Calculations of Energy Loss and Multiple Scattering ELMS in Molecular Hydrogen

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Calculations of Energy Loss and Multiple
Scattering (ELMS) inMolecular Hydrogen

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• W W M Allison, Oxford
• Presented at
• NuFact02 Meeting, July 2002

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• 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 PT transfer (scattering) and PL
  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.

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• This is a report on some work in progress at
  Oxford
• We have derived from first principles the cross
  section for transfers, PT and PL, 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.

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Distributions in Pt and energy loss - the ELMS
program
Input data Photoabsorption cross section of the
medium Density Incident particle momentum,
P Incident particle mass (muon)
Input theory Maxwells 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)
recent data for H2
all known
not done yet
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
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• Plan of talk
• Theory of the double differential cross section
• The photoabsorption data for atomic and molecular
  hydrogen
• The ELMS Monte Carlo program
• Energy loss and multiple scattering distributions
  for H and H2 (preliminary).
• Verification and estimation of systematic
  uncertainties
• Further work
• Preliminary conclusions

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1. Theory of the double differential cross section
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 forcevelocity and thus the mean rate
of energy change with distance is the force
itself From the solution of Maxwells Equations
for the moving point charge in a medium
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Integrating over ? this gives This mean energy
loss is due to the average effect of collisions
with probability per unit distance N ds for a
target density N. The energy loss dE - S ??
- S ?? N dx ds . Thus equating integrands
where ?1 and ?2 are the real and imaginary parts
of e(k,?). ?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 s(?). Following
Allison Cobb, Ann Rev Nucl Part Sci (1980) we
may write where m is the electron mass.
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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.
The resulting cross section covers both collision
with an atom as a whole, and with constituent
electrons at higher Q2.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 pL and pT.
in a different part of phase space so that there
is no interference!
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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!
nuclear small angle scattering (suppressed by
screening)
nuclear backward scattering in CM (suppressed by
nuclear form factor)
electrons at high Q2
whole atoms at low Q2 (dipole region)
Log cross section (30 decades)
Log pL or energy transfer (16 decades)
electrons backwards in CM
Log pT transfer (10 decades)
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Zooming in to look for Cherenkov Radiation just
below ionisation threshold...... 500MeV/c ? in
Argon gas .... No
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2. Photoabsorption cross section for atomic and
molecular hydrogen
New data compilation Atomic and Molecular
Photoabsorption, J Berkowitz, Academic Press
(2002).
m2 per atom
m2 per atom
10 100
1000eV
10 100
1000eV
Atomic H photoabsorption cross section (all
theory)
Molecular H photoabsorption cross
section (theory and experiment)
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The Rutherford cross section is modified at both
high and low Q2. Form factors corrections to
electron constituent scattering For scattering
from constituent electrons at low Q2 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 Q2 in scattering off constituent
electrons is small as can be seen from the
given formula. At maximum Q2 there is pure
point-like µ-e Dirac scattering. Form factor
corrections to proton constituent scattering at
high Q2 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 Q2 corrections are not
very important but we get them right anyway. Form
factor corrections to proton constituent
scattering at low Q2 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 Q2 ....
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F(Q2)
Formfactors (squared) for Hydrogen. At high Q2,
Green the Rosenbluth form factor. At low Q2,
Red atomic hydrogenic wave function form factor
log Q2 m-2
nucleus screening by electrons at 10-10 m
proton structure effect at 10-15 m
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The double differential cross section in
molecular hydrogen
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3. The ELMS Monte Carlo Program
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 106 collisions per
m.) We cut the problem up into elements of
probability. Thus the chance of a collision with
transverse momentum between ?kT and ?(kTdkT) and
longitudinal momentum between ?kL and ?(kLdkL)
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.
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4. Energy loss and multiple scattering
distributions for H and H2 (preliminary).
Input ELMS calculation of 105 traversals of 180
MeV/c muons. Absorber 0.5m liquid H2, density
0.0586 g cm-3. bubble chamber value Result
Preliminary comp. with range/mtm relation in H2
bubble chambers. OK
Actual value at this density 1.093 (bubble
chamber data)
Expected value (PDG) 16.94 mrad with radiation
length 61.28 g cm2
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Results for some other elements. Muons at
180MeV/c.
Most elements agree to 2-3 - except hydrogen
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180MeV/c muons in 500mm molecular H at liquid
density. 105 samples. Red curve normal dist
Left plots projected pt, right plots dE/dx or pL.
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Contributions of each component of cross section.
180MeV/c muon. Liquid Molecular Hydrogen.
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Correlations. 105 sample. 180MeV/c muons. The
scatter plot of magnitude of 2D Pt (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.
PT
PL or E
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5. Verification and estimation of systematic
uncertainties

• 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 H2?
• 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.

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6. Further work

• Calculate range in liquid H2 of µ from p 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.

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7. 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 61.28 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.