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,

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.

!!


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


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.


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


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


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


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


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?


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...


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


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)


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

QQ


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


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)


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


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

QQ

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


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.


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.


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


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)


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


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


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


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...


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)


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


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.


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


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


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


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


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


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.


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


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)


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 ...


Energy-loss spectra for different momenta and absorber thickesses


... and 2-D transverse momentum transfer spectra for same


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”!


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


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%


ELMS

GEANT

105 incident muonsMomentum 200MeV/cThickness 10cmLinear plot of energy transfer, MeV


ELMS

GEANT

105 incident muonsMomentum 200MeV/cThickness 10cmLog plot of energy transfer, MeV


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


Comparison of Atomic H and Molecular H2

The different binding changes the energy loss but not the scattering


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


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.


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


Projected Pt dist. for tracks selected by energy loss decile

200 MeV/c1mm


Comparing first and last decile ...specific dedx and Pt correlations 1mm H2, 200MeV/c


The same for higher momentum and thicker...1m H2, 20GeV/c


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.


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


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


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


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


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.


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

Documents

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,