Discrete resonance electron constituent nuclear constituent All mechanisms

3 July 2002ELMS calculations in Molecular Hydrogen 20 Correlations sample. 180MeV/c muons. The scatter plot of magnitude of 2D P t (left to lower right ) against  E (left to upper right). Note that the main peak has been very heavily truncated. Correlations largely confined to hard single scatters. PTPT P L or E

3 July 2002ELMS calculations in Molecular Hydrogen 21 Results independent of balance between folding and MC over 3 orders of magnitude The deviation of the mean MC dE/dx value from value derived from the probability table should be described by the calculated error on the MC mean value.  Results independent of modest variation of the momentum transfer value at which resonance scattering is replaced by electron constituent scattering. Preliminary investigation suggests that this test is passed.  Atomic Formfactor. Molecular effects on electron screening of nuclear scattering. Effect of density on electron screening. Probably small, but largest for H 2 ?  Effect of constituent electron fermi momentum. Arguments suggest that this is small.  Bremsstrahlung effects. Supposed small at energies of interest for cooling.  Other than for H, the magnetic effect of nuclear spin. Certainly negligible. 5. Verification and estimation of systematic uncertainties

3 July 2002ELMS calculations in Molecular Hydrogen 22  Calculate range in liquid H 2 of μ from π decay and compare with Bubble Chamber data at appropriate density. (done but check again)  Further work on materials other than Hydrogen  Extend ELMS to thick targets. It is currently assumed that the medium is thin such that the path length in the target and the cross sections are not affected by the scattering or energy loss.  Then further extend ELMS to transport a 6-D phase space distribution through a given absorber. 6. Further work

3 July 2002ELMS calculations in Molecular Hydrogen Preliminary conclusions  It has been shown that robust calculations of Energy Loss and Multiple Scattering distributions, and their correlations can be made.  Such calculations have been made for molecular hydrogen at liquid density based on the latest available atomic physics data.  While the calculations for other elements roughly agree with expected Multiple Scattering (based on Radiation Length values), in molecular hydrogen the calculations are low by about 14%, compared with predictions for a Radiation Length of g cm -2.  Further comparisons of calculated Energy Loss distributions with other estimates will be interesting.  Comparison with MUSCAT data will provide a good check.  Correlations between MS and dE/dx are understood and are largely confined to the correlations that occur as a result of hard single scatters.