Title: The Hybrid Scheme of Simulations of the Electron- photon and Electron-hadron Cascades In a Dense Medium at Ultra-high Energies
1The Hybrid Scheme of Simulations of the Electron-
photon and Electron-hadron Cascades In a Dense
Medium at Ultra-high Energies
- L.G. DedenkoM.V. Lomonosov Moscow State
University,119992 Moscow, Russia
2Content
- Introduction
- Hybrid multilevel scheme
- The 5-level scheme for the atmosphere
- Examples
- Conclusion
3GOALS
- Simulations of cascades at ultra-high energies
- Acoustical (radio) signals production
- Transport of acoustical (radio) signals in the
real matter - Detections of signals
4ENERGY SCALE
5SPACE SCALE
6Transport equations for hadrons
- here k1,2,....m number of hadron types
- - number of hadrons k in bin
EEdE and depth bin xxdx ?k(E)
interaction length Bk decay constant
Wik(E',E) energy spectra of hadrons of type k
produced by hadrons of type i.
7The integral form
- here
- E0 energy of the primary particle Pb (E,xb)
boundary condition xb point of interaction
of the primary particle.
8- The decay products of neutral pions are regarded
as a source function S?(E,x) of gamma quanta
which give origins of electron-photon cascades in
the atmosphere - Here a number of
neutral pions decayed at depth x dx with
energies E?dE?
9- The basic cascade equations for electrons and
photons can be written as follows -
-
- where ?(E,t), P(E,t) the energy spectra of
photons and electrons at the depth t ß the - ionization losses µe, µ? the absorption
coefficients Wb, Wp the bremsstrahlung and - the pair production cross-sections Se, S? the
source terms for electrons and photons.
10- The integral form
-
-
- where
- At last the solution of equations can be found by
the method of subsequent approximations. It is
possible to take into account the Compton effect
and other physical processes.
11- Source functions for low energy electrons and
gamma quanta - xmin(E0E/e)
12- For the various energies
- Emin Ei Eth (Emin1 MeV, Eth10 GeV)
- and starting points of cascades
- 0XkX0 (X01020 gcm-2)
- simulations of 2108 cascades in the atmosphere
with help of CORSIKA code and responses (signals)
of the scintillator detectors using GEANT 4 code - SIGN?(Rj,Ei,Xk)
- SIGN?(Rj,Ei,Xk)
- 10mRj2000m
- have been calculated
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14SIGNAL ESTIMATION
15- Responses of scintillator detectors at distance
Rj from the shower core (signals S(Rj)) -
- Eth10 GeV
- Emin1 MeV
16ENERGY DEPOSITION
17POSITIVE CHARGE (GEANT4)
18NEGATIVE CHARGE (GEANT4)
19FOR HADRON CASCADESFLUCTUATIONS ARE OF
IMPORTANCE
20CHARGE EXCESS (GEANT4)
21THIS FUNCTIONS SHOULD BE ESTIMATED WITH THE
GEANT4 CODE WITH STATISTICS OF 106
22FOR E1012 GEV NEARLY1012 PARTICLES SHOULD
BETAKEN INTO ACCOUNT
23FOR ELECTRON-PHOTON CASCADES FLUCTUATIONS ARE
VERY IMPORTANT DUE TO THE LPM-EFFECT
24EXAMPLES
or
25The Poisson formulae
26Energy deposition QdE/dV in water
27Energy deposition in water
28Energy deposition in water
29Energy deposition in water
30ENERGY DEPOSITION IN WATER
31ENERGY DEPOSITION IN WATER
32ENERGY DEPOSITION IN WATER
33ENERGY DEPOSITION IN WATER
34ENERGY DEPOSITION IN WATER
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36Charge excess
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38Lateral distributions of gammas, electrons and
positrons
39ENERGY DEPOSITION in detector
40Energy distributions of gammas, electrons,
positrons
41Ratio of a signal to a charge particle density
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43el_ed.jpg
44ga_ed.jpg
45pos_ed.jpg
46Conclusion
- The hybrid multilevel scheme has been suggested
to estimate acoustical (radio) signals produced
by e? and eh cascades in dense medium.
47Acknowledgements
- We thank G.T. Zatsepin for useful discussions,
the RFFI (grant 03-02-16290), INTAS (grant
03-51-5112) and LSS-1782.2003.2 for financial
support.
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50- Number of muons in a group with hk(xk) and Ei
-
- here P(E,x) from equations for hadrons D(E,Eµ)
decay function limits Emin(Eµ), Emax(Eµ)
W(Eµ,Ethr,x,x0) probability to survive.
51Transverse impulse distribution
52The angle a
-
- here hk hk(xk) production height for hadrons.
53- Direction of muon velocity is defined by
directional cosines -
- All muons are defined in groups with bins of
energy EiEi?E angles ajaj?aj, - dm dm? dm and height production hk hk ?hk.
The average values have been used , ,
and . Number of muons and
were regarded as some weights.
54The relativistic equation
-
- here mµ muon mass e charge ? lorentz
factor t time geomagnetic field.
55The explicit 2-d order scheme
- here
- Ethr , E threshold energy and muon energy.
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58Ratio with to without magnetic field