The Theory of Charged Particle Energy Loss and Multiple Scattering in Materials and its Application...
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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
20th January 2004
20 Jan 2004 Oxford Seminar 2
From: %%%%%%%%%%%%%%%%%%%Sent: 10 December 2003 02:20To: %%%%%%%%%%%Subject: [Mice-sofware] Muon multiple Coulomb scatteringHi %%%%%%%,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.
!!
20 Jan 2004 Oxford Seminar 3
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 pt and pt +Δpt
is
However the cross section is large so that multiple collisions occur.
EE
N tt
pp
22
3
dd
d
)(dd
d2
3
PEtp
20 Jan 2004 Oxford Seminar 4
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, althoughthe analysis could be extended to other materials.
20 Jan 2004 Oxford Seminar 5
Single collision. Double differential cross section
for energy and transverse mtm. transfers, E and pT :
PpE Tdd
d2
Input data (Low energy) Photoabsorption
cross section of mediumDensity & refractive indexIncident particle momentum, PIncident particle mass (muon)
OverviewInput theoryMaxwell’s equations Causality and dispersionDipole approx. Oscillator strength sum rule Point charge scattering with rel. recoil kinematics &i) hi-Q2 μ-e scattering (Dirac)ii) lo-Q2 H atomic form factor (exact H wave fn.)iii) hi-Q2 μ-p scattering (Rosenbluth)Radiative energy loss (Bremsstrahlung)
all known
recent data for
H2
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
20 Jan 2004 Oxford Seminar 6
Plan of talk1. Theory of the double differential cross section2. The photoabsorption data input for atomic and molecular
Hydrogen3. The Energy Loss and Multiple Scattering in thin absorbers 4. Energy Loss and Multiple Scattering distributions for
general absorbers of H2
5. Estimation of systematic uncertainties and errors6. Correlations between Energy Loss and Multiple Scattering
distributions7. Conclusions
20 Jan 2004 Oxford Seminar 7
It is just an inelastic scattering process...σ as a function of (Q2,ν) or (Q2,x) or (pT, 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
20 Jan 2004 Oxford Seminar 8
In a collision 3-mtm (PT,PL) 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 relation between 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
2222222222 ////// kpp cEcEc
c
Q
ckL
2
2
EkP LLTT and ,kP
22222222 // ckkckQ TL
M
Q
2
2
12
22
M
Qx
20 Jan 2004 Oxford Seminar 9
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
.ββE tcte
x
E,
d
d
Scattering by the atom as a whole and its constituent electrons
What is the value of this electric field?
20 Jan 2004 Oxford Seminar 10
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:
dde
~i
~i
2
1),( 3.i
2 kkAβE βk tcttct
ck
cze
ck
ze
βkβ
kA
βkk
.2
),(~
and
./1
2),(
~ where
200
20
200
20
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...
20 Jan 2004 Oxford Seminar 11
22222
2221
2
224
2222
LT2
3 1
1dd
d
LTL
TL
TL kkk
kk
kkNkk
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.
kk
k
kk
0
3222
222
20
3
2
d1
Imβ
Im2
2
d
d
kk
cck
k
ce
x
E
where 1 and 2 are the real and imaginary parts of ε(k,ω).
kk
33
3
dd
d
d
d
N
x
E
We may equate integrands and deduce the cross section for collisions with transverse momentum transfer kT and longitudinal momentum transfer kL
20 Jan 2004 Oxford Seminar 12
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.
.2
d)(2
H)(),(0
22
2
m
k
m
kNck
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)
20 Jan 2004 Oxford Seminar 13
The resulting differential cross section covers both collision with an atom as a whole, and with constituent electrons at higher k2.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
m
k
2
2
m
Q
2
2
22222222 // ckkckQ TL
At high Q2 this cross section becomes the relativistic Rutherford form for spinless point charges:
42
2
2
4
d
d
20 Jan 2004 Oxford Seminar 14
As such it describes Deep Inelastic Scattering from atoms with stationary constituent spinless electrons for which
Near the kinematic upper limit in Q2 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 pL
(or E) and pT rather than x and Q2.
Let us have a look…
122
2222
m
Q
m
Qx
2sin22cos 2
22
222
cm
Q
20 Jan 2004 Oxford Seminar 15
Illustration: Calculated cross section for 500MeV/c in Argon. Note that this is a log-log-log plot
log kL
2
18
17
7
log kT
whole atom, low Q2 (dipole
region) electron, at high
Q2
electron, μ backwards
in CM
nuclear small angle scattering (suppressed
by screening)
nuclear backward scattering in CM
(suppressed by nuclear form factor)
Log pL or energy transfer
(16 decades)
Log pT transfer (10 decades)
Log cross
section (30
decades)
20 Jan 2004 Oxford Seminar 16
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 chargedistribution modified by F(Q2).
At low Q2 the nuclear charge is screened by atomic electrons. Qmin21/a0
2 1020 m-
2
In the case of Hydrogen the electron wavefunction is particularly well known. At high Q2 the finite nuclear size gives Qmax
21/r02 1030 m-2.
At high Q2 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.).
2242
22
2
4
d
dQF
Q
Z
Q
M
Q
2
2
Nuclear constituent coulomb scattering
20 Jan 2004 Oxford Seminar 17
log10 Q2 m-2
Here lies the source of the statistical problem:dσ/dQ2 varies over 20 orders of magnitude!
Further, the contribution to energy loss, for example, goes like
F(Q2)
24
22
2d
1
2~d
d
d~d~ Q
QM
QE
prop. to area under this curve, which is log dependent on the max/min Q2. 1. the importance of very rare collisions at high Q2 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 mproton structure at 10-15 m
Nuclear formfactor F(Q2) vs. log10Q2
20 Jan 2004 Oxford Seminar 18
2. Data input for atomic and molecular hydrogen
Particularly well known for H and H2. 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 10 100 1000eV
m2 per atom
m2 per atom
We need the Photoabsorption cross section.
20 Jan 2004 Oxford Seminar 19
Dielectric PermittivityExperimental value 1.236
Mean Ionisation Potential(not used!)
Calculated values unshifted with shift -0.8eV unshifted with shift -0.8eV
Molecular H2 1.214 1.236 19.22 18.59
Atomic H 1.353 15.01
But that is for low density Hydrogen.
Condensed Matter effects a) broaden the binding energies of discrete lines, andb) reduce their binding energies (essentially because ee/√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 Jan 2004 Oxford Seminar 20
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
20 Jan 2004 Oxford Seminar 21
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(Q2)
20 Jan 2004 Oxford Seminar 22
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 .
33
33
3
3
dd
)(d|),,(|d
d
)(d|),,(|
),,(),,(
pp
PpPrpPJp
p
pPPrPJ
rPJrP
tNNt
tt
t
3. Energy Loss and Multiple Scattering in thin H2 absorbers (ELMSA)
),,()(),,( tt rPPvrPJ 222 / is )( mPc PPv
20 Jan 2004 Oxford Seminar 23
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 H2 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 kT and (kT+ΔkT) and longitudinal momentum between kL and (kL+ΔkL) is:
LTLT
kkkk
N dd
dyProbabilit
2
20 Jan 2004 Oxford Seminar 24
The size of the cells is chosen so that the fractional range of k covered is small, 2% or less. Typically there are 5-10 104 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 PT”
We also consider the 1-D projected momentum transfer, Px
(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.)
EE
EN dd
d
20 Jan 2004 Oxford Seminar 25
The database generated by EMSA contains 105 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...
20 Jan 2004 Oxford Seminar 26
Cross section of 200 MeV/c muons in liquid H2 2 GeV/c
Horiz axisLog Energy transfer from3E-08 to 3E09 eV
Vert axis Log Transverse momentum transfer from1E01 to 2E09 eV/c
High PT resonance region. Possible
multipole contributions
PT-independent atomic scattering region (Dipole Approximation)
20 Jan 2004 Oxford Seminar 27
Study of the contribution of different mechanismsMuon momentum, MeV/c 200 2000 20000
Collisions, 106 m-1 0.970 0.893 0.892
Mean dEdx, MeV cm2 g-1 (calc. from cross section, not MC)
all mechanisms 4.302 4.272 4.896
nuclear recoil 0.002 0.002 0.012
electron recoil 2.180 2.391 2.997
resonance, continuum 1.108 0.971 0.973
resonance, discrete 0.994 0.835 0.834
cherenkov 0.018 0.071 0.072
bremsstrahlung 0.000 0.001 0.019
Projected PT in 10mm of liquid H2, RMS98, MeV/c
all mechanisms 0.3941 0.3573 0.3583
nuclear recoil 0.2675 0.2316 0.2309
electron recoil 0.2652 0.2444 0.2414
resonance continuum 0.0606 0.0540 0.0538
resonance discrete 0.0549 0.0489 0.0487
Note: Half the scattering is
due to constituent electrons
Note: Half the energy loss is
due to constituent electrons
20 Jan 2004 Oxford Seminar 28
Cross section of 200 MeV/c muons in liquid H2 (just below threshold)
Horiz axisLong. momentum transfer on linear scale from0 to 20 eV/c
Vert axis Trans. momentum transfer on linear scale from0 to 20 eV/c
2 GeV/c
... Cherenkov Radiation already included
PT
PL, also freq.
20 Jan 2004 Oxford Seminar 29
Bremsstrahlung energy loss, MeV cm2 g-1
Momentum Mean total Brem, cf 4.2-4.9
Contribution by primary collision type
GeV/c nuclear constituent
electron constituent
atomic resonance
2 0.001 0.001
3 0.003 0.002 0.001
6 0.006 0.003 0.003
13 0.012 0.007 0.005
26 0.024 0.013 0.011
51 0.051 0.026 0.024 0.001
102 0.105 0.052 0.052 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 Q2
at higher energy the nuclear form factor reverses the relative contributions
20 Jan 2004 Oxford Seminar 30
ELMS total cross section as number of collisions/106 per metre against log10P, with muon P in MeV/c (note, this is not a Monte Carlo result)
Non relativistic region Relativistic region. Note suppressed zero
log10Plog10P
20 Jan 2004 Oxford Seminar 31
Sp. Mean Energy Loss
Muon in liquid H2
MeV cm2 g-1 vs. log10P 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)
log10P
20 Jan 2004 Oxford Seminar 32
Sp. Mean Energy Loss
Muon in liquid H2
MeV cm2 g-1 vs. log10P 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)
log10P
20 Jan 2004 Oxford Seminar 33
4. Energy loss and multiple scattering distribut-ions for general absorbers of H2 (ELMSB)
Momentum GeV/c 0.1 0.2 0.4 1.0 2.0 4.0 10.0 20.0 stat err
Collisions/105 1.455 0.970 0.907 0.894 0.893 0.893 0.892 0.892
Mean dEdx (σ) 6.334 4.303 3.967 4.091 4.272 4.462 4.707 4.896
Mean dEdx (MC) 6.222 4.293 3.964 4.106 4.294 4.471 4.715 4.860 0.3%
Median dEdx 6.450 4.093 3.606 3.494 3.470 3.473 3.486 3.482 0.1%
90%ile dEdx 7.165 5.134 4.901 4.963 5.004 5.036 5.068 5.053 0.3%
99%ile dEdx 7.954 7.223 9.801 14.39 16.82 18.31 19.4 19.3
RMS dEdx 0.47 0.72 1.3 3.0 5.6 9.7 28.7 32.9 ?
Mean 2D-Pt 2.090 1.683 1.572 1.564 1.573 1.583 1.594 1.597 ~1%
RMS proj Px/y 1.797 1.466 1.358 1.383 1.531 1.398 1.443 1.482 ?
RMS98 proj Px/y 1.645 1.323 1.239 1.233 1.235 1.241 1.247 1.246 0.3%
105 muons tracked through 10cm liquid H2 absorber
20 Jan 2004 Oxford Seminar 34
1mm 2mm 4mm 1cm 2cm 4cm 10cm 20cm 40cm 1m
100MeV/c .169 .237 .341 .546 .770 1.088 1.796 2.530
200MeV/c .130 .189 .284 .447 .736 1.029 1.467 2.292 3.300 4.408
400MeV/c .130 .180 .252 .405 .624 .880 1.359 1.914 2.714 4.275
1GeV/c .219 .219 .410 .432 .616 .872 1.383 1.953 2.751 4.313
2GeV/c .133 .191 .270 .419 .605 .864 1.531 2.056 3.090 4.554
4GeV/c .135 .191 .269 .427 .615 .878 1.398 1.973 2.795 4.408
10GeV/c .147 .208 .294 .466 .673 .952 1.444 2.042 2.887 4.562
ELMS values of RMS 1-D projected PT (MeV/c)
Consider any column Numbers fluctuating unstably by several %, even with these statistics!
Samples of 105 muons for a variety of thicknesses and momenta.
20 Jan 2004 Oxford Seminar 35
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 X0 = 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.
00
ln038.016.1398XX
RMS
20
3 10/10 X
20 Jan 2004 Oxford Seminar 36
1mm 2mm 4mm 1cm 2cm 4cm 10cm 20cm 40cm 1m
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 X0 = 61.28 g cm-2 (corrected for energy loss)
20 Jan 2004 Oxford Seminar 37
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 ...
20 Jan 2004 Oxford Seminar 40
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”!
20 Jan 2004 Oxford Seminar 41
ELMS
105 incident muonsMomentum 200MeV/cThickness 10cmLinear plot of projected transverse momentum transfer, MeV/c
Normal distribution with same variance as ELMS for the least 98%
GEANT
20 Jan 2004 Oxford Seminar 42
ELMS
105 incident muonsMomentum 200MeV/cThickness 10cmLog plot of projected transverse momentum transfer, MeV/c
GEANT
Normal distribution with same variance as ELMS for the least 98%
20 Jan 2004 Oxford Seminar 43
ELMS
GEANT
105 incident muonsMomentum 200MeV/cThickness 10cmLinear plot of energy transfer, MeV
20 Jan 2004 Oxford Seminar 44
ELMS
GEANT
105 incident muonsMomentum 200MeV/cThickness 10cmLog plot of energy transfer, MeV
20 Jan 2004 Oxford Seminar 45
Comparison of Elms and GEANT (4.5.2 Patch 02, released 3 October 2003)105 muons 200MeV/c passing through 10cm LH2, ρ = 0.0708 g cm-3
ELMS GEANT auto ratio ELMS/GEANT GEANT 1mm GEANT 2mm
Mean dE, MeV 3.046 3.093 0.985 3.089 3.090
Median dE 2.888 2.928 0.986 2.924 2.926
90%ile dE 3.623 3.715 0.975 3.703 3.707
99%ile dE 5.099 5.315 0.959 5.321 5.356
Mean Pt, MeV/c 1.681 1.774 0.948 1.585 1.640
RMS98 proj. 1.321 1.405 0.941 1.264 1.306
Statistical errors <1%, but somewhat more for “dedx 99%ile”GEANT overestimates energy loss by 2%GEANT (auto stepsize) overestimates PT by 5-6%, although predictions vary up to 10% depending on stepsize
20 Jan 2004 Oxford Seminar 46
Comparison of Atomic H and Molecular H2
The different binding changes the energy loss but not the scattering
20 Jan 2004 Oxford Seminar 47
5. Estimation of systematic uncertainties and errors
Look at Energy Loss and scattering of 200 MeV/c muons in 10 cm liquid H2. Simulate with variations in the cross section. How much difference do they make?
Percentage changes to ELMS values due to variations
Theory Photoabsorption data Form factor & spin
multipole Halved shift, -0.4eV
Doubled linewidth0.4eV
Mott (spinless)
Thomas Fermi
Saxon Woods
Rutherford (espinless)
Collisions m-1
+2.5 -1.5 0 0 +5.2 0 0
Specific Energy Loss, MeV g cm-2
Tabulated +1.9 +0.1 0 0 0 0 +4.5
Projected Transverse Momentum Transfer, MeV/c
RMS98 +1.5 +0.5 +0.9 -1.2 +1.3 -0.2 +1.7
20 Jan 2004 Oxford Seminar 48
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 Q2 part of the resonance region. This has been estimated crudely by increasing the cross section there by 25%, ie by factor 1+0.25*(Q2a0
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.
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6. Correlations between the energy loss and scattering distributions
For a single incident momentum and absorber thickness let us look at the PT distributions for each of ten energy-loss deciles.
First, 200 MeV/c 1mm absorber 105 muons, 104per decile
[Or plot the energy loss in each of ten 3D-PT deciles]
decile? 0-10%, 10-20%, 20-30%,etc of distribution
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Projected Pt dist. for tracks selected by energy loss decile
200 MeV/c1mm
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Comparing first and last decile ...specific dedx and Pt correlations 1mm H2, 200MeV/c
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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.
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Nuclear target:Negligible energy
loss, so no correlation
Electron target:Complete
correlation between energy loss
and scatteringResonant atomic target:No correlation in Dipole
approximation
Cross section for Hydrogen, log PT vs. log energy transfer
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200 MeV/c 1mm of LH2 ELMSMean 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
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ELMS
GEANT
mean energy transfer, MeV
Mean 2-D transverse
momentum transfer, MeV/c
200 MeV/c 10cm Mean 2D PT transfer MeV/c (y)
vs. mean energy transfer, MeV (x) for each energy transfer decile
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Momentum
Thickness 200MeV/c 2GeV/c 20GeV/c
1mm 0.30 0.46 0.47
10mm 0.19 0.34 0.48
100mm 0.19 0.30 0.47
1m 0.16 0.14 0.44
10m 0.09 0.15
100m 0.18
Putting numbers on the correlations
Values of dimensionless correlation parameter
Electron constituent scattering is ~100% correlated between PT 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.
iiiiii
iiii
dPtdPtdPtdEdEdE
dPtdEdPtdE
22
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7. Conclusions1. 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 Q2 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.