calculations of energy loss and multiple scattering (elms) in molecular hydrogen w w m allison,...

Calculations of Energy Loss and Multiple Scattering (ELMS) in

Molecular Hydrogen

W W M Allison, Oxford

Presented at

NuFact02 Meeting, July 2002

3 July 2002 ELMS 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, orwe 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.


3

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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Input data Photoabsorption cross section

of the mediumDensityIncident particle momentum, PIncident particle mass (muon)

Distributions in Pt and energy loss - the ELMS program

Input theoryMaxwell’s equations and point charge scatteringCausalityOscillator strength sum ruleDipole 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, pL and pT :

Ppp TL dd

d2

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

H2


5

Plan of talk1. 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 H2 (preliminary).

5. Verification and estimation of systematic uncertainties

6. Further work

7. Preliminary conclusions


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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

wheredde

~i

~i

2

1),( 3.i

2

kkAβE βk tcttct

.ββE tcte

x

E,

d

d

ck

czec

k

zeβk

βkAβkk .

2),(

~ and .

/1

2),(

~2

002

02

002

0

1. Theory of the double differential cross section


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22222

2221

2

224

2222

LT2

3 1

1dd

d

LTL

TL

TL kkk

kk

kkNkk

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

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).

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.

.2

d)(2

H)(),(0

22

2

m

k

m

kNck


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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.

2242

22

2

4

d

dQF

Q

z

Q

22222222 // ckkckQ TL

M

Q

2

2

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.

.2

d)(2

H)(),(0

22

2

m

k

m

kNck

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!

log kL

2

18

17

7

log kT

whole atoms at low Q2 (dipole region)

electrons at high

Q2

electrons 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)


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50GeV/c in Argon gas .... Yes, at larger anglesPT

PL or E

5GeV/c in Argon gas. ....Yes PT

PL or E

Zooming in to look for Cherenkov Radiation just below ionisation threshold......

500MeV/c in Argon gas .... NoPT

PL or E


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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).

Atomic H photoabsorption cross section

(all theory)

Molecular H photoabsorption cross section

(theory and experiment)

10 100 1000eV 10 100 1000eV

m2 per atom

m2 per atom


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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 ....

mQ 2/22


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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

F(Q2)


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The double differential cross section 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 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 (kT+dkT) and longitudinal momentum between kL and (kL+dkL) is:

(The size of the cells is chosen so that the fractional range of k covered is small.)

1-2

mdddd

dLT

LT

kkkk

NP

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


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Refractive index Mean dE/dx

RMS projected θMS (98% lowest)

MeV cm2 g-1 mrad

Molecular Hydrogen 1.085 3.474 14.52

Atomic Hydrogen 1.137 3.714 14.63

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:

4. Energy loss and multiple scattering distributions for H and H2 (preliminary).

Actual value at this density 1.093 (bubble chamber

data)

Expected value (PDG) 16.94 mrad

with radiation length 61.28 g cm2

Preliminary comp. with range/mtm

relation in H2 bubble chambers. OK


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elementmm

Values from105 ELMS samples PDG book values

RMS deviation Pl

MeV/c

RMS deviation

2D Pt

MeV/c

Correl-ation (MeV/c)2

RMS98 proj ang

RMS 98

/X0 Ratio ELMS/PDGmrad mrad

Molecular H

500 1.136 2.390 0.496 14.52 16.94 0.0478 0.857

Lithium 1 0.098 0.378 0.224 1.58 1.60 0.0006 0.988

Beryllium 1 0.247 0.904 0.469 3.49 3.63 0.0028 0.961

Carbon 1 0.232 1.122 0.305 4.94 5.11 0.0053 0.967

Aluminium 1 0.466 1.703 0.463 7.68 7.71 0.0113 0.996

Iron 1 0.518 3.400 0.419 18.81 18.62 0.0569 1.010

Lithium 10 1.054 1.133 0.239 5.44 5.69 0.0065 0.956

Beryllium 10 0.621 2.179 0.216 12.01 12.75 0.0284 0.942

Carbon 10 0.758 3.073 0.282 17.06 17.93 0.0530 0.951

Aluminium 10 0.887 4.452 0.341 26.53 26.94 0.1125 0.985

Iron 10 1.604 9.210 0.334 63.62 64.65 0.5686 0.984

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.

Molecular Hydrogen 500mm Atomic Hydrogen 500mm

Mean PtMean dEdx

correl-ation

RMS98 ang

mean Pt

mean dEdx

correl-ation

RMS98ang

MeV/cMeV

cm2 g-1 (MeV/c)2 mrad MeV/cMeV

cm2 g-1 (MeV/c)2 mrad

Cherenkov 0.002 0.057 0.000 0.007 0.002 0.073 0.000 0.009

Discrete 0.022 0.004 0.000 0.089 0.022 0.003 0.000 0.092

resonance 0.498 1.156 0.000 2.070 0.519 1.333 0.000 2.153

electron constituent

2.180 2.243 0.145 9.086 2.254 2.296 0.129 9.313

nuclear constituent

2.639 0.011 0.248 10.75 2.662 0.011 0.175 10.97

All mechanisms

3.542 3.474 0.496 14.52 3.584 3.715 0.984 14.63


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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.

PTPL or E


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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.

5. Verification and estimation of systematic uncertainties


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Calculate range in liquid H2 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


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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.

calculations of energy loss and multiple scattering (elms) in molecular hydrogen w w m allison,...

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